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F3A: Chapter 1A

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Book 3A Lesson Worksheet 1A (Refer to §1.1A)

1.1 Factorization Using Identities

1.1A Using the Difference of Two Squares Identity

a2 – b2 ≡ (a + b)( a – b)

Example 1 Instant Drill 1 Factorize Factorize 2 2 Factorize: convert the 2 2 (a) x – 3 , polynomial into (a) p – 4 , 2 2 2 2 (b) 5 – y . the product of its (b) 8 – q . factors. a = ___, b = ___ Sol (a) x2 – 32 a = x, b = 3 Sol (a) p2 – 42 = (x + 3)( x – 3) = ( )( )

a = ___, b = ___ (b) 52 – y2 a = 5, b = y (b) 82 – q2 = (5 + y)(5 – y) = ( )( ) ○○○→→→ Ex 1A 1, 2

Example 2 Instant Drill 2 Factorize Factorize 2 Do you remember 2 (a) x – 4, the square (a) h – 9, (b) y2 – 36. numbers 1, 4, 9, (b) k2 – 49. 16, 25, ? 2 4 = 2 2 9 = ( ) Sol (a) x2 – 4 Sol (a) h2 – 9 = x2 – 22 = ( ) 2 – ( ) 2 = (x + 2)( x – 2) = ( )( )

36 = 62 (b) y2 49 = ( ) 2 – 36 (b) k2 – 49 y2 2 = – 6 = ( ) 2 ( ) 2 y y = ( + 6)( – 6) = ( )( )

1. Factorize (a) x2 – 64, (b) u2 – 100, (c) w2 – 121. 10 2 = ___ 11 2 = ___

2. Factorize (a) 25 – y2, (b) 36 – p2, (c) 81 – n2.

○○○→→→ Ex 1A 3–5

Example 3 Instant Drill 3 Factorize Factorize (a) 4x2 – 1, (a) 9x2 – 16, (b) 4x2 – y2. (b) 9x2 – 16 y2. a = ( ) Sol (a) 4x2 – 1 Sol (a) 9x2 – 16 a = 2 x b = ( ) = (2 x)2 – 12 b = 1 = ( ) 2 – ( ) 2 = (2 x + 1)(2 x – 1) =

a = ( ) (b) 4x2 – y2 (b) 9x2 – 16 y2 2 2 a = 2 x 2 2b = ( ) = (2 x) – y b = y = ( ) – ( ) = (2 x + y)(2 x – y) =

3. Factorize 4. Factorize (a) 1 – 36 x2, (a) 4x2 – 25 y2, 4x2 = ( ) 2 (b) 25 p2 – 49, (b) 9h2 – 64 k2, 25 y2 = ( ) 2 (c) 64 – 81 s2. (c) 49 m2 – 100 n2.

○○○→→→ Ex 1A 8–14

5. Factorize 2 2 2 6. Factorize x y = ( xy) 2 2 (a) x2y2 (a) x2 y2z2 4x = ( ) – 16, 4 – , 2 2 2 (b) p2q2 – 81. (b) s2 – 25 p2q2. y z = ( )

○○○→→→ Ex 1A 15–18

 Level Up Questions Factorize the following polynomials. [Nos. 7 −−−8] 7. (a) 49 b2 – 36 a2c2 (b) –9x2y2 + 16 w2 = = 16 w2 – ( ) =

8. (a) 8x2 – 8 Take out the (b) 6x2 – 6y2 common The common factor is = 8( ) factor 8 = ( )( ______. ) = first. =

New Century Mathematics (2nd Edition) 3A

1 More about Factorization of Polynomials

Consolidation Exercise  1A

Level 1 Factorize the following polynomials. [Nos. 1–18] 1. x2 − 22 2. 42 − y2 3. z2 − 64

4. c2 − 100 5. 36 − x2 6. −y2 + 1

7. 1 − 16 u2 8. 25 m2 − 49 9. 16 a2 − 9

10. −81 x2 + 25 11. 36 a2 − b2 12. 4p2 − 25 q2

13. c2d2 − 9 14. −49 + h2k2 15. 16 p2 − q2r2

16. x2 − 100 y2z2 17. 11 x2 − 11 18. 3m2 − 3n2

Level 2 Factorize the following polynomials. [Nos. 19–33] 19. (3 + x)2 − 1 20. (y − 3) 2 − 25 21. 121 − (m − 2n)2

22. (3 + x)2 − (1 + 2 x)2 23. (2 x + 3 y)2 − (x − 2y)2 24. (a − 2b)2 − (2 a + b)2

25. 3 − 75 x2 26. 18 c2 − 72 d2 27. 6ab 2 − 24 ac 2

28. 5( p + q)2 − 45 29. 32 h2 − 2( k − 3) 2 30. 18( x + 2 y)2 − 2( x − y)2

31. 16 − a2 + 4 b − ab 32. 2x + 7 y + 4 x2 − 49 y2 33. p2 + 3 q − 3p − q2

Answer Consolidation Exercise 1A 1. (x + 2)( x − 2) 2. (4 + y)(4 − y) 3. (z + 8)( z − 8) 4. (c + 10)( c − 10) 5. (6 + x)(6 − x) 6. (1 + y)(1 − y) 7. (1 + 4 u)(1 − 4u) 8. (5 m + 7)(5 m − 7) 9. (4 a + 3)(4 a − 3) 10. (5 + 9 x)(5 − 9x) 11. (6 a + b)(6 a − b) 12. (2 p + 5 q)(2 p − 5q) 13. (cd + 3)( cd − 3) 14. (hk + 7)( hk − 7) 15. (4 p + qr)(4 p − qr ) 16. (x + 10 yz )( x − 10 yz ) 17. 11( x + 1)( x − 1) 18. 3( m + n)( m − n) 19. (4 + x)(2 + x) 20. (y + 2)( y − 8) 21. (11 + m − 2n)(11 − m + 2 n) 22. (4 + 3 x)(2 − x) 23. (3x + y)( x + 5y) 24. −(3 a − b)(a + 3b) 25. 3(1 + 5 x)(1 − 5x) 26. 18( c + 2 d)( c − 2d) 27. 6a(b + 2 c)( b − 2c) 28. 5( p + q + 3)( p + q − 3) 29. 2(4 h + k − 3)(4 h − k + 3) 30. 2(4 x + 5 y)(2 x + 7y) 31. (4 − a)(4 + a + b) 32. (2 x + 7 y)(1 + 2x − 7y) 33. (p − q)( p + q − 3)

F3A: Chapter 1B

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Book 3A Lesson Worksheet 1B (Refer to §1.1B)

1.1B Using the Perfect Square Identities (I) Square of the Sum of Two Numbers

a2 + 2 ab + b2 ≡ (a + b)2

Example 1 Instant Drill 1 Factorize x2 + 2 x + 1. Factorize x2 + 16 x + 64.

a = ___, b = ___ Sol x2 + 2 x + 1 a = x, b = Sol x2 + 16 x + 64 = x2 + 2( x)(1) + 1 2 1 = ( ) 2 + 2( )( ) + ( ) 2 = (x + 1) 2 = ( )2

1. Factorize 2. Factorize (a) x2 + 14 x + 49, (a) 4 + 4 m + m2, (b) s2 + 20 s + 100. (b) 16 + 8 x + x2.

○○○→→→ Ex 1B 1, 3–5, 11, 12

Example 2 Instant Drill 2 Factorize Factorize (a) 9x2 + 6 x + 1, (a) 4h2 + 20 h + 25, (b) 4y2 + 12 y + 9. (b) 9k2 + 24 k + 16. 9x2 = (3 x)2 Sol (a) 9x2 + 6 x + 1 Sol (a) 4h2 + 20 h + 25 = (3 x)2 + 2(3 x)(1) + 1 2 = ( ) 2 + 2( )( ) + ( ) 2 = (3 x + 1) 2 = ( )2 (b) 4y2 + 12 y + 9 (b) 9k2 + 24 k + 16 = (2 y)2 + 2(2 y)(3) + 3 2 = = (2 y + 3) 2

3. Factorize 4. Factorize (a) 81 x2 + 18 x + 1, (a) 25 y2 + 60 y + 36, Rearrange the (b) 9n2 + 42 n + 49. (b) 49 m2 + 4 + 28 m. terms.

○○○→→→ Ex 1B 6, 13

(II) Square of the Difference of Two Numbers

a2 – 2ab + b2 ≡ (a – b)2

Example 3 Instant Drill 3 Factorize x2 – 8x + 16. Factorize x2 – 14 x + 49. Sol x2 – 8x + 16 a = x, b = Sol x2 – 14 x + 49 a = ___, b = ___ 4 = x2 – 2( x)(4) + 4 2 = ( ) 2 – 2( )( ) + ( ) 2 = (x – 4) 2 = ( )2

5. Factorize 6. Factorize (a) x2 – 4x + 4, (a) 1 – 2y + y2, (b) s2 – 16 s + 64. (b) 36 – 12 k + k2.

○○○→→→ Ex 1B 2, 7–9

Example 4 Instant Drill 4 Factorize Factorize (a) 16 x2 – 8x + 1, (a) 25 h2 – 30 h + 9, (b) 9y2 – 12 y + 4. (b) 36k2 – 60 k + 25. Sol (a) 16 x2 – 8x + 1 Sol (a) 25 h2 – 30 h + 9 = (4 x)2 – 2(4 x)(1) + 1 2 = ( ) 2 – 2( )( ) + ( ) 2 = (4 x – 1) 2 = ( )2 (b) 9y2 – 12 y + 4 (b) 36k2 – 60 k + 25 = (3 y)2 – 2(3 y)(2) + 2 2 = = (3 y – 2) 2

7. Factorize 8. Factorize (a) 81 x2 – 36 x + 4, (a) 64 h2 – 48 h + 9, 2 2 Rearrange the (b) 16 p – 56 p + 49. (b) 49 y + 36 – 84 y. terms.

○○○→→→ Ex 1B 10, 14

9. Factorize 2 2 10. Factorize 2 2 a + 2 ab + b 2 2 (a) x + 8 xy + 16 y , ≡ (a) x + 4 y + 4 xy , 2 2 2 2 (b) h – 6hk + 9 k . 2 −( 2 (b) –10 pq + 25 p + q . a 2ab + b ≡ (

○○○→→→ Ex 1B 15–18  Level Up Questions Factorize the following polynomials. [Nos.11 –14] 11. 25 a2 – 40 ab + 16 b2

12. 100 x2 + 140 xy + 49 y2

2 2 2 2 13. (a) 49 x + 4 y + 28 xy (b) – 48 ab + 9 b + 64 a

2 14. 3x – 18 x + 27 Take out the common factor of all the terms first.

New Century Mathematics (2nd Edition) 3A

1 More about Factorization of Polynomials

Consolidation Exercise  1B

Level 1 Factorize the following polynomials. [Nos. 1–18] 1. x2 + 2( x)(2) + 2 2 2. x2 − 2( x)(7) + 7 2 3. k2 + 2 k + 1

4. r2 + 16 r + 64 5. 49 c2 + 14 c + 1 6. m2 − 6m + 9

7. u2 − 18 u + 81 8. 64 y2 − 16 y + 1 9. 100 + 20 t + t2

10. p2 + 16 − 8p 11. 9k2 + 42 k + 49 12. 25 − 40 x + 16 x2

13. 36 x2 + 12 xy + y2 14. u2 − 22 uv + 121 v2 15. 144 a2 + 24 ab + b2

16. 25 p2 − 110 pq + 121 q2 17. 56 cd + 49 c2 + 16 d2 18. 4m2 + 81 n2 − 36 mn

Level 2 Factorize the following polynomials. [Nos. 19–33] 19. 3x2 + 18 x + 27 20. −4k2 − 28 k − 49

21. −100t2 + 120 t − 36 22. −a2 + 12 ab − 36 b2

23. 2m2 − 28 mn + 98 n2 24. −112 x2 − 168 xy − 63 y2

25. x3 − 4x2 + 4 x 26. −p3 + 10 p2q − 25 pq 2

27. 12 y + 12 xy + 3 x2y 28. (x − 2) 2 + 6( x − 2) + 9

29. 25( m + n)2 + 10( m + n) + 1 30. 16( a + b)2 − 8c(a + b) + c2

31. (a) m2 − 8m + 16 (b) m2 − 8m + 16 − n2

32. (a) p2 + 18 pq + 81 q2 (b) p2 + 18 pq + 81 q2 − 25

33. (a) 36 a2 − 12 ab + b2 (b) 36 a2 − 12 ab + b2 − 66 a + 11 b

Answer Consolidation Exercise 1B 1. (x + 2) 2 2. (x − 7) 2 3. (k + 1) 2 4. (r + 8) 2 5. (7 c + 1) 2 6. (m − 3) 2 7. (u − 9) 2 8. (8 y − 1) 2 9. (10 + t)2 10. (p − 4) 2 11. (3 k + 7) 2 12. (5 − 4x)2 13. (6 x + y)2 14. (u − 11 v)2 15. (12 a + b)2 16. (5 p − 11 q)2 17. (7 c + 4 d)2 18. (2 m − 9n)2 19. 3(x + 3) 2 20. −(2 k + 7) 2 21. −4(5 t − 3) 2 22. −(a − 6b)2 23. 2( m − 7n)2 24. −7(4 x + 3 y)2 25. x(x − 2) 2 26. −p(p − 5q)2 27. 3y(2 + x)2 28. (x + 1) 2 29. (5 m + 5 n + 1) 2 30. (4 a + 4 b − c)2 31. (a) (m − 4) 2 (b) (m − 4 + n)( m − 4 − n) 32. (a) (p + 9q)2 (b) (p + 9q + 5)( p + 9q − 5) 33. (a) (6 a − b)2 (b) (6 a − b)(6 a − b − 11)

F3A: Chapter 1C

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Book 3A Lesson Worksheet 1C (Refer to §1.2A)

1.2 Factorization Using the Cross-method

1.2A Factorization of Polynomials in the Form of x2 + bx + c

Cross-method Using the fact that 2 (x + m)( x + n) = x + ( m + n)x + mn x + m to factorize polynomials. ×) x + n x2 + mx +) nx + mn x2 + (m + n)x + mn

Example 1 Instant Drill 1 Factorize x2 + 4 x + 3. Factorize x2 + 8 x + 7. Sol [Step 111: Write the constant term +3 as a Sol [Step 111: Write the constant term +7 as a product of two factors. product of two factors. +3 = (+1)(+3) +7 = (+1)( ) +3 = (–1)(–3) +7 = (–1)( ) Step 222: Test each possible pair of factors Step 222: Test each possible pair of factors by the cross-method. by the cross-method. x +1 x –1 x +1 x –1 x +3 x –3 x x +x + 3x = +4 x –x – 3x = –4x +x + ___ = ___ –x _____ = ___ ] ] Find the x term. Which can give +8 x? Can it give +4 x? x2 + 4 x + 3 = (x + 1)( x + 3) x2 + 8 x + 7 = (x )( x )

1. Factorize x2 – 3x + 2. 2. Factorize x2 – 12 x + 11.

111 111: +2 = ( )( ) : +11 = ( )( ) +11 = ( )( ) +2 = ( )( ) 222 222: x ( ) x ( ) : x ( ) x ( ) x ( ) x ( ) x ( ) x ( )

x2 – 3x + 2 =

○○○→→→ Ex 1C 2, 3

Example 2 Instant Drill 2 Factorize x2 + 2 x – 3. Factorize x2 + 6 x – 7. Sol [Step 111: Write the constant term –3 as a Sol [Step 111: Write the constant term ( ) as product of two factors. a product of two factors.

–3 = (–3)(+1) ( ) = ( )( ) –3 = (+3)(–1) ( ) = ( )( ) Step 222: Test each possible pair of factors Step 222: Test each possible pair of factors by the cross-method. by the cross-method. x –3 x +3 x ( ) x ( ) x +1 x –1 x ( ) x ( ) x x x x x x –3 + = –2 +3 – = +2 ] This pair can give ] 2 +2 x. x + 2 x – 3 = (x + 3)( x – 1) x2 + 6 x – 7 = (x )( x )

3. Factorize the following polynomials. (a) x2 + 12 x – 13 (b) x2 – 12 x – 13

111: –13 = ( )( )

–13 = ( )( ) 222: x ( ) x ( ) x ( ) x ( )

4. Factorize the following polynomials. 2 2 (a) x – 16 x – 17 x ( ) (b) x + 18 x – 19 x ( )

○○○→→→ Ex 1C 4

Example 3 Instant Drill 3 Factorize x2 + 6 x + 8. Factorize x2 + 7 x + 6. Sol [Step 111: Write the constant term +8 as a Sol [Step 111: +6: (+1)(+6), product of two factors: (+2)(+3) We can skip +8: (+1)(+8), (+2)(+4), writing (–1)(–6)

(–1)(–8), (–2)(– 4) and (–2)(–3).

Since the coefficient of x is +6, do we Step 222: Test each possible pair of factors

need to test (–1)(–8), (–2)(– 4)? by the cross-method. Why? x ( ) ( ) Step 222: Test each possible pair of factors x ( ) ( ) by the cross-method. x +1 +2 ] x +8 +4 +x + 8x +2 x + 4 x x2 + 7 x + 6 = (x )( x ) = +9 x = +6 x ] x2 + 6 x + 8 = (x + 2)( x + 4) 20

5. Factorize x2 – 7x + 10. 6. Factorize x2 – 9x + 14.

111: +10: ( )( ), ( )( ) x ( )

222: x ( ) ( ) x ( ) x ( ) ( )

7. Factorize x2 + 8 x + 15. 8. Factorize x2 + 10 x + 24.

○○○→→→ Ex 1C 5–9

9. Factorize x2 + 8x – 9. 10. Factorize –15 + x2 + 2 x. Arrange the terms in descending order first.

11. Factorize x2 – 4x – 21. 12. Factorize x2 – 22 – 9x.

○○○→→→ Ex 1C 10–12, 18, 19

 Level Up Questions

13 . Factorize 15 x + x2 + 26.

14. Factorize x2 – 3x – 18.

New Century Mathematics (2nd Edition) 3A

1 More about Factorization of Polynomials

Consolidation Exercise  1C

Level 1 1. (a) List out all the possible ways of writing −10 as a product of two factors. (The first one is already done as an example for you.) −10 = (+1)( −10), −10 = ( )( ), −10 = ( )( ), −10 = ( )( ) (b) Using the result of (a) , factorize the following polynomials. (i) x2 + 9 x − 10 (ii) x2 − 9x − 10 (iii) x2 − 3x − 10

Factorize the following polynomials. [Nos. 2–19] 2. x2 + 4 x + 3 3. x2 − 3x + 2 4. x2 + 6 x − 7

5. r2 + 5 r + 4 6. k2 + 13 k + 22 7. a2 − 10 a + 9

8. m2 − 12 m + 35 9. h2 − 7h + 12 10. w2 + 8 w − 9

11. b2 + 3 b − 10 12. p2 + 7 p − 18 13. c2 − c − 20

14. y2 − 12 y − 28 15. q2 + 13 q + 40 16. −2v + v2 − 15

17. 4n − 21 + n2 18. −10 s + 24 + s2 19. 42 + z2 − 13 z

Level 2 Factorize the following polynomials. [Nos. 20–34] 20. −x2 + 10 x + 11 21. −x2 − 14 x − 13 22. −x2 + 5 x − 4

23. −x2 − 4x + 32 24. −2a + 35 − a2 25. 11 y − y2 + 12

26. −20 + 12 z − z2 27. b2 + 24 b + 128 28. m2 − 27 m − 90

29. u2 + 8 u − 84 30. −14 q − q2 + 72 31. x2 − 12 xy + 11 y2

32. r2 + 4 rs − 21 s2 33. −p2 − 11 pq + 26 q2 34. −b2 + 48 c2 + 8 bc

35. (a) Factorize k2 + 10 k − 39. (b) Hence, factorize hk − 3h − k2 − 10 k + 39.

Answer Consolidation Exercise 1C 1. (a) −10 = ( −1)(+10), −10 = (+2)( −5), −10 = ( −2)(+5) (b) (i) (x − 1)( x + 10) (ii) (x + 1)( x − 10) (iii) (x + 2)( x − 5) 2. (x + 1)( x + 3) 3. (x − 1)( x − 2) 4. (x + 7)( x − 1) 5. (r + 1)( r + 4) 6. (k + 11)( k + 2) 7. (a − 9)( a − 1) 8. (m − 7)( m − 5) 9. (h − 3)( h − 4) 10. (w + 9)( w − 1) 11. (b + 5)( b − 2) 12. (p + 9)( p − 2) 13. (c + 4)( c − 5) 14. (y − 14)( y + 2) 15. (q + 5)( q + 8) 16. (v + 3)( v − 5) 17. (n + 7)( n − 3) 18. (s − 4)( s − 6) 19. (z − 6)( z − 7) 20. −(x + 1)( x − 11) 21. −(x + 1)( x + 13) 22. −(x − 1)( x − 4) 23. −(x + 8)( x − 4) 24. −(a + 7)( a − 5) 25. −(y − 12)( y + 1) 26. −(z − 10)( z − 2) 27. (b + 16)( b + 8) 28. (m + 3)( m − 30) 29. (u + 14)( u − 6) 30. −(q + 18)( q − 4) 31. (x − 11 y)( x − y) 32. (r + 7 s)( r − 3s) 33. −(p + 13 q)( p − 2q) 34. −(b + 4 c)( b − 12 c) 35. (a) (k − 3)( k + 13) (b) (k − 3)( h − k − 13)

F3A: Chapter 1D

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Book 3A Lesson Worksheet 1D (Refer to §1.2B)

1.2B Factorization of Polynomials in the Form of ax 2 + bx + c

Example 1 Instant Drill 1 Factorize 2x2 + 7 x + 5. Factorize 3x2 + 5 x + 2. Sol [Step 111: Write 2 x2 as a product of two Sol [Step 111: Write 3 x2 as a product of two factors. factors. 2x2 = ( x)(2 x) 3x2 = ( )( ) Step 222: The constant term +5 can be Step 222: The constant term ( ) can be written as: Do not skip written as: (+1)(+5), (+5)(+1) anyone ( )( ), ( )( ) Step 333: Test each possible pair ofof factors Step 333: Test each possible pair of factors by the cross-method. by the cross-method. x +1 +5 ( ) ( ) ( ) 2x +5 +1 ( ) ( ) ( ) +2x + 5x +10 x + x = +7 x = +11 x ] ] 2x2 + 7 x + 5 = (x + 1)(2 x + 5) 3x2 + 5 x + 2 = ( )( )

1. Factorize 5 x2 – 34 x – 7. 2. Factorize 7x2 – 4x – 11.

111: 5x2: ( )( )

222: –7: ( )( ),

( )( )

333: ( ) ( )

( )

2 ( ) ( ) ( ) 5x – 34 x – 7 =

3. Factorize 2 x2 + 3 x – 5. 4. Factorize 3x2 – 10 x + 3.

○○○→→→ Ex 1D 1–7, 18

Example 2 Instant Drill 2 Factorize 6x2 – 11 x + 3. Factorize 8x2 + 14 x + 5. Sol [Step 111: The term 6 x2 can be written as: Sol [Step 111: The term 8 x2 can be written as: (x)(6 x), (2 x)(3 x) ( )( ), ( )( ) Step 222: The constant term +3 can be Step 222: The constant term ( ) can be written as: written as: (–1)(–3), (–3)(–1) ( )( ), ( )( ) Step 333: Test each possible pair of factors Step 333: Test each possible pair of factors by the cross-method. by the cross-method.

x –1 –3 6x –3 –1 –6x – 3x –18 x – x = –9x = –19 x 2x –1 –3 3x –3 –1 –3x – 6x –9x – 2x = –9x = –11 x ] ] 6x2 – 11 x + 3 = (2 x – 3)(3 x – 1) 8x2 + 14 x + 5 = ( )( )

5. Factorize 4 x2 + 4 x – 3. 6. Factorize 10 x2 – 11 x + 3.

111 2 : 4x : ( )( ), ( )( ) 222: –3: ( )( ), ( )( ) 333:

7. Factorize 14 x2 – 19 x – 3.

○○○→→→ Ex 1D 8–11, 17, 19 Example 3 Instant Drill 3 Factorize 6x2 + 31 x + 14. Factorize 10 x2 + 17 x – 6. Sol [Step 111: The term 6 x2 can be written as: Sol [Step 111: The term 10 x2 can be written as: (x)(6 x), (2 x)(3 x) ( )( ), ( )( ) Step 222: The constant term +14 can be Step 222: The constant term ( ) can be written as: written as: (+1)(+14), (+14)(+1), ( )( ), ( )( ), (+2)(+7), (+7)(+2) ( )( ), ( )( ) Step 333: Test each possible pair of factors Step 333: Test each possible pair of factors by the cross-method. by the cross-method. x +1 +14 +2 +7 6x +14 +1 +7 +2 +6 x + 14 x +84 x + x +12 x + 7 x +42 x + 2 x = +20 x = +85 x = +19 x = +44 x 2x +1 +14 +2 +7 3x +14 +1 +7 +2 +3 x + 28 x = +31 x ] 6x2 + 31 x + 14 = (2 x + 1)(3 x + 14) ]

8. Factorize 9x2 – 16 x – 4. 9. Factorize 15 x2 – 22 x + 8.

111 2 : 9x : ( )( ), ( )( ) 222 : – 4: ( )( ), ( )( ), ( )( ), ( )( ) 333:

10. Factorize 21 x2 + 41 x + 10. 11. Factorize 13 x + 6x2 – 28.

○○○→→→ Ex 1D 12–16, 20, 21

 Level Up Questions 12 . Factorize 12 x2 – 37 x + 21.

2 13. Factorize –7x – 6x + 13. –7x2 – 6x + 13 28 = –( )

New Century Mathematics (2nd Edition) 3A

1 More about Factorization of Polynomials

Consolidation Exercise  1D

Level 1 Factorize the following polynomials. [Nos. 1–18] 1. 2x2 + 3 x + 1 2. 3y2 + 7 y + 2 3. 5z2 + 16 z + 3

4. 3a2 + a − 2 5. 2b2 − 13 b − 7 6. 5n2 − 13 n + 6

7. 8y2 + 25 y + 3 8. 7u2 − 12 u − 4 9. 10 t2 + 3 t − 1

10. 28 d2 − 11 d + 1 11. 21 x2 − 5x − 6 12. 8m2 + 22 m + 15

13. 15 r2 − 23 r + 4 14. 18 y2 + 9 y − 14 15. 11 + 14 c + 3 c2

16. 10 k2 − 7 − 9k 17. −2 + 25 x2 − 5x 18. −z − 35 + 6 z2

Level 2 Factorize the following polynomials. [Nos. 19–33] 19. −2x2 + 5 x − 2 20. −3y2 − 40 y − 13 21. −11 t + 3 − 20 t2

22. 8x − 4x2 + 21 23. 2p2 − 24 p + 22 24. 5k2 + 90 − 45 k

25. −35 c + 14 c2 − 126 26. −12 a2 + 9 a + 30 27. 58 u − 14 − 48 u2

28. 11 m2 − 32 mn − 3n2 29. 5x2 − 18 xy − 8y2 30. 18 r2 + 45 rs − 38 s2

31. ab − 63 a2 + 12 b2 32. 3p2 + 108 q2 − 39 pq 33. 4h2 + 42 k2 − 34 hk

34. (a) Factorize 8 x2 + 2 xy − 3y2. (b) Hence, factorize 3 y2 − 8x2 + 5 y − 10 x − 2xy .

Answer Consolidation Exercise 1D 1. (2x + 1)( x + 1) 2. (3y + 1)( y + 2) 3. (5z + 1)( z + 3) 4. (3a − 2)( a + 1) 5. (2b + 1)( b − 7) 6. (5n − 3)( n − 2) 7. (8y + 1)( y + 3) 8. (7u + 2)( u − 2) 9. (5 t − 1)(2 t + 1) 10. (7d − 1)(4d − 1) 11. (7x + 3)(3x − 2) 12. (4m + 5)(2m + 3) 13. (5r − 1)(3r − 4) 14. (3 y − 2)(6 y + 7) 15. (3c + 11)( c + 1) 16. (5 k − 7)(2 k + 1) 17. (5x + 1)(5x − 2) 18. (3z + 7)(2z − 5) 19. −(2x − 1)( x − 2) 20. −(3y + 1)( y + 13) 21. −(4 t + 3)(5 t − 1) 22. −(2x + 3)(2 x − 7) 23. 2( p − 11)( p − 1) 24. 5( k − 6)( k − 3) 25. 7(2 c − 9)( c + 2) 26. −3(4 a + 5)( a − 2) 27. −2(8 u − 7)(3 u − 1) 28. (11 m + n)( m − 3n) 29. (5x + 2 y)( x − 4y) 30. (3 r − 2s)(6 r + 19 s) 31. −(7 a + 3 b)(9 a − 4b) 32. 3( p − 9q)( p − 4q) 33. 2( h − 7k)(2 h − 3k) 34. (a) (2 x − y)(4 x + 3 y) (b) (y − 2x)(5 + 4 x + 3 y)

F3A: Chapter 1E

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 Book 3A Lesson Worksheet 1E (Refer to §1.3)

1.3 Factorization Using the Difference and Sum of Two Cubes Identities (A) Difference of the Two Cubes Identity

a3 −−− b3 ≡ (a −−− b)(a2 + ab + b2)

Example 1 Instant Drill 1 Factorize Factorize (a) x3 − 1, (a) x3 − 64, (b) y3 − 27. (b) y3 − 125. 3 − 3 − Sol (a) x 1 Sol (a) x 64 43 = ______= x3 − 13 = x3 − ( ) 3 = (x − 1)[ x2 + x(1) + 1 2] = ( − )[( )2 + ( )( ) + ( )2] = (x − 1)( x2 + x + 1) = (b) y3 − 27 (b) y3 − 125 53 = ______= y3 − 33 = y3 − ( ) 3 = (y − 3)[ y2 + y(3) + 3 2] = = (y − 3)( y2 + 3 y + 9)

1. Factorize 2. Factorize (a) p3 – 216, (a) 8 – k3, (b) h3 – 1 000. (b) 343 – n3.

(a) p3 – 216 = ( ) 3 − ( ) 3 =

(b) h3 – 1 000 =

Try to memorize the following cube numbers: 3. Factorize 4. Factorize 13 = 1 63 = 216 3 3 3 3 23 = 8 3 73 =3 343 (a) 27 x – 1, (b) 8h – 125 . (a) x – 64 y , (b)3 512 p – 3q . 3 = 27 8 = 512 (a) 27 x3 – 1 43 = 64 93 = 729 3 3 = ( ) 3 – 13 5 = 125 10 = 1 000 =

(b) 8h3 – 125 =

○○○→→→ Ex 1E 2, 4, 6, 8, 10, 11, 13

(B) Sum of the Two Cubes Identity

a3 + b3 ≡ (a + b)(a2 −−− ab + b2)

Example 2 Instant Drill 2 Factorize Factorize (a) x3 + 1, (b) y3 + 8. (a) x3 + 27, (b) y3 + 125. Sol (a) x3 + 1 Sol (a) x3 + 27 = x3 + 13 = x3 + ( ) 3 = (x + 1)[ x2 − x(1) + 1 2] = ( + )[( )2 − ( )( ) + ( )2] = (x + 1)( x2 − x + 1) = (b) y3 + 8 (b) y3 + 125 Pay attention to the = y3 + 23 = ( ) 3 + ( ) 3 sign! = (y + 2)[ y2 − y(2) + 2 2] = = (y + 2)( y2 − 2y + 4)

5. Factorize 6. Factorize (a) p3 + 216, (b) h3 + 729. (a) 64 + k3, (b) 343 + n3.

7. Factorize 8. Factorize (a) 64 p3 + 1, (a) m3 + 512 n3, (b) 343 h3 + 27. (b) 1 000 h3 + k3.

○○○→→→ Ex 1E 1, 3, 5, 7, 9, 12, 14

9. Factorize

(a) 27 x3 – 125 y3, (b) 8p3 + 729 q3.

○○○→→→ Ex 1E 15–18

 Level Up Questions 1 3 3 10. Factorize x + . 1 1   8 = =   8 23  

11. Factorize x3y3 − 512 z3. x3y3 = ( xy )3

New Century Mathematics (2nd Edition) 3A

1 More about Factorization of Polynomials

Consolidation Exercise   1E

Level 1 Factorize the following polynomials. [Nos. 1–15] 1. (3 k)3 + 1 2. 1 − (5 r)3 3. (11 x)3 + 1

4. y3 − 27 5. 64 + z3 6. 1 − 8w3

7. 216 c3 + 1 8. 125 − x3y3 9. a3 + 343 b3

10. 125 m3 − 8 11. 343 + 27 s3 12. 512 x3 − 729

13. 27 x3 − 64 y3 14. 125 a3 + 729 b3 15. 1 000 p3 − 343 q3

Level 2 Factorize the following polynomials. [Nos. 16–27] 3 3 1 3 1 3 y 16. x − 17. 8y + 18. 27 x − 64 27 125

19. 4k3 + 108 20. −448 r3 + 7 21. 686 a3 − 54 b3

22. 500 x3 − 32 y3 23. ab 4 − a4b 24. 135 xy 3 − 40 x4

25. (x − 2) 3 − 729 26. 64 x3 + ( x − 1) 3 27. (1 − 3x)3 − (1 + 3 x)3

28. (a) Factorize 9 x2 − 4. (b) Hence, factorize 729 x6 − 64.

Answer Consolidation Exercise 1E 1. (3k + 1)(9k2 − 3k + 1) 2. (1 − 5r)(1 + 5 r + 25 r2) 3. (11 x + 1)(121 x2 − 11 x + 1) 4. (y − 3)( y2 + 3 y + 9) 5. (4 + z)(16 − 4z + z2) 6. (1 − 2w)(1 + 2 w + 4 w2) 7. (6 c + 1)(36 c2 − 6c + 1) 8. (5 − xy )(25 + 5 xy + x2y2) 9. (a + 7 b)( a2 − 7ab + 49 b2) 10. (5 m − 2)(25 m2 + 10 m + 4) 11. (7 + 3 s)(49 − 21 s + 9 s2) 12. (8 x − 9)(64 x2 + 72 x + 81) 13. (3 x − 4y)(9 x2 + 12 xy + 16 y2) 14. (5 a + 9 b)(25 a2 − 45 ab + 81 b2) 15. (10 p − 7q)(100 p2 + 70 pq + 49 q2)  1  x 1  16.  x −  x2 + +   4  4 16   1  2y 1  17. 2y+ 4y2 − +   3  3 9   2   y  2 3xy y  18. 3x − 9x + +   5  5 25  19. 4( k + 3)( k2 − 3k + 9) 20. 7(1 − 4r)(1 + 4 r + 16 r2) 21. 2(7 a − 3b)(49 a2 + 21 ab + 9 b2) 22. 4(5 x − 2y)(25 x2 + 10 xy + 4 y2) 23. ab (b − a)( b2 + ab + a2) 24. 5x(3 y − 2x)(9 y2 + 6 xy + 4 x2) 25. (x − 11)( x2 + 5 x + 67) 26. (5 x − 1)(13 x2 + 2 x + 1) 27. −18 x(3 x2 + 1) 28. (a) (3 x + 2)(3 x − 2) (b) (3 x + 2)(3 x − 2)(81 x4 + 36 x2 + 16)

F3A: Chapter 2A

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Book 3A Lesson Worksheet 2A (Refer to §2.1)

2.1A Zero Index

a0 = 1, where a ≠ 0. e.g. 3 0 = 1, (–5) 0 = 1, (2 x)0 = 1

1. (a) Evaluate the following without using a calculator. (i) 70 (ii) (–8) 0 (iii) –90

(b) Simplify the following expressions, where b, c, d, e ≠ 0.  0 b 0 3 4 0 (i)   (ii) –5c (iii) (–d e )  4 

○○○→→→ Ex 2A 1, 2, 16, 17

2.1B Negative Integral Indices

1 1 1 1 1 a–n = , where a ≠ 0 and n is a positive integer. e.g. 3 –1 = , 4 –2 = = , (6 x)–1 = an 3 42 16 6x

2. (a) Evaluate the following without using a calculator. (i) 7–1 (ii) 2–3 (iii) (–8) –2

(b) Simplify the following expressions (where r, s, t ≠ 0) and express the answers with positive indices. –4 –1 –2 (i) r (ii) (–s) (iii) (8 t)

○○○→→→ Ex 2A 3, 4, 21

Without using a calculator, evaluate the following and give the answers in fractions. [Nos. 3–4] 3. 4–1 + 2 –1 4. 3–2 × (–9) 0

○○○→→→ Ex 2A 5–9

2.1C Laws of Integral Indices

If m and n are integers and a, b ≠ 0, then (a) am × an = am + n (b) am ÷ an = am – n

Example 1 Instant Drill 1 Without using a calculator, evaluate Without using a calculator, evaluate –4 3 –2 –4 5 –3 –7 –8 (a) 5 × 5 , (b) 7 ÷ 7 . (a) 6 × 6 , (b) 9 ÷ 9 . –4 3 5 –3 Sol (a) 5 × 5 Sol (a) 6 × 6 –4 + 3 m n m + n ( ) ( ) = 5 a × a = a = 6 = 5–1 = 1 1 = a–1 = 5 a –2 –4 –7 –8 (b) 7 ÷ 7 (b) 9 ÷ 9 –2 – (–4) m n m – n ( ) ( ) = 7 a ÷ a = a = 9 = 7–2 + 4 = = 7 2 = 49 ○○○→→→ Ex 2A 7–11

Simplify the following expressions (where p, q, r, s ≠ 0) and express the answers with positive indices. [Nos. 5–6] –8 2 5 –3 5. (a) p × p 6. (a) q ÷ q m × n m + n am ÷ an = am – n 7 –5 a a = a –4 3 (b) 3r × r (b) 16 s ÷ 2s

Remember to express the answers with positive ○○○→→→ Ex 2A 22–24 indices ! Example 2 Instant Drill 2 Without using a calculator, evaluate Without using a calculator, evaluate (a) (2 –3)–2, (a) (4 –1)3, − −  4  1  5  1 (b)   . (b)   .  7   3  Sol (a) (2 –3)–2 Sol (a) (4 –1)3 × ( ) ( ) = 2–3 (–2) (am)n = am × n = ( ) = 26 = = 64

− −  4  1  5  1 (b)   (b)    7   3  −1 n ( ) 4  a  a n ( ) = −   = = 7 1  b  bn ( )( ) −1 7 4 71 = = −1 1 = 4 7 4 ○○○→→→ Ex 2A 12–15

Simplify the following expressions and express the answers with positive indices. [Nos. 7–8] (All the letters in the expressions represent non-zero numbers. ) −1 2 –5  2  n n 7. (a) (k ) m n m × n  a  a (a ) = a 8. (a)     = (b) (y–3)–6  h   b  bn −2  p  (b)    q 

○○○→→→ Ex 2A 25–27 ○○○→→→ Ex 2A 30

Example 3 Instant Drill 3 Simplify the following expressions (where Simplify the following expressions (where x, y ≠ 0) and express the answers with positive r, s ≠ 0) and express the answers with positive indices. indices. –2 –1 4 (a) (4 x) (a) (–3r ) (b) (–5y–1)3 (b) (2 s2)–5 –2 –1 4 Sol (a) (4 x) Sol (a) (–3r ) = 4–2x–2 (ab )n = anbn = ( ) ( ) ( ) ( ) 1 1 = a–1 = 16 x 2 a =

(b) (–5y–1)3 (b) (2 s2)–5 × = (–5) 3y–1 3 = = (–5) 3y–3 125 = − y 3

Simplify the following expressions and express the answers with positive indices. [Nos. 9–14] (All the letters in the expressions represent non-zero numbers. ) 9. (a) (7 g4)–3 10. (a) (–hk )–9 –5 2 n n n –2 –4 (b) (–6t ) (ab ) = a b (b) (xy )

○○○→→→ Ex 2A 31

24 3( d 3 ) −2 11. (a) − 12. (a) (− s)2 3 9d − 51 3 (−e ) 5 f (b) (b) − e6 5( f ) 24

○○○→→→ Ex 2A 28, 29

–1 0 3 –2 2 13. (a) (4 h k ) 14. (a) (6 rt ) −2 4 –3  − 3p0  (b) (–5ab )   (b)  −4   q 

○○○→→→ Ex 2A 32, 33  Level Up Questions

15 . Evaluate the following without using a calculator. (a) 4–1 ÷ 2–3 = (2 ( ) )–1 ÷ 2–3 =

2 –4 (b) 27 × (–3) Convert 27 into the powers of 3. i.e. 27 = 3 ( )

16. Simplify the following expressions (where x, y ≠ 0) and express the answers with positive indices. 5 −4 −1 –3 2 –4 (x y ) (a) (–x y ) (b) x −6

x 0 y −7 (c) (−x −2 y −1 ) −5

New Century Mathematics (2nd Edition) 3A

2 Laws of Indices

Consolidation Exercise  2A

Level 1

Without using a calculator, find the values of the following expressions and give the answers in integers or fractions. [Nos. 1 −−−12] 1. −70 2. (−7) 0 3. (19 0)−2

1 4. 5. 6−1 + (3 + 1)0 6. 3−2 × (−2−1) 8−1

7. 80 ÷ (−5) −2 8. 97 × 9–5 9. 7−3 ÷ 7−2

− −  2  1  3  2 10. −   11. −  12. (2 −2)−2  3   5 

Simplify the following expressions and express the answers with positive indices. [Nos. 13 −−−27] (All the letters in the expressions represent non-zero numbers .)

13. (8 p)0 14. 5q0 15. −(r−9)0

16. (a3b−3)0 17. (−x0)−6 18. (−y)−9

19. c−4 × c5 20. g ÷ g−4 21. (s−3)5

−2 4  1  22. (−k −3)−4 23. 24.   b)5( −2  z5 

−  4  3 −3  u  −2 2  1  25.   26. (3 c d) 27. −   2   2mn 

Level 2 Without using a calculator, find the values of the following expressions and give the answers in integers or fractions. [Nos. 28−−−33] −4 −  1  28. 25 −2 ÷ 5−3 29. − )4( 1 ×−  30. 5−4 ÷ 125 −4 × 25 −5  2 

31. 24 ÷ 6−2 × 2−4 32. (2 −4 × 125 0) ÷ 12 −2 33. 62 − 6–11 ÷ 6−12

Simplify the following expressions and express the answers with positive indices. [Nos. 34−−−45] (All the letters in the expressions represent non-zero numbers .)

34. (x −3y −2)−3 35. (7 −1a−5b3)−1 36. (−5−1r3s−2)−2

−2 −1 2 − −13  0   −20  (m n ) − 5c   ts  37. −4 38.  6  39.  − 42  n  d   ts 

− 35 x −9 ( 2ba − )21 40. 41. (mn 3)−2(nm −4)−1 42. 32 x −7 (a−1b)3

−1 3 − 243 26  −5 −1   − −5  2( yx ) −33 × (a )  a b  × b  43. −5 1 −− 2 44. 6( a ) −6 45.    2 −1  − (4 x y ) a)2(  8a   4 ba 

Answer Consolidation Exercise 2A 1. −1 27. −8m3n3 2. 1 1 28. 3. 1 5 4. 8 29. −4 7 1 5. 30. 6 25 1 6. − 31. 36 18 32. 9 7. 25 33. 30 8. 81 34. x9y6 1 9. 7a5 7 35. b3 − 3 10. 25 s 4 2 36. 6 25 r 11. 9 n7 37. 12. 16 m2 13. 1 d 12 38. 14. 5 25 − 6 15. 1 t 39. 16. 1 s2 17. 1 27 1 40. − 18. − x 2 y9 m2 41. 19. c n7 5 20. g 7 a 1 42. 21. b5 s15 y6 22. k12 43. − x4 23. 100 b2 8a9 10 44. 24. z 27 8 25. 1 12 45. − u 8b11 9d 2 26. c 4

F3A: Chapter 2B

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Book 3A Lesson Worksheet 2B (Refer to §2.2)

2.2 Scientific Notation

2.2A Introduction

A positive number expressed in scientific notation is in the form of a × 10 n, where 1 ≤ a < 10, and n is an integer.

Example 1 Instant Drill 1 Express each of the following numbers in Express each of the following numbers in scientific notation. scientific notation. (a) 5 000 (a) 60 000 (b) 10 500 000 (b) 4 020 000

Sol (a) 5 000 5 000 . Sol (a) 60 000 3 ( ) = 5 × 10 Move to the left = 6 × 10 60 000 . for 3 digits ___ digits (b) 10 500 000 (b) 4 020 000 × 7 = 1.05 10 10 500 000 . = 4 020 000 . 7 digits ___ digits

1. Express each of the following numbers in scientific notation. (a) 300 000 (b) 87 000 000 (c) 923.1

○○○→→→ Ex 2B 2(a)–(c)

Example 2 Instant Drill 2 Express each of the following numbers in Express each of the following numbers in scientific notation. scientific notation. (a) 0.004 (a) 0.000 3

(b) 0.000 062 0.004 (b) 0.000 008 4

Sol (a) 0.004 Move to the right Sol (a) 0.000 3 × –3 × ( ) = 4 10 for 3 digits = 3 10 0.000 3 ___ digits (b) 0.000 062 (b) 0.000 008 4 × –5 = 6.2 10 0.000 062 = 0.000 008 4 5 digits ___ digits

2. Express each of the following numbers in scientific notation. (a) 0.000 07 (b) 0.001 34

○○○→→→ Ex 2B 2(d)–(f)

Example 3 Instant Drill 3 Express the following numbers as integers or Express the following numbers as integers or decimals. decimals. (a) 7 × 10 3 (a) 9.2 × 10 5 (b) 6 × 10 –5 (b) 2 × 10 –7 (c) 2.1 × 10 –6 (c) 3.8 × 10 –4 Sol (a) 7 × 10 3 Sol (a) 9.2 × 10 5 . = 7 × 1 000 7 000 = 9.2 × ( ) = 7 000 Move to the right = 9.2______for 3 digits _ (b) 6 × 10 –5 (b) 2 × 10 –7 ___ digits 0000 0 6. = 6 × 0.000 01 = Move to the left = 0.000 06 ______2 . for 5 digits ___ digits (c) 2.1 × 10 –6 (c) 3.8 × 10 –4 = 2.1 × 0.000 001 0000 = = 0.000 002 1 00 2.1 ______3 .8 6 digits ___ digits

3. Express the following numbers as integers or decimals. (a) 8 × 10 4 (b) 7.3 × 10 6

○○○→→→ Ex 2B 4

2.2B Applications of Scientific Notation I. Simplifying Operations

Example 4 Instant Drill 4

Without using a calculator, evaluate Without using a calculator, evaluate 5 × 10 6 + 3.8 × 10 7 and express the answer in 6.3 × 10 8 + 2 × 10 7 and express the answer in scientific notation. scientific notation. × 7 × 6 2 10 6 7 5 10 8 7 1 Sol × × Sol × × = [( ) × 10 ] × 5 10 + 3.8 10 = (0.5 × 10 1) × 10 6 6.3 10 + 2 10 × 7 × 7 10 7 = 0.5 10 + 3.8 10 × 7 = = 0.5 10 8 = (0.5 + 3.8) × 10 7 = ( ) × 10 = 4.3 × 10 7

Example 5 Instant Drill 5 Without using a calculator, evaluate Without using a calculator, evaluate (4 × 10 2) × (3 × 10 6) and express the answer in (6 × 10 –5) × (7 × 10 9) and express the answer in scientific notation. scientific notation.

Sol (4 × 10 2) × (3 × 10 6) Sol (6 × 10 –5) × (7 × 10 9) 10 –5 × 10 9 2 + 6 = 10 ( ) + = (4 × 3) × 10 m × n m + = a a = a ( ) = 12 × 10 8 n = 1.2 × 10 9 12 = 1.2 × 10 1

Without using a calculator, evaluate the following expressions and express the answers in scientific notation. [Nos. 4–7] Express each 4. 7.2 × 10 6 – 4.2 × 10 5 5. 2 000 000 + 40 000 000 term in scientific notation

○○○→→→ Ex 2B 5(a), (b)

am ÷ an = am – 9 –13 –4 5 6. (6.2 × 10 ) × (2 × 10 ) 7. (8.1 × 10 ) ÷ (3 × 10 ) n

○○○→→→ Ex 2B 5(c), (d) II. Practical Applications

8. Express the following data in scientific notation. (a) The radius of moon is about 17 381 000 m. (b) The length of an Amoeba is about 0.000 22 m. (c) The world population in 2017 is about 7 510 000 000. (d) The diameter of a human red blood cell is about 0.000 006 2 m. ○○○→→→ Ex 2B 7

‘Explain Your Answer’ Question

9. The diameter of the Earth is about 1.27 × 10 4 km. The distance between the Jupiter and the Earth is about 629 000 000 km. Susan claims that the distance between the Jupiter and the Earth is more than 50 000 times the diameter of the Earth. Do you agree? Explain your answer. Distance between the Jupiter and the Earth = 629 000 000 km = ( ) × 10 ( ) km The required number of times = Express this data in scientific ∵ ______( > / = / < ) 50 000 notation ∴ The claim is (agreed / disagreed).

 Level Up Questions 10 . Round off the following numbers to 3 significant figures and express the results in scientific notation. (a) 2 468.3 = ( ), cor. to 3 sig. fig. Round off the number i.e. 2 468.3 = , cor. to 3 sig. fig.first.

(b) 0.000 517 29

11. Without using a calculator, evaluate 7.4 × 10 6 – 3.62 × 10 5 + 8 × 10 4 and express the answer in scientific notation.

New Century Mathematics (2nd Edition) 3A

2 Laws of Indices

Consolidation Exercise  2B

Level 1 1. Determine whether each of the following is expressed in scientific notation. If not, express the number in scientific notation. (a) −3.14 × 10 0 (b) 4.27 × (−10) 5 (c) 8.73 × 10 −8 (d) −65.3 × 10 −5

2. Express each of the following numbers in scientific notation. (a) 8 000 000 000 (b) 2 296.03 (c) 9 580 000 (d) 0.003 109 (e) 0.000 401 (f) 0.000 098

3. Round off the following numbers to 3 significant figures and express the results in scientific notation. (a) 9 753.1 (b) 907 684.27 (c) 0.040 742 (d) 0.000 246 89 27 5 (e) 360 (f) 73 51 11

4. Express each of the following numbers as an integer or a decimal. (a) 5 × 10 3 (b) 9.53 × 10 6 (c) −6.19 × 10 5 (d) −10 −3 (e) 6.1 × 10 –1 (f) 3.93 × 10 −5

5. Without using a calculator, evaluate the following expressions and express the answers in scientific notation. (a) 256 000 000 000 + 8 300 000 000 (b) 0.000 000 045 − 0.000 000 144 (c) (5 × 10 36 ) × (2 × 10 −24 ) (d) (6 × 10 –4) ÷ (3 × 10 2)

6. Use a calculator to evaluate the following expressions and express the answers in scientific notation. (a) 8.5 × 10 6 + 7.3 × 10 7 (b) 2.3 × 10 15 − 6.9 × 10 14 (c) (5.4 × 10 3) × (9.5 × 10 −6) (d) (2.43 × 10 −12 ) ÷ (9 × 10 −7)

7. Express the following data in scientific notation. (a) The surface area of the moon is about 38 000 000 km 2. (b) Hong Kong citizens dispose about 15 000 000 000 g of garbage every day. (c) The average diameter of human hair is about 0.000 05 m. (d) The diameter of a water molecule is about 0.000 000 29 mm.

Level 2 8. Without using a calculator, evaluate the following expressions and express the answers in scientific notation. (a) 4 × 10 2 + 0.7 × 10 4 − 2 × 10 3 (b) 6.6 × 10 −6 + 20 × 10 −8 + 1 × 10 −7 (c) 34 × 10 3 + 5.8 × 10 4 + 0.72 × 10 5 (d) –840 × 10 −2 + 98 × 10 −1 + 7 × 10 1

9. Without using a calculator, find the values of the following expressions and express the answers in scientific notation. 25 000 000 ×5 000 000 (a) 000.0 000 000 5 3 000 000 2 − 8 300 000 000 000 (b) 2 800 000 000 −1

10. Use a calculator to find the values of the following expressions correct to 2 significant figures, and express the answers in scientific notation. − 13 000 32 000  3 (a)  +  (b) 83 ÷ (4 001 −3 × 4 007 8)  7 9 

2  × −3  2 331  −2 − 4.6 10  (c) −6 (d) 10 2  6.4 ×10  2×10 

11. A jet has travelled for 39.7 hours. If the average speed of the jet is 2.8 × 10 2 m/s, find the distance travelled in m. (Give the answer correct to 3 significant figures and express the result in scientific notation .)

12. △ABC is a right-angled triangle with ∠ABC = 90 °. If AB = 1.3 × 10 10 m and BC = 9.6 × 10 9 m, find the length of AC in m. (Give the answer correct to 2 significant figures and express the result in scientific notation .)

Answer Consolidation Exercise 2B 1. (a) yes (b) no, −4.27 × 10 5 (c) yes (d) no, −6.53 × 10 −4 2. (a) 8 × 10 9 (b) 2.296 03 × 10 3 (c) 9.58 × 10 6 (d) 3.109 × 10 −3 (e) 4.01 × 10 −4 (f) 9.8 × 10 −5 3. (a) 9.75 × 10 3 (b) 9.08 × 10 5 (c) 4.07 × 10 −2 (d) 2.47 × 10 −4 (e) 3.61 × 10 2 (f) 7.35 × 10 1 4. (a) 5 000 (b) 9 530 000 (c) −619 000 (d) −0.001 (e) 0.61 (f) 0.000 039 3 5. (a) 2.643 × 10 11 (b) −9.9 × 10 −8 (c) 1 × 10 13 (d) 2 × 10 −6 6. (a) 8.15 × 10 7 (b) 1.61 × 10 15 (c) 5.13 × 10 −2 (d) 2.7 × 10 −6 7. (a) 3.8 × 10 7 km 2 (b) 1.5 × 10 10 g (c) 5 × 10 −5 m (d) 2.9 × 10 −7 mm 8. (a) 5.4 × 10 3 (b) 6.9 × 10 −6 (c) 1.64 × 10 5 (d) 7.14 × 10 1 9. (a) 2.5 × 10 23 (b) 1.96 × 10 21 10. (a) 6.3 × 10 −12 (b) 8.0 × 10 −17 (c) 2.3 × 10 4 (d) 9.9 × 10 −5 11. 4.00 × 10 7 m 12. 1.6 × 10 10 m

F3A: Chapter 2C

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 Book 3A Lesson Worksheet 2C (Refer to §2.3)

2.3A Denary System and Denary Numbers

(a) Denary system consists of ten numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 e.g. 23 and 489 are denary numbers. 10 10 ‘10’ indicates (b) Taking 8 542 10 as an example, that they are Digit 8 5 4 2 denary numbers. Place value 10 3 10 2 10 1

× 10 × 10 × 10 3 2 ∴ 8 542 10 = 8 ××× 10 + 5 ××× 10 + 4 ××× 10 + 2 ××× 1 The expression is called the expanded form of 8 542 10 .

Example 1 Instant Drill 1 Consider the denary number 7 058 10 . Consider the denary number 6 904 10 . (a) Write down the place value of each digit in (a) Write down the place value of each digit in the number. the number. (b) Hence, express 7 058 10 in the expanded (b) Hence, express 6 904 10 in the expanded form. form. Sol (a) The place value of each digit is listed Sol (a) The place value of each digit is listed as follows: as follows: Digit 7 0 5 8 Digit 6 9 0 4 Place value 10 3 10 2 10 1 Place value 10 1 (b) 7 058 10 (b) 6 904 10 = 7 ××× 10 3 + 0 ××× 10 2 + 5 ××× 10 + 8 ××× 1 = 6 × 10 ( ) + 9 × ( ) +

1. Consider the denary number 298 10 . 2. Express the denary number 18 307 10 in the (a) Write down the place value of each expanded form. digit in the number.

(b) Hence, express 298 10 in the expanded Digit 1 8 3 0 7 form. Place v (a) The place value of each digit is listed a as follows: Digit Place value (b) ○○○→→→ Ex 2C 1, 6, 9 Represent each of the following expressions as a denary number. [Nos. 3–4]

3. 5 × 10 2 + 6 × 10 + 7 × 1 4. 8 × 10 3 + 0 × 10 2 + 5 × 10 + 9 × 1

Digit

Place value 10 2 10 1

○○○→→→ Ex 2C 12

2.3B Binary System and Binary Numbers

(a) Binary system consists of two numerals: 0 and 1 e.g. 101 2 and 100110 2 are binary numbers. ‘2’ indicates (b) Taking 1110 as an example, that they are 2 binary numbers. Digit 1 1 1 0 Place value 23 22 2 1

× 2 × 2 × 2 3 2 ∴ 1110 2 = 1 ××× 2 + 1 ××× 2 + 1 ××× 2 + 0 ××× 1 The expression is called the expanded form of 11102.

Example 2 Instant Drill 2 Consider the binary number 1011 2. Consider the binary number 1111 2. (a) Write down the place value of each digit in (a) Write down the place value of each digit in the number. the number. (b) Hence, express 1011 2 in the expanded (b) Hence, express 1111 2 in the expanded form. form. Sol (a) The place value of each digit is listed Sol (a) The place value of each digit is listed as follows: as follows: Digit 1 0 1 1 Digit 1 1 1 1 Place value 23 22 2 1 Place value 2 1 (b) 1011 2 (b) 1111 2 = 1 ××× 23 + 0 ××× 22 + 1 ××× 2 + 1 ××× 1 = 1 × ( ) + 1 × ( ) +

5. Consider the binary number 100 2. 6. Consider the binary number 11001 2. (a) Write down the place value of each (a) Write down the place value of each digit in the number. digit in the number. (b) Hence, express 100 2 in the expanded (b) Hence, express 11001 2 in the form. expanded form.

(a) The place value of each digit is listed as follows: Digit

Place value (b)

○○○→→→ Ex 2C 2, 7

Express the following binary numbers in the expanded form. [Nos. 7–8] 7. 1001 2 8. 10101 2

Digit 1 0 0 1 Digit 1 0 1 0 1 Place Place v v a a

○○○→→→ Ex 2C 10

Represent each of the following expressions as a binary number. [Nos. 9–10] 9. 1 × 22 + 1 × 2 + 0 × 1 10. 1 × 24 + 0 × 23 + 0 × 22 + 1 × 2 + 1 × 1

Digit

Place value 22 2 1

○○○→→→ Ex 2C 13

2.3C Hexadecimal System and Hexadecimal Numbers

(a) Hexadecimal system consists of sixteen numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F, where the values of A to F are as follows: A B C D E F 10 11 12 13 14 15 e.g. 1A2B 16 and 56CD89 16 are hexadecimal numbers. ‘16’ indicates (b) Taking C50D as an example, that they are 16 hexadecimal Digit C 5 0 D numbers. Place value 16 3 16 2 16 1

× 16 × 16 × 16 3 2 ∴ C50D 16 = 12 ××× 16 + 5 ××× 16 + 0 ××× 16 + 13 ××× 1 The expression is called the expanded form of C50D 16 . 60

Example 3 Instant Drill 3 Consider the hexadecimal number 34AB 16 . Consider the hexadecimal number 2CE 16 . (a) Write down the place value of each digit in (a) Write down the place value of each digit in the number. the number. (b) Hence, express 34AB 16 in the expanded (b) Hence, express 2CE 16 in the expanded form. form. Sol (a) The place value of each digit is listed Sol (a) The place value of each digit is listed as follows: as follows: Digit 3 4 A B Digit 2 C E Place value 16 3 16 2 16 1 Place value 1 (b) 34AB 16 (b) 2CE 16 = 3 ××× 16 3 + 4 ××× 16 2 + 10 ××× 16 + 11 ××× 1 = 2 × ( ) + ( ) × ( ) +

11. Consider the hexadecimal number F05 16 . 12. Consider the hexadecimal number D7A8 16 . (a) Write down the place value of each (a) Write down the place value of each digit in the number. digit in the number. (b) Hence, express F05 16 in the expanded (b) Hence, express D7A8 16 in the form. expanded form.

(a) The place value of each digit is listed as follows: Digit Place value (b)

○○○→→→ Ex 2C 3, 8

Express the following hexadecimal numbers in the expanded form. [Nos. 13–14] 13. 14. 369 16 B6570 16 Digit 3 6 9 Digit B 6 5 7 0 Place value Place v a

○○○→→→ Ex 2C 11

Represent each of the following expressions as a hexadecimal number. [Nos. 15–16] 15. 8 × 16 2 + 14 × 16 + 12 × 1 16. 13 × 16 3 + 0 × 16 2 + 15 × 16 + 11 × 1

Digit

Place value 16 2 16 1

○○○→→→ Ex 2C 14

 Level Up Questions 17. Represent 5 × 10 4 + 3 × 10 2 + 2 × 10 as a denary number.

Digit 5 0 Place value 10 4 10 3 10 2 10 1 Add the place holder ‘0’ in suitable places. 5 × 10 4 + 3 × 10 2 + 2 × 10 = 5 × 10 4 + ( ) × 10 3 + 3 × 10 2 + 2 × 10 + ( ) × ( ) =

18. Represent 25 + 2 3 + 1 as a binary number.

19. Represent 16 5 + 16 2 + 16 + 1 as a hexadecimal number.

New Century Mathematics (2nd Edition) 3A

2 Laws of Indices

Consolidation Exercise   2C

Level 1 1. Write down the place value of each digit in 3 045 10 . Digit 3 0 4 5 Place value

2. Write down the place value of each digit in 10110 2. Digit 1 0 1 1 0 Place value

3. Write down the place value of each digit in A4CF 16 . Digit A 4 C F Place value

4. Write down the place value of the underlined digit in each of the following numbers. Number Place value

(a) 4610

(b) 10011 2

(e) D24B 16

5. Consider the denary number 3 579 10 . (a) Write down the place value of each digit in the number. (b) Hence, express 3 579 10 in the expanded form.

6. Consider the binary number 1011 2. (a) Write down the place value of each digit in the number. (b) Hence, express 1011 2 in the expanded form.

7. Consider the hexadecimal number E4B2 16 . (a) Write down the place value of each digit in the number. (b) Hence, express E4B2 16 in the expanded form.

8. Express the following denary numbers in the expanded form. (a) 23 10 (b) 153 10 (c) 2 345 10

9. Express the following binary numbers in the expanded form. (a) 110 2 (b) 1101 2 (c) 11001 2

10. Express the following hexadecimal numbers in the expanded form. (a) B5 16 (b) C3F 16 (c) A5DF 16

11. Represent each of the following expressions as a denary number. (a) 4 × 10 + 7 × 1 (b) 7 × 10 2 + 6 × 10 + 0 × 1 (c) 8 × 10 2 + 0 × 10 + 1 × 1

12. Represent each of the following expressions as a binary number. (a) 1 × 22 + 0 × 2 + 1 × 1 (b) 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1

13. Represent each of the following expressions as a hexadecimal number. (a) 9 × 16 2 + 5 × 16 + 3 × 1 (b) 11 × 16 2 + 10 × 16 + 2 × 1

Level 2 14. Represent each of the following expressions as a denary number. (a) 5 × 10 2 + 3 × 10 (b) 2 × 10 5 + 5 × 10 2 + 4 × 10 (c) 8 × 100 + 9 × 100 000 + 3 × 1 000 + 4

15. Represent each of the following expressions as a binary number. (a) 24 + 2 3 (b) 1 × 16 + 0 × 8 + 1 × 2 + 0 × 1 + 0 × 4 (c) 1 × 32 + 3 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 2 × 1

16. Represent each of the following expressions as a hexadecimal number. (a) 2 × 16 5 + 10 × 16 4 + 256 (b) 15 × 16 + 12 × 16 3 + 2 × 16 4 + 4 × 16 2 + 13 (c) 18 × 16 + 14 × 16 3 + 3 × 16 4 + 2 × 16 2 + 19

17. Use each of the digits 0, 2, 3 and 6 once only to form the smallest 4-digit denary number and the largest 4-digit denary number respectively. Write these two numbers in the expanded form.

18. (a) Find the smallest 4-digit binary number in which only one digit is 0. Write the number in the expanded form. (b) Find the largest 4-digit binary number in which only one digit is 0. Write the number in the expanded form.

19. Evaluate each of the following expressions and express the answer as a denary number. (a) 52 16 + 16 10 + 10 2 (b) (345 16 − 678 10 ) ÷ 11 2

20. If x is a digit between 0 and 9 inclusive such that x5A16 = 2 394 10 , find the value of x.

21. If y is a digit between 0 and 9 inclusive such that 1 yAB 16 + 101 2 = 5 040 10 , find the value of y.

Answer Consolidation Exercise 2C 1. Digit 3 0 4 5 3 2 Place value 10 10 10 1

2. Digit 1 0 1 1 0 4 3 2 Place value 2 2 2 2 1

3. Digit A 4 C F 3 2 Place value 16 16 16 1

4. (a) 1 (b) 22 (c) 16 (d) 25 (e) 16 3

5. (a) Digit in 3 579 10 3 5 7 9 3 2 Place value 10 10 10 1

(b) 3 × 10 3 + 5 × 10 2 + 7 × 10 + 9 × 1

6. (a) Digit in 1011 2 1 0 1 1 3 2 Place value 2 2 2 1

(b) 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1

7. (a) Digit in E4B2 16 E 4 B 2 3 2 Place value 16 16 16 1

(b) 14 × 16 3 + 4 × 16 2 + 11 × 16 + 2 × 1 8. (a) 2 × 10 + 3 × 1 (b) 1 × 10 2 + 5 × 10 + 3 × 1 (c) 2 × 10 3 + 3 × 10 2 + 4 × 10 + 5 × 1 9. (a) 1 × 22 + 1 × 2 + 0 × 1 (b) 1 × 23 + 1 × 22 + 0 × 2 + 1 × 1 (c) 1 × 24 + 1 × 23 + 0 × 22 + 0 × 2 + 1 × 1 10. (a) 11 × 16 + 5 × 1 (b) 12 × 16 2 + 3 × 16 + 15 × 1 (c) 10 × 16 3 + 5 × 16 2 + 13 × 16 + 15 × 1

11. (a) 47 (b) 760 (c) 801

12. (a) 101 2 (b) 1011 2

13. (a) 953 16 (b) BA2 16

14. (a) 530 (b) 200 540 (c) 903 804

15. (a) 11000 2 (b) 100102 (c) 1011110 2

16. (a) 2A0100 16 (b) 2C4FD16 (c) 3E333 16 17. smallest: 2 036, 2 × 10 3 + 0 × 10 2 + 3 × 10 + 6 × 1 largest: 6 320, 6 × 10 3 + 3 × 10 2 + 2 × 10 + 0 × 1 3 2 18. (a) 1011 2, 1 × 2 + 0 × 2 + 1 × 2 + 1 × 1 3 2 (b) 1110 2, 1 × 2 + 1 × 2 + 1 × 2 + 0 × 1

19. (a) 100 10 (b) 53 10 20. 9 21. 3 67

F3A: Chapter 2D

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 Book 3A Lesson Worksheet 2D (Refer to §2.4A)

2.4A Conversion of Binary or Hexadecimal Numbers into Denary Numbers

Example 1 Instant Drill 1 Convert 110 2 into a denary number. Convert 1011 2 into a denary number.

Express 110 2 in the expanded form first. Digit 1 0 1 1 Digit 1 1 0 Place Sol 1102 Place value 22 2 1 Sol 1011 2 val = 1 × 22 + 1 × 2 + 0 × 1 = 1 × ( ) + 0ue × ( ) + ______= 4 + 2 + 0 ______= 6 =

1. Convert 10001 2 into a denary number. 2. Convert 11001 2 into a denary number.

Digit 1 0 0 0 1 Digit 1 1 0 0 1 Place Place v v a a

○○○→→→ Ex 2D 1–6

Example 2 Instant Drill 2 Convert 12B 16 into a denary number. Convert 1A5 16 into a denary number.

Express 12B in the expanded 16 form first. Digit 1 A 5

Digit 1 2 B Place Sol 12B 16 Sol 1A5 16 value Place value 16 2 16 1 2 = 1 × 16 + 2 × 16 + 11 × 1 = 1 × ( ) + ( ) × ( ) + = 256 + 32 + 11 ______= 299 =

3. Convert CD 16 into a denary number. 4. Convert 1F0 16 into a denary number.

Digit C D Digit 1 F 0 Place value Place value

A = 10, B = 11, …

○○○→→→ Ex 2D 7–15

‘Explain Your Answer’ Question 5. Paul claims that 111110 2 must be greater than BE 16 because 1111102 has more digits than BE 16 . Do you agree? Explain your answer.

1111102 = 1 × ( ) + ______First, convert 1111102 and BE 16 into denary ______numbers. Then, do = comparison.

BE 16 =

∵ i.e. 111110 2 (> / = / <) BE 16 ∴ The claim is (agreed / disagreed).

 Level Up Questions 6. Convert 1110111 2 into a denary number.

Digit 1 1 1 0 1 1 1 Place v a

7. Convert DEC 16 into a denary number.

New Century Mathematics (2nd Edition) 3A

2 Laws of Indices

Consolidation Exercise   2D

Level 1 Convert the following binary numbers into denary numbers. [Nos. 1 −−−5]

1. (a) 10 2 (b) 100 2

2. (a) 111 2 (b) 1011 2

3. (a) 11000 2 (b) 110012

4. (a) 100112 (b) 111112

5. (a) 110010 2 (b) 100111 2

Convert the following hexadecimal numbers into denary numbers. [Nos. 6 −−−12]

6. (a) 29 16 (b) D7 16

7. (a) 2F 16 (b) E0 16

8. (a) 357 16 (b) 60E 16

9. (a) B30 16 (b) C0F 16

10. (a) 2017 16 (b) 6C5B 16

11. (a) DBA 16 (b) DFAC 16

12. (a) A3BD 16 (b) BDFC 16

Level 2 Convert the following denary numbers into binary numbers. [Nos. 13 −−−15]

13. (a) 30 10 (b) 32 10

14. (a) 112 10 (b) 237 10

15. (a) 377 10 (b) 393 10

Convert the following denary numbers into hexadecimal numbers. [Nos. 16 −−−18]

16. (a) 64 10 (b) 243 10

17. (a) 426 10 (b) 625 10

18. (a) 2 575 10 (b) 3 664 10

19. Convert the following binary numbers into hexadecimal numbers.

(a) 1100100 2 (b) 10011011 2

20. Convert the following hexadecimal numbers into binary numbers.

(a) 42 16 (b) CD 16

21. Consider the following three numbers:

11111111 2, 24010 , F4 16 Arrange them in descending order.

Answer Consolidation Exercise 2D 1. (a) 2 (b) 4 2. (a) 7 (b) 11 3. (a) 24 (b) 25 4. (a) 19 (b) 31 5. (a) 50 (b) 39 6. (a) 41 (b) 215 7. (a) 47 (b) 224 8. (a) 855 (b) 1 550 9. (a) 2 864 (b) 3 087 10. (a) 8 215 (b) 27 739 11. (a) 3 514 (b) 57 260

12. (a) 41 917 (b) 48 636

13. (a) 11110 2 (b) 100000 2

14. (a) 1110000 2 (b) 11101101 2

15. (a) 101111001 2 (b) 110001001 2

16. (a) 40 16 (b) F3 16

17. (a) 1AA 16 (b) 271 16

18. (a) A0F 16 (b) E50 16

19. (a) 64 16 (b) 9B 16

20. (a) 1000010 2 (b) 11001101 2

21. 11111111 2 > F4 16 > 240 10

F3A: Chapter 3A

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Book 3A Lesson Worksheet 3A (Refer to §3.1)

3.1 Simple Interest

If the principal is $ P and the annual interest rate is R%, then the simple interest ($ I) received after T years is given by: I = P × R% × T

Example 1 Instant Drill 1 $4 000 is deposited in a bank at an annual $6 000 is deposited in a bank at an annual interest rate of 3%. Find the simple interest interest rate of 4%. Find the simple interest received after received after (a) 2 years, (b) 5 years. (a) 3 years, (b) 8 years. Sol (a) Simple interest received Sol (a) Simple interest received = $4 000 × 3% × 2 = $( ) × ( )% × ( ) = $240 = $ P = 4 000, R = 3, P = ? (b) Simple interest received T = 2. Find I. (b) Simple interest received R = $4 000 × 3% × 5 = $( ) × ( )% × ( ) = ? = $600 = $ T

1. $30 000 is deposited in a bank at an 2. $15 000 is deposited in a bank at an interest interest rate of 10% p.a. Find the simple rate of 8% p.a. Find the simple interest interest received after received after (a) 7 years, (a) 6 months, (b) 2.5 years. (b) 9 months. 6 months (a) Simple interest received (a) Simple interest received ( ) ‘p.a.’ means = years = $( ) × ( ) ‘per× ( ) ( ( ) ) = $( ) × ( ) × = year ’. ( )

(b) (b)

9 months ( ) = years ( ) ○○○→→→ Ex 3A 1

Example 2 Instant Drill 2 Mr Chan deposits $2 000 in a bank at an Miss Wong deposits $5 000 in a bank at an interest rate of 5% p.a. Find the time required interest rate of 6% p.a. Find the time required to receive a simple interest of $600. to receive a simple interest of $1 200. Sol Let T years be the time required. Sol Let T years be the time required. 600 = 2 000 × 5% × T ( ) = ( ) × ( ) × T 600 = 100 T Set up an equation = T = 6 to find T. ∴ The time required is 6 years. ∴ The time required is years.

3. Gloria deposits $9 000 in a bank at an 4. A sum of money is deposited in a bank at interest rate of 3% p.a. How long will it an interest rate of 2% p.a. The simple take to receive a simple interest of $945? interest received after 4 years will be $260. Find the sum of money deposited.

Let $ P be the sum of money deposited.

( ) = ( ) × ( ) × ( ) =

∴ The sum of money deposited is .

○○○→→→ Ex 3A 4, 5

5. Ben borrows a sum of money from a bank. 6. $75 000 is deposited in a bank. The simple The interest rate is 9% p.a. If he has to pay interest received after 8 years will be a simple interest of $10 800 after 5 years, $42 000. Find the interest rate per annum. find the sum of money borrowed.

○○○→→→ Ex 3A 6 ○○○→→→ Ex 3A 7, 8

(a) Amount ($ A) = principal ($ P) + interest ($ I) i.e. A = P + I (b) Since I = P × R% × T, we have A = P + I A = P(1 + R% × T) = P + P × R% × T = P(1 + R% × T) Example 3 Instant Drill 3 Edward deposits $700 in a bank at a simple Joey deposits $3 000 in a bank at a simple interest rate of 4% p.a. Find the amount interest rate of 6% p.a. Find the amount received after 2 years. received after 5 years. Sol Interest = $700 × 4% × 2 I = P × R% × T Sol Interest = $( ) × ( ) × ( ) = $56 = Amount = $(700 + 56) A = P + I Amount = $[( ) + ( )] = $756 = Alternative Alternative A = P(1 + R% × T) Amount = $700 × (1 + 4% × 2) Amount = $( ) × [1 + ( ) × ( )] = $756 =

7. Mr Poon deposits $5 000 in a bank at a 8. Teresa borrows $32 000 from a bank at a simple interest rate of 3% p.a. Find the simple interest rate of 10% p.a. How much amount received after 4.5 years. will she repay after 3 months?

A = P + I or A = P(1 + R% × T) ○○○→→→ Ex 3A 2, 3

9. $1 800 is deposited in a bank at a simple 10. Samuel deposits a sum of money in a bank interest rate of 5% p.a. How long will it at a simple interest rate of 2% p.a. If he take to receive an amount of $2 160? receives an amount of $51 300 after 7 years, find the sum of money deposited.

○○○→→→ Ex 3A 10, 11 ○○○→→→ Ex 3A 12

11. Flora deposits $6 000 in a bank at a certain simple interest rate. If she receives an amount of $9 240 after 9 years, what is the interest rate per annum?

Method 1 Method 2 A = P(1 + R% × T) Interest = $[( ) – ( )] Let R% be the interest rate per annum. = ( ) = ( ) × ( ) Let R% be the interest rate per annum.

× × = ( ) = ( ) R% ( ) =

○○○→→→ Ex 3A 13

‘Explain Your Answer’ Question 12 . Mr Hung invests $20 000 in a bond which offers simple interest at 4.7% p.a. Will the simple interest received after 20 years be greater than his original principal? Explain your answer. Simple interest received =

∵ $ ______( < / = / > ) $20 000 ∴ The simple interest received (will be / will not be) greater than his original principal.

 Level Up Question 13. A sum of money $ P is deposited in a bank at a simple interest rate of 5% p.a. How long will it take to receive an amount which is 3 times the original principal? Express the amount in terms of Amount = $______P. Let T years be the time required. Then se t up an equation to find

( ) = ( ) × [( ) + ( ) × ( )] =

New Century Mathematics (2nd Edition) 3A

3 Percentages (II)

Consolidation Exercise  3A

Level 1 1. Complete the following table. Principal Interest rate (p.a. ) Time Simple interest (a) $2 000 5% 4 years (b) $3 000 6.25% 8 years (c) $47 000 3% 2.5 years (d) $96 000 8% 13 months

2. $175 000 is deposited in a bank at a simple interest rate of 4% p.a. Find the amount received after 9 months.

3. $8 000 is deposited in a bank at an interest rate of 3.5% p.a. How long will it take to receive a simple interest of $560?

4. $96 000 is deposited in a bank at an interest rate of 1% p.a. How many months will it take to receive a simple interest of $4 320?

5. A sum of money is deposited in a bank at an interest rate of 2% p.a. The simple interest received after 7.5 years will be $4 119. Find the sum of money deposited.

6. $30 000 is deposited in a bank. The simple interest received after 6 years will be $14 400. Find the interest rate per annum.

7. A sum of money is deposited in a bank at an interest rate of 3% p.a. The simple interest received after 9 months will be $1 800. Find the amount received.

8. $24 000 is deposited in a bank at a simple interest rate of 7% p.a. How long will it take to receive an amount of $29 040?

9. Sam invests $60 000 at a simple interest rate of 4.8% p.a. How long will it take to receive an amount of $75 840?

10. Mandy deposits a sum of money in a bank at a simple interest rate of 4% p.a. If she receives an amount of $7 410 after 42 months, find the sum of money deposited.

11. Samson plans to deposit $50 000 in either bank H or bank K. The table below shows the simple interest rates per annum offered by the two banks. Bank H Bank K Interest rate 5.4% p.a. 3.7% p.a. Samson will take out the amount after 4 years. (a) Which bank should Samson choose in order to earn more interest? Explain your answer. (b) If Samson deposits the money in the bank in (a) , find the interest he will earn.

Level 2 12. A railway company plans to borrow $5 000 000 from a bank. The amount will be repaid after 6 years. Bank M charges simple interest at 4% p.a. while bank N charges simple interest at 5.5% p.a. How much more interest will the company pay if it borrows the money from bank N rather than bank M?

13. Susan deposits $36 000 in bank A at a simple interest rate of 2% p.a. and $5 000 in bank B at a simple interest rate of 1.8% p.a. Find the total amount she will receive after 7 years.

14. Annie deposits $380 000 in a bank at a simple interest rate of 6% p.a. (a) Find the amount received after 9 months. (b) If she deposits the amount in (a) in another bank which offers a simple interest rate of 4% p.a., how many months will it take to receive an amount of $401 071?

15. When a sum of money is deposited in a bank at a simple interest rate of 5% p.a., how long will it take to triple the original sum of money?

16. Jessica borrows $30 000 from a bank at a simple interest rate of R% p.a. Find the value of R in each of the following. (a) The interest is 21% of the original principal after 3.5 years. (b) The amount is 2 times the original principal after 20 years.

17. William deposits $18 000 in a bank at a simple interest rate of R% p.a. for T years. If he deposits the money for further 6 months, he will receive $360 more. Find the value of R.

18. Rick wants to borrow $41 000 from a bank for 3 years. The bank offers the following loan scheme.

For a loan term of more than 4 years, the interest rate is 9% p.a. Otherwise, the interest rate is 7% p.a. Note: Simple interest is charged for all loans.

(a) Find the interest that Rick will pay after 3 years. (b) If Rick saves a fixed amount of money every month during these 3 years to repay the loan and the interest found in (a) , can the amount of money saved each month be less than $1 300? Explain your answer.

19. Mrs Chan deposits $100 000 in a bank at a simple interest rate of 3% p.a. for 2 months. Meanwhile, she makes an investment of $30 000 and loses 1.25%. On the whole, does she make a profit or a loss? Explain your answer.

Answer Consolidation Exercise 3A 1. (a) $400 (b) $1 500 (c) $3 525 (d) $8 320 2. $180 250 3. 2 years 4. 54 5. $27 460 6. 8% 7. $81 800 8. 3 years 9. 5.5 years 10. $6 500 11. (a) bank H (b) $10 800 12. $450 000 13. $46 670 14. (a) $397 100 (b) 3 months 15. 40 years 16. (a) 6 (b) 5 17. 4 18. (a) $8 610 (b) no 19. a profit

F3A: Chapter 3B

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Book 3A Lesson Worksheet 3B (Refer to §3.2)

[In this worksheet, give the answers correct to the nearest dollar if necessary. ]

3.2A Formula for Compound Interest

If the principal is $ P and the interest rate per period is R%, then the amount ($ A) received after n periods is given by: A = P(1 + R%) n

Example 1 Instant Drill 1 If $500 is deposited in a bank at an interest rate If $1 000 is deposited in a bank at an interest of 4% p.a. compounded yearly, find the amount rate of 7% p.a. compounded yearly, find the received after amount received after (a) 5 years, (b) 10 years. (a) 4 years, (b) 8 years. Sol (a) Amount Sol (a) Amount = $500 × (1 + 4%) 5 = $( ) × [1 + ( )%] ( ) P = ? = $608, cor. to the nearestP dollar= 500, R = 4, = $ , cor. to the nearest dollar R (b) Amount n = 5. Find (b) Amount = ? A. n = $500 × (1 + 4%) 10 = $( ) × [1 + ( )%] ( ) = $740, cor. to the nearest dollar = $ , cor. to the nearest dollar

1. If $20 000 is deposited in a bank at an 2. Grace deposits $ P in a bank at an interest interest rate of 5% p.a. compounded rate of 3% p.a. compounded yearly. If the yearly, find the amount received after amount received after 2 years is $63 654, (a) 6 years, (b) 4.5 years. find the value of P.

(a) Amount ( ) = P × ( ) ( ) Set up an × ( ) equat = $( ) ( ) = = ion to

(b)

○○○→→→ Ex 3B 8

The compound interest ($ I) is given by: n I = P[(1 + R%) – 1] ∵ A = P + I ∴ I= A – P = P(1 + R%) n – P = P[(1 + R%) n – 1]

Example 2 Instant Drill 2 If $2 000 is deposited in a bank at an interest If $8 000 is deposited in a bank at an interest rate of 6% p.a. compounded yearly, find the rate of 2% p.a. compounded yearly, find the compound interest received after 8 years. compound interest received after 5 years. Sol Amount Sol Amount × 8 × ( ) = $2 000 (1 + 6%) A = P(1 + R%) n = $( ) ( ) = $3 187.70, cor. to the nearest $0.01 = $( ), cor. to the nearest $0.01 Compound interest Compound interest = $(3 187.70 – 2 000) = $[( ) – ( )] = $1 188, cor. to the nearest dollarI = A – P = $ , cor. to the nearest dollar Alternative Alternative Compound interest Compound interest = $2 000 × [(1 + 6%) 8 – 1] = $( ) × [( ) ( ) – 1] n = $1 188, cor. to the nearestI = P dollar[(1 + R %) – = $ , cor. to the nearest dollar 1]

3. If $50 000 is deposited in a bank at an 4. Leo deposits a sum of money in a bank at interest rate of 8% p.a. compounded an interest rate of 10% p.a. compounded yearly, find the compound interest yearly. If he receives a compound interest received after 6 years. of $5 296 after 3 years, find the sum of money deposited.

Let $ P be the sum of money deposited.

( ) = P × [( ) ( ) – 1]

∴ The sum of money deposited is .

○○○→→→ Ex 3B 9 ○○○→→→ Ex 3B 1–3

5. Complete the following table. ( The interest is compounded yearly. ) Interest rate Compound Principal Time Amount (p.a. ) interest (a) $7 000 3% 4 years (b) $15 000 7% 5 years (c) 5% 3 years $46 305 (d) 4% 2 years $16 320

3.2B Comparison between Compound Interest and Simple Interest

Example 3 Instant Drill 3 $4 000 is deposited in a bank at an interest rate $6 500 is deposited in a bank at an interest rate of 3% p.a. for 10 years. Find the amount of 4% p.a. for 15 years. Find the amount received in each of the following situations. received in each of the following situations. (a) Simple interest is calculated. (a) Simple interest is calculated. (b) The interest is compounded yearly. (b) The interest is compounded yearly. Sol (a) Amount Sol (a) Amount = $4 000 × (1 + 3% × 10)For simple = = $5 200 interest, (b) Amount A = P(1 + R% × T) (b) Amount = $4 000 × (1 + 3%) 10 For compound = = $5 376, cor. to the nearest dollarinterest, A = P(1 + R%) n

6. Nancy deposits $30 000 in a bank at an interest rate of 2.5% p.a. for 4 years. Find the interest received in each of the following situations. (a) Simple interest is calculated. (b) The interest is compounded yearly. n I = P × R% × T I = P[(1 + R%) – 1]

○○○→→→ Ex 3B 10, 11

3.2C Interest Compounded at Different Periods

Suppose $ P is deposited in a bank at an interest rate of R% per period , and the compound interest is calculated once per period . Then the amount ($ A) after n periods is given by: A = P(1 + R%)n total number of periods interest rate per period

Example 4 Instant Drill 4 $1 000 is deposited in a bank at an interest rate $5 000 is deposited in a bank at an interest rate of 6% p.a. for 5 years. Find the amount of 8% p.a. for 3 years. Find the amount received if the interest is compounded received if the interest is compounded half-yearly . quarterly . Sol Sol P = 1 000 P = 5 000 ∵ Number of periods per year = 2 ∵ Number of periods per year = ( ) %8 %6 ∴ ∴ Interest rate per period ( R%) = R% = 2 ( ) × × Total number of periods ( n) = 5 2 n = 3 ( ) Interest rate for half a year Interest rate per quarter %6 = = 2 = 3% Taking half a year as a period, Taking a quarter as a period, number of periods in 5 years number of periods in ( ) years = = 5 × 2

= 10 amount amount = = $1 000 × (1 + 3%) 10 = $1 344, cor. to the nearest dollar

7. $9 000 is deposited in a bank at an interest 8. Hubert borrows $25 000 from a bank at an rate of 10% p.a. compounded half-yearly. interest rate of 14% p.a compounded Find the amount after 4 years. quarterly. Find the amount to be repaid after 6 years.

○○○→→→ Ex 3B 12, 13

‘Explain Your Answer’ Question 9. Anna wants to deposit $100 000 in either bank A or bank B for 3 years. Bank A offers an interest rate of 4.5% p.a. compounded yearly while bank B offers an interest rate of 4% p.a. compounded half-yearly. Which bank pays a higher interest? Explain your answer.

 Level Up Questions 10. At the beginning of 2017, Miss Lau deposits $4 000 in a bank at an interest rate of 8% p.a. compounded yearly. Find the amount she will receive at the beginning of 2024.Find the time for Time for deposit (in years) = 2024 – ( ) deposit first. = Amount =

11. Winnie borrows a sum of money at an interest rate of 12% p.a. compounded monthly. If she has to pay an interest of $482.4 after 2 months, find the sum of money borrowed.

New Century Mathematics (2nd Edition) 3A

3 Percentages (II)

Consolidation Exercise  3B

[In this exercise, give the answers correct to the nearest dollar if necessary .]

Level 1 Complete the following table. [Nos. 1–3] (The interest is compounded yearly .) Interest rate Compound Principal Time Amount (p.a. ) interest 1. $2 000 3% 4 years 2. $8 000 5% 6 years 3. $15 000 7% 8 years

Complete the following table, given that the interests are compounded half-yearly. [Nos. 4–6] Interest rate Interest rate Number of Principal Time Amount (p.a. ) per period periods 4. $1 000 2% 1% 3 years 5. $300 000 4% 5 years 10 6. $60 000 8% 42 months

7. Mr Wong deposits $ P in a bank at an interest rate of 6% p.a. compounded yearly. If the amount received after 5 years is $70 000, find the value of P, correct to the nearest integer.

8. $4 000 is deposited in a bank at an interest rate of 3% p.a. for 6 years. Find the interest received in each of the following situations. (a) Simple interest is calculated. (b) The interest is compounded yearly.

9. $90 000 is deposited in a bank at an interest rate of 7% p.a. for 5 years. Find the difference between the interests calculated on the bases of simple interest and compound interest (compounded yearly).

10. $36 000 is borrowed from a bank at an interest rate of 4% p.a. compounded quarterly. Find the amount to be repaid after 2 years.

11. $68 000 is deposited in a bank at an interest rate of 5% p.a. compounded monthly. Find the amount after 36 months.

12. Jackie borrows $10 000 from a bank at an interest rate of 16% p.a. compounded every 4 months. Find the interest he will pay after 5 years.

Level 2 13. Ann deposits $14 000 in a bank for 2 years. The interest is compounded yearly. (a) Find the compound interest received if the interest rate is (i) 3% p.a., (ii) 6% p.a. (b) Is the compound interest found in (a)(ii) twice the interest found in (a)(i) ? Explain your answer.

14. Judie plans to borrow $60 000 for 4 years. She can borrow the money from bank A at a simple interest rate of 8% p.a. or from bank B at an interest rate of 7% p.a. compounded yearly. Which bank should she choose in order to pay less interest? Explain your answer.

15. David borrows a sum of money from a bank at an interest rate of 9% p.a. compounded quarterly. If the interest he pays after 5 years is $22 000, find the sum of money borrowed.

16. At the beginning of 2014, Mr Tam deposited a sum of money in a bank at an interest rate of 6% p.a. compounded yearly. If he received an interest of $13 000 at the beginning of 2017, find (a) the sum of money deposited, (b) the amount received at the beginning of 2019.

17. 3 years ago, Mary deposited $400 000 in a bank, and a simple interest of $96 000 is just received. Now, she deposits the amount obtained in another bank at the same annual interest rate, but the interest will be compounded monthly. Find the amount she can obtain after another 3 years.

18. Roy wants to deposit $20 000 in a bank for 3 years. He can deposit the money in bank A at an interest rate of 4% p.a. compounded half-yearly, or in bank B at 3% p.a. compounded daily. Which bank should he choose in order to earn more interest? Explain your answer. (Assume there are 365 days in a year .)

19. Sandy plans to deposit $96 000 in a bank for 4 years. The saving schemes offered by bank H and bank K are as follows:

Bank H: interest rate of 4% p.a. compounded monthly with a cash reward of $200 per year. Bank K: interest rate of 4.5% p.a. compounded quarterly.

Which bank should she choose in order to earn more interest? Explain your answer.

20. Anderson is going to borrow $300 000 from a bank for 8 years. Bank A charges interest at 9% p.a. compounded yearly. Bank B charges interest at 7% p.a. compounded monthly. (a) Which bank should he choose in order to pay less interest? Explain your answer. (b) Suppose the interest rate charged by bank B changes to 8.5% p.a. (i) Which bank should he choose in order to pay less interest? Explain your answer. (ii) When compared to the original interest rate of 7% p.a., how much more interest will be charged by bank B now?

Answer Consolidation Exercise 3B 1. amount = $2 251, interest = $251 2. amount = $10 721, interest = $2 721 3. amount = $25 773, interest = $10 773 4. number of periods = 6, amount = $1 062 5. interest rate per period = 2%, amount = $365 698 6. interest rate per period = 4%, number of periods = 7, amount = $78 956 7. 52 308 8. (a) $720 (b) $776 9. $4 730 10. $38 983 11. $78 980 12. $11 802 13. (a) (i) $852.6 (ii) $1 730.4 (b) no 14. bank B 15. $39 250 16. (a) $68 057 (b) $91 076 17. $630 038 18. bank A 19. bank K 20. (a) bank B (b) (i) bank B (ii) $66 398

F3A: Chapter 3C

Date Task Progress

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Book 3A Lesson Worksheet 3C (Refer to §3.3A)

3.3A Increasing at a Constant Rate

If a value P increases at a constant rate of R% per period, its new value A after n periods is given by: A = P(1 + R%) n where (1 + R%) is called the growth factor.

Example 1 Instant Drill 1 John weighs 50 kg this year. If his weight A metal rod is at 78 °C now. If its temperature increases at a constant rate of 9% per year, find increases at a constant rate of 5% per hour, find his weight after 4 years. its temperature after 3 hours. (Give the answer correct to the nearest 0.1 kg. ) (Give the answer correct to the nearest 0.1 °C. ) P = ___ Sol John’s weight after 4 years R = Sol Temperature after 3 hours ___ × 4 ( ) = 50 (1 + 9%) kg P = 50 = ( ) × ( ) °C n = R = = 70.6 kg, cor. to the nearest 0.1 kg = , 9 n =

1. Peter buys a gold coin at $4 000. If its 2. In a city, the number of hotels was 600 in value increases steadily at a rate of 8% per 2012. If the number of hotels in the city year, what will its value be after 10 years? increases at a steady rate of 1.5% per year, (Give the answer correct to the nearest find the number of hotels in 2017, correct $10. ) to the nearest integer.

Number of periods = ( ) – ( ) = Number of hotels in 2017 =

○○○→→→ Ex 3C 1–6

3. The height of a tree increases at a constant 4. After a promotion campaign, the number rate of 20% per year. If the present height of blood donors increases at a constant rate of the tree is 16 m, find its height 2 years of 4% per day. ago. (a) Find the growth factor. (Give the answer correct to the nearest (b) If there are 1 200 blood donors today, 0.01 m. ) how many blood donors were there Let P m be its height 2 years ago. one week ago? (Give the answer correct to the ( ) = P × ( ) ( ) nearest integer. ) =

○○○→→→ Ex 3C 10–13

5. The profit of a company is $100 000 this 6. In an experiment, the number of bacteria year. It is estimated that the profit of the increases by 3% every half an hour. If company will increase by 16% every there are 5 000 bacteria now, how many 2 years. Estimate the profit after 10 years. bacteria will there be after 12 hours? (Give the answer correct to the (Give the answer correct to the nearest 2 significant figures. ) integer. )

Taking _____ years as a period, number of periods in 10 years 10 = ( ) = Estimated profit after 10 years =

○○○→→→ Ex 3C 7, 8

‘Explain Your Answer’ Question 7. The population of a town is 75 000 this year. If its population grows at a steady rate of 2.3% per year, will the population of the town exceed 90 000 after 8 years? Explain your answer.

 Level Up Questions 8. Due to technological improvement, the rice production of a farm increases at a constant rate of 10% every 5 years. Suppose its rice production is 65 tonnes this year. Find the rice production of the farm 20 years ago, correct to the nearest 0.1 tonnes. 1 tonne = 1 000 kg

9. In 2010, the monthly salary of Fred was $20 000. His monthly salary increased at a constant rate of 3% per year from 2010 to 2016. (a) Find the growth factor. (b) Find the increase in his monthly salary from 2010 to 2016. (Give the answer correct to 3 significant figures. )

New Century Mathematics (2nd Edition) 3A

3 Percentages (II)

Consolidation Exercise  3C

Level 1 Complete the following table. [Nos. 1–3] Growth rate Original value Growth factor Time New value per year 1. 300 50% 1.5 1 year 2. 6 000 40% 3 years 3. $20 000 30% 5 years

4. The water consumption of a village is 30 units in a certain week, and it increases steadily at a rate of 6% per week. Find the water consumption after 4 weeks, correct to 3 significant figures.

5. The population of a city was 540 000 in 2015 and it increases steadily at a rate of 3.8% per year. Find the population in 2021. ( Give the answer correct to the nearest integer .)

6. The present value of an oil painting is $256 000. If its value increases by 30% every 4 years, what will its value be after 12 years?

7. A computer virus spreads through a certain network. The number of infected computers increases at a steady rate of 300% every 10 minutes. If 2 computers are infected initially, find the number of infected computers after 1 hour.

8. The average temperature of a town in January is 20°C. It increases steadily at a rate of 4% every month until August. Will the average temperature in June be higher than 25 °C? Explain your answer.

Complete the following table. [Nos. 9–10] Growth rate Original value Growth factor Time New value per year 9. 10% 1.1 1 year 1 210 10. 60% 3 years 49 152

Level 2 11. The weight of a dog was 115 g at the beginning of 2013. Then, its weight increased steadily at a rate of 2% per year from 2013 to 2016. (a) Find the weight of the dog at the end of 2016. (b) If the weight of the dog increases by 4% every year from 2017 onwards, find its weight at the end of 2018. (Give the answers correct to the nearest g.)

12. A tree was 210 cm tall 4 years ago. Then, its height increases steadily at a rate of 3% per year. (a) Find its height at present. (b) Find the percentage increase in its height over these 4 years. (Give the answers correct to 3 significant figures .)

13. The average stock price of a company increased by 1% every month over the past 9 months. It is known that the average stock price is $78 this month. (a) Find the growth factor. (b) Find the increase in the average stock price over the past 9 months. (Give the answer correct to 3 significant figures .)

14. Over the past 3 years, the monthly income of Joe increased at a constant rate of 25% per year. It is known that the monthly income of Joe at present is $50 000. (a) What was the monthly income of Joe 3 years ago? (b) Find the increase in the monthly income of Joe over the past 3 years.

15. In 2015, the number of fish in a pond was 260. In 2016, the number of fish was 273. (a) Find the growth factor. (b) Suppose the growth factor remains unchanged. Find the number of fish in the pond in 2020. (Give the answer correct to the nearest integer.)

16. From 2014 to 2016, company A’s profit increased by 3% per year and company B’s profit increased by 4% per year. It is given that both companies made the same profit in 2016. (a) Which company’s profit was higher in 2014? Explain your answer. (b) If the difference of profits of companies A and B in 2014 was $40 000, find the profit of each company in 2016. (Give the answer correct to the nearest dollar .)

Answer Consolidation Exercise 3C 1. 450 2. growth factor = 1.4, new value = 16 464 3. growth factor = 1.3, new value = $74 258.6 4. 37.9 units 5. 675 426 6. $562 432 7. 8 192 8. no 9. 1 100 10. original value = 12 000, growth factor = 1.6 11. (a) 124 g (b) 135 g 12. (a) 236 cm (b) 12.6% 13. (a) 1.01 (b) $6.68 14. (a) $25 600 (b) $24 400 15. (a) 1.05 (b) 332 16. (a) company A (b) $2 217 332

F3A: Chapter 3D

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Book 3A Lesson Worksheet 3D (Refer to §3.3B)

3.3B Decreasing at a Constant Rate

If a value P decreases at a constant rate of R% per period, its new value A after n periods is given by: A = P(1 – R%) n where (1 – R%) is called the decay factor.

Example 1 Instant Drill 1 A pool contains 300 m 3 of water originally. If The monthly charge of a mobile data plan is the volume of water decreases at a constant $180. If the monthly charge decreases at a rate of 10% per hour, find the volume of water constant rate of 5% per year, find the monthly in the pool after 6 hours. charge after 3 years. 3 (Give the answer correct to the nearest 0.1 m .) (Give the answer correct to the nearestP =dollar. _____ ) Sol Volume of water in the pool after 6 Sol Monthly charge after 3 years R = hours = $( ) × ( ) ( ) _____ 6 3 n = = 300 × (1 – 10%) m = , = 159.4 m 3, cor. to the P = 300 nearest 0.1 m 3 R = 10 n = 6

1. The number of tigers in a country is 1 400 2. In a shop, the profit from selling a model at present. It is known that the number of of calculator was $50 000 in 2013. The tigers decreases by 4% per year. profit from selling this model decreases at (a) Find the decay factor. a rate of 15% per year. Find the profit from (b) Find the number of tigers after 5 selling this model in 2017. years. (Give the answer correct to the nearest (Give the answer correct to the $100. ) nearest integer. ) Number of periods

= ( ) – ( ) = Profit from selling this model in 2017 =

○○○→→→ Ex 3D 1, 2, 5–7

100

3. Suppose the weight of a block of dry ice 4. The number of newborn babies in a city decreases steadily at a rate of 2% per decreases steadily at 4% per year. If the minute. If its present weight is 200 g, find number of newborn babies in the city is its weight 5 minutes ago. 18 000 this year, find the number of (Give the answer correct to the nearest newborn babies 9 years ago. 0.01 g. ) (Give the answer correct to the nearest thousand. ) Let P g be its weight 5 minutes ago.

( ) = P × ( ) ( )

○○○→→→ Ex 3D 3, 4

(a) Depreciation is the decrease in value of a product after it has been used for a period of time, where Depreciation is a depreciation = original value – new value kind of decrease depreciati on at a constant rate. depreciation rate = × 100% original value (b) For a product with the original value $ P and depreciation rate R% per period, its new value $ A after n periods is given by: A = P(1 – R%) n

Example 2 Instant Drill 2 The original price of a watch is $4 000. If its The original price of a machine is $30 000. If value depreciates by 7% per year, what will the its value depreciates by 10% per year, what value of the watch be after 2 years? will the value of the machine be after 3 years?

Sol Value of the watch after 2 years Sol Value of the machine after years = $4 000 × (1 – 7%) 2 = $( ) × ( ) ( ) = $3 459.6 =

101

5. The present value of a sofa is $5 600. If its 6. The value of a smartphone depreciates by depreciation rate is 11% per year, find its 20% every year. If its present value is value after 5 years, correct to the nearest $3 072, find its value 3 years ago. dollar.

○○○→→→ Ex 3D 10

7. The original price of a piano is $47 000. If 8. The value of a camera depreciates by 5% its value depreciates at 18% every 2 years, every 6 months. Its value was $7 500 in find the value of the piano after 6 years, 2015. Find the depreciation of the camera correct to the nearest $100. from 2015 to 2017, correct to the nearest $10.

○○○→→→ Ex 3D 11 depreciation ○○○→→→ Ex 3D 8, 9 = original value – new value ‘Explain Your Answer’ Question 9. Steven bought a printer 2 years ago. The depreciation rate is 25% per year and its present value is $810. Steven claims that he spent less than $1 500 for buying the printer. Do you agree? Explain your answer.

 Level Up Questions 10. The expenditure of a family in March is $35 200. The family decreases their expenditure at a rate of 5% every 2 months since March. Find the expenditure of the family in (a) July, (b) September in the same year.

102

11. Four years ago, the number of traffic accidents in a district was 400. Suppose the number of traffic accidents decreases steadily at a rate of 3.7% per year. (a) Find the decay factor. (b) Find the percentage decrease in the number of traffic accidents over the past four years. (Give the answer correct to 2 significant figures. ) Find the number of traffic accidents this year first.

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New Century Mathematics (2nd Edition) 3A

3 Percentages (II)

Consolidation Exercise  3D

Level 1 Complete the following table. [Nos. 1–3] Rate of decrease Original value Decay factor Time New value per year 1. 800 30% 4 years 2. $7 500 0.75 2 years 3. 20% 3 years 1 600 cm 3

4. A tank contains 20 L of water. Owing to an accident, the volume of water in the tank decreases at a rate of 2% per minute. Find the volume of water in the tank after 4 minutes. (Give the answer correct to the nearest L .)

5. In February, Ron joins a slimming programme and his weight decreases by 3% per month. If his weight in February is 120 kg, find his weight in July, correct to the nearest kg.

6. The original price of a mobile phone is $6 000. If its value depreciates by 4% every 3 months, find the value of the mobile phone after 1 year. (Give the answer correct to the nearest dollar. )

7. The value of a book decreases by 20% every year. If its present value is $400, find its value 2 years ago.

8. The value of a bike depreciates by 10% every 3 years. Its value was $8 748 in 2016. Find its value in 2007.

Level 2 9. The distance between a car and a building is 80 km. The car is now moving towards the building so that the distance between them decreases by 6% every 10 minutes. Find the distance between them after (a) 20 minutes, (b) 1 hour. (Give the answers correct to 3 significant figures .)

104

10. Thomas bought a car 5 years ago. Its value depreciates by 7% every year. Its present value is $142 000. (a) Find the original price of the car, correct to the nearest dollar. (b) When the depreciation of the car is greater than $60 000, Thomas will buy a new car. Will he buy a new car now? Explain your answer.

11. The value of a computer was $4 000 in 2015. Its value then decreases by 13% every year. (a) Find the value of the computer in 2017. (b) What is the percentage decrease in the value of the computer from 2015 to 2017?

12. Kate bought a piano for $64 000 in 2015. (a) If the value of the piano decreases by 6% every 6 months, find its value in 2017. (b) If the value of the piano decreases at a rate of 12% every year, will its value in 2017 be the same as that found in (a) ? Explain your answer. If not, what is the difference? (Give the answers correct to the nearest dollar if necessary .)

13. The population of a city was 350 000 in 1961. It decreased to 346 150 in 1962. (a) Find the percentage change in the population over that year. (b) If the population continues to change at the same rate as in (a) per year, find the population in (i) 1980, (ii) 2000. (Give the answers correct to the nearest integer. )

14. The value of an ebook reader depreciates by x% every 8 months. It is known that the present value of the reader is $1 100 and its value will be $770 after 8 months. (a) Find the value of x. (b) Find the depreciation of the ebook reader after 2 years as compared to the present value.

105

Answer Consolidation Exercise 3D 1. decay factor = 0.7, new value = 192.08 2. rate of decrease per year = 25%, new value = $4 218.75 3. original value = 3 125 cm 3, decay factor = 0.8 4. 18 L 5. 103 kg 6. $5 096 7. $625 8. $12 000 9. (a) 70.7 km (b) 55.2 km 10. (a) $204 114 (b) yes 11. (a) $3 027.6 (b) 24.31% 12. (a) $49 968 (b) no, $406 13. (a) −1.1% (b) (i) 283 660 (ii) 227 365 14. (a) 30 (b) $722.7

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F3A: Chapter 3E

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Book 3A Lesson Worksheet 3E (Refer to §3.4)

3.4A Successive Percentage Changes

If a value N increases by x% and then decreases by y%, then new value = N(1 + x%)(1 – y%)

Example 1 Instant Drill 1 75 increases by 20% and then decreases by 120 increases by 50% and then decreases by 30%. Find the new value. 15%. Find the new value. increase by 50% : (______) Sol New value Sol New value decrease by 15 % : = 75 × (1 + 20%) ×increase (1 – 30%) by 20% : (1 + 20%) = 120 × ( ) × ( ) decrease by 30% : (1 – 30%) = 75 × 1.2 × 0.7 = = 63

1. $300 decreases by 10% and then increases 2. 5 kg increases by 30% and then increases by 40%. Find the new value. by a further 70%. Find the new value.

○○○→→→ Ex 3E 1

3. In a tutorial school, there were 1 250 4. On Monday, Helen practised on violin for students in 2013. The number of students 80 minutes. Her practising time decreased increased by 10% in 2014 and then by 20% on Tuesday, and then decreased by decreased by 4% in 2015. Find the number a further 25% on Wednesday. Find her of students in 2015. practising time on Wednesday.

○○○→→→ Ex 3E 3, 4

109

5. A shop sold 200 TVs in May. The number 6. The original price of a suit is $5 600. The of TVs sold decreased by 50% in June and price of the suit first increases by 20% and then increased by 80% in July. then decreases by 10%. (a) Find the number of TVs sold in July. (a) Find the final price of the suit. (b) What was the increase or decrease in (b) Find the increase or decrease in the the number of TVs sold from May to price of the suit as compared to the July? original price.

○○○→→→ Ex 3E 5, 6

7. The production cost of a tennis racket is $ P. If the production cost is first increased by 16% and then decreased by 25%, the new production cost will be $609. Find the value of P. P( )( ) = ( ) =

○○○→→→ Ex 3E 7–9

110

3.4B Percentage Changes of Different Components

To find the overall percentage change in a quantity with different components: Step 111: Find the original value of the quantity. Step 222: Find the new value of each component and hence the new value of the quantity. change Step 333: Use the formula ‘percentage change = × 100%’ to find original value the overall percentage change.

Example 2 Instant Drill 2 Last year, there were 10 boys and 30 girls in a There are 40 large tables and 20 small tables in choir. This year, the number of boys increases a hall. Now, the number of large tables by 20% and the number of girls decreases by increases by 25% while the number of small 40%. Find the percentage change in the total tables decreases by 5%. Find the percentage number of children in the choir. change in the total number of tables in the hall. Sol Original total number of children Step 111 Sol Original total number of tables Step 111 = 10 + 30 = ( ) + ( ) = 40 = New number of boys Step 222 New number of large tables Step 222 = 10 × (1 + 20%) = ( ) × ( ) = 12 = New number of girls New number of small tables = 30 × (1 – 40%) = ( ) × ( ) = 18 = New total number of children New total number of tables = 12 + 18 = ( ) + ( ) = 30 = ∴ Percentage change in the Step 333 ∴ Percentage change in the Step 333 total number of children total number of tables 30 − 40 ( ) − ( ) = × 100% = × 100% 40 ( )

= –25% =

111

8. Last month, the income and the expenditure of a flower shop were $24 000 and $18 000 respectively. This month, the income decreases by 10% while the expenditure increases by 10%. Find the percentage change in the profit of the shop. Profit of the shop last month = $[( ) – ( )] = Income of the shop this month =

Expenditure of the shop this month =

Profit of the shop this month =

∴ Percentage change in the profit of the shop =

○○○→→→ Ex 3E 10, 11

9. In the figure, the length and the width of a rectangle are 25 cm and 18 25 cm cm respectively. (a) Find the area of the rectangle. (b) If the length of the rectangle increases by 80% while the width 18 cm decreases by 40%, find the percentage change in the area of the rectangle.

○○○→→→ Ex 3E 14, 15

112

10. A professional examination consists of papers I, II and III. In the first attempt, Calvin scored 50, 70 and 80 in papers I, II and III respectively. (a) Find the total score of Calvin in the first attempt. (b) In the second attempt, Calvin’s scores in papers I and II both increase by 20% and his score in paper III remains unchanged. Find the percentage change in his total score as compared to the first attempt. Paper I Paper II Paper III 1st atte 50 70 80 mpt 2nd atte

○○○→→→ Ex 3E 12, 13 ‘Explain Your Answer’ Question 11 . A bowl contains 300 mL of water originally. The volume of water in the bowl increases by 30% and then decreases by 40%. Sandy claims that the overall percentage change of the volume of water in the bowl is –10%. Do you agree? Explain your answer. New volume of water = 300 × ( ) × ( ) mL =

Overall percentage change =

∵ ______( = / ≠ ) –10% ∴ The claim is (agreed / disagreed).

 Level Up Questions 12. A value decreases by 30% and then increases by 50%. Find the overall percentage change.

Let P be the original value. New value =

∴ Overall percentage change 113

13. After the fare of a bus route increases from $6 to $6.3, the number of passengers reduces by 14%. Find the percentage change in the revenue obtained from the bus route.

Let x be the original number of passengers. Original revenue = $6 x New number of passengers = x × ( ) = New revenue = $( ) × ( ) = $( ) ∴ The required percentage change =

114

New Century Mathematics (2nd Edition) 3A

3 Percentages (II)

Consolidation Exercise  3E

Level 1 1. Find the new value in each of the following situations. (a) 200 increases by 30% and then increases by a further 40%. (b) 8 m decreases by 50% and then decreases by a further 10%. (c) $1 600 decreases by 25% and then increases by 60%.

2. Find the original value in each of the following situations. (a) After increasing by 6% and then increasing by a further 7%, the new value is 90 736. (b) After increasing by 30% and then decreasing by 20%, the new value is 208 g. (c) After decreasing by 40% and then increasing by 60%, the new value is 76.8 cm 3.

3. In January, the number of visitors of a theme park is 50 000. The number decreases by 24% from January to February, and then increases by 7% from February to March. Find the number of visitors in March.

4. The income of a taxi driver on Friday is 40% more than that on Saturday. The income on Saturday is 8% less than that on Sunday. If the income on Sunday is $2 000, find the income on Friday.

5. Ron’s first examination result is 60 marks. His second examination result is 20% higher than his first one, while his third examination result is 25% lower than his second one. (a) What is Ron’s third examination result? (b) What is the increase or decrease in Ron’s third examination result as compared with his first examination result?

6. A hawker sells 200 apples on Thursday. The number of apples sold increases by 40% from Thursday to Friday and then decreases by 30% from Friday to Saturday. Is the number of apples sold on Saturday less than that on Thursday? Explain your answer.

7. The cost of a product decreases by 20% and then increases by 25%. The final cost of the product is $77. Find the original cost of the product.

115

8. Last year, there were 800 male students and 900 female students in a university. This year, the number of male students increases by 30% and the number of female students decreases by 15%. (a) (i) Find the number of male students this year. (ii) Find the number of female students this year. (b) Find the percentage change in the total number of students, correct to 3 significant figures.

9. Last year, 6 000 candidates took an accountant examination and 1 500 of them failed the examination. This year, the number of candidates who pass the examination increases by 18%, and the number of candidates who fail the examination decreases by 16%. Find the percentage change in the total number of candidates.

10. Last week, the number of books sold in bookstores P, Q and R were 150, 240 and 320 respectively. As compared to last week, the number of books sold this week in bookstore P decreases by 2%, that in bookstore Q increases by 5% and that in bookstore R remains unchanged. Find the percentage change in the total number of books sold in the three bookstores, correct to 3 significant figures.

11. The length of a rectangle is 10 cm and the width is 7 cm. (a) Find the area of the rectangle. (b) If the length of the rectangle decreases by 10% and the width increases by 10%, find (i) the new area, (ii) the percentage change in the area.

12. Last month, there were 500 workers in a factory and the wage of each worker was $12 000. This month, the number of workers decreases by 12% and the wage of each worker increases by 6%. Find the percentage change in the sum of wages of all workers over these two months.

Level 2 13. Find the percentage change in each of the following situations. (a) A value increases by 80% and then decreases by 25%. (b) A value decreases by 17% and then increases by 17%.

14. The profit of a company increased by 40% from 2014 to 2015, and then decreased by 35% from 2015 to 2016. What was the percentage change in the profit of the company from 2014 to 2016?

15. In a shop, the price of a product is decreased by 20% and then increased by 30%. Find the percentage change in the price of the product.

116

16. The tax revenue of a government increased by 20%, 10%, 35% respectively in the first three quarters of 2016, and then decreased by 45% in the fourth quarter of the year. What was the percentage change in the quarterly tax revenue over the whole year 2016?

17. If a number is decreased by 90% and then increased by x%, the overall percentage change is −83%. Find x.

18. The operating cost ($ F) of a yoga club can be calculated by the following formula: F = 150 000 + 20 000 N + R, where N is the number of yoga instructors and $ R is the monthly rent of the yoga club. (a) Last month, there were 20 yoga instructors and the monthly rent was $300 000. Find the operating cost. (b) This month, the number of yoga instructors decreases by 25% and the monthly rent increases by 15%. Does the operating cost decrease by 10% as compared to last month? Explain your answer.

19. The numbers of tourists of three cities A, B and C in 2015 are listed below: City A B C Number of tourists 2 500 900 1 600 In 2016, the numbers of tourists of cities B and C increased by 40% and 20% respectively. If the total number of tourists of the three cities increased by 4%, find the percentage change in the number of tourists of city A.

20. If the base of a parallelogram increases by 18% and its height decreases by 30%, find the percentage change in its area.

21. Last week, the numbers of male customers and the female customers in a shopping mall were in the ratio 1 : 3. This week, the number of male customers increases by 16% while the number of female customers decreases by 12%. Is there an increase in the total number of customers as compared to last week? Explain your answer.

117

Answer Consolidation Exercise 3E 1. (a) 364 (b) 3.6 m (c) $1 920 2. (a) 80 000 (b) 200 g (c) 80 cm 3 3. 40 660 4. $2 576 5. (a) 54 marks (b) a decrease of 6 marks 6. yes 7. $77 8. (a) (i) 1 040 (ii) 765 (b) +6.18% 9. +9.5% 10. +1.27% 11. (a) 70 cm 2 (b) (i) 69.3 cm 2 (ii) −1% 12. −6.72% 13. (a) +35% (b) −2.89% 14. −9% 15. +4% 16. −1.99% 17. 70 18. (a) $850 000 (b) no 19. −19.2% 20. −17.4% 21. no

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F3A: Chapter 3F

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Book 3A Lesson Worksheet 3F (Refer to §3.5)

3.5A Rates

Rates for the year = rateable value × rates percentage charge rates for the year Rates for a quarter = 4

(In this worksheet, the rates percentage charge is set as 5%. ) Example 1 Instant Drill 1 The rateable value of a flat is $300 000. Find The rateable value of a flat is $85 000. Find the the rates payable in a year. rates payable in a year. Sol Rates payable in a year Sol Rates payable in a year = $300 000 × 5% = ( ) × ( ) = $15 000 = ○○○→→→ Ex 3F 1, 2

1. The rateable value of a building is 2. The rateable value of a piece of land is $1 600 000. Find the rates for a quarter of $2 304 000. Find the rates for a quarter of a year. a year.

Rates for a quarter of a year

= Rateable value ↓ × 5%

Rates for a year ↓ ÷ 4 Rates for a quarter ○○○→→→ Ex 3F 3, 4

3. Mr Luk pays $9 000 for the annual rates 4. The owner of a building pays $57 100 of on his flat. Find the rateable value of the rates quarterly. Find the rateable value of flat. the building.

Let $ P be the rateable value of the flat. Rates for a quarter

P × ( ) = ( ) rates for the year = = 4

○○○→→→ Ex 3F 5–7

3.5B Salaries Tax

To calculate the salaries tax payable: Step 111: Split up the net chargeable income Net chargeable income Tax rates into several parts, which are called On the first $40 000 2% ‘tax bands’. (see Table 1 ) On the next $40 000 7% 222 On the next $40 000 12% Step : Calculate the tax for each part by Remainder 17% 120 Table 1

multiplying the corresponding tax rates. Step 333: Salaries tax payable In this worksheet, refer to = sum of the taxes in Step 222 Table 1 when calculating salaries tax. Note: The government may adjust the tax bands and the tax rates.

Example 2 Instant Drill 2 The net chargeable income of May is $125 000. The net chargeable income of Alex is $130 000. Find her salaries tax payable. Find his salaries tax payable. Sol Step 111: Sol Step 111: Net chargeable income Net chargeable income = $125 000 = $130 000 = $(40 000 + 40 000 + 40 000 + 5 000) = $(40 000 + 40 000 + ______) Step 222: Step 222: Net chargeable Net chargeable Rate Tax Rate Tax income income On the first $40 000 × 2% On the first $40 000 × 2% 2% 2% $40 000 = $800 $40 000 = $800 On the next $40 000 × 7% On the next $40 000 × 7% 7% 7% $40 000 = $2 800 $40 000 = ______On the next $40 000 × 12% On the next 12% ____ $40 000 = $4 800 ______Remainder $5 000 × 17% Remainder 17% ____ $5 000 = $850 ______Step 333: Step 333: Her salaries tax payable His salaries tax payable = $(800 + 2 800 + 4 800 + 850) = $(800 + ______) = $9 250 =

121

5. The net chargeable income of Ann is $150 000. Find her salaries tax payable. Net chargeable income =

Net chargeable income Rate Tax On the first $40 000 2% $40 000 × ______= ______

Her salaries tax payable =

○○○→→→ Ex 3F 8–10

Net chargeable income = annual income – allowances

Example 3 Instant Drill 3 The annual income of Kary is $220 000. If she The annual income of Tim is $240 000. If he has a total allowance of $120 000, find has a total allowance of $150 000, find (a) her net chargeable income, (a) his net chargeable income, (b) her salaries tax payable. (b) his salaries tax payable. Sol (a) Net chargeable income Sol (a) Net chargeable income = $(220 000 – 120 000) = $[( ) – ( )] = $100 000 = $ (b) Net chargeable income (b) Net chargeable income = $100 000 = $( ) = $(40 000 + 40 000 + 20 000) = $(40 000 + 40 000 + ______) Net chargeable Net chargeable Rate Tax Rate Tax income income On the first $40 000 × 2% On the first $40 000 × 2% 2% 2% $40 000 = $800 $40 000 = ______On the next $40 000 × 7% 7% $40 000 = $2 800 Remainder $20 000 × 12% 12% $20 000 = $2 400 Her salaries tax payable His salaries tax payable = $(800 + 2 800 + 2 400) = = $6 000

6. The annual income of Charles is $408 000. If he has a total allowance of $132 000, find his salaries tax payable. Net chargeable income =

Net chargeable income Rate Tax On the first $40 000 2% $40 000 × ______= ______

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His salaries tax payable =

○○○→→→ Ex 3F 12, 13

 Level Up Question 7. The average monthly income of Vanessa is $13 000 and her total allowance is $120 000. Find her salaries tax payable.

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New Century Mathematics (2nd Edition) 3A

3 Percentages (II)

Consolidation Exercise  3F

[In this exercise, when calculating salaries tax, refer to the tax rates as shown in Table 1 on P.3.44 of the textbook .]

Level 1 1. The rateable value of a flat is $360 000. Find the rates payable in a year.

2. The rateable value of a shopping centre is $3 000 000. Find the rates payable in a quarter of a year.

3. The rateable value of a piece of land is $76 900 000. Find the rates payable in a quarter of a year.

4. Mr Kan pays $18 000 for the annual rates on his property. Find the rateable value of the property.

5. David pays $7 200 for the quarterly rates on his flat. Find the rateable value of the flat.

6. The owner of an apartment pays $22 000 of rates quarterly. What is the rateable value of the apartment?

Find the salaries tax payable for each of the persons below. [Nos. 7–9] 7. A taxi driver has a net chargeable income of $32 000. 8. Carrie is a dancer with a net chargeable income of $79 000. 9. Jon is an artist with a net chargeable income of $101 000.

10. The net chargeable incomes of Ramsey and Paul are $8 000 and $24 000 respectively. Is the salaries tax paid by Paul 3 times that paid by Ramsey? Explain your answer.

11. The annual income of Ellen is $186 000. If she has a total allowance of $110 000, find her salaries tax payable.

Level 2 12. Priscilla paid $6 500 for the rates last year. At the beginning of this year, she moves to a new flat and the rates payable decreases by 10% as compared to last year. What is the rateable value of the new flat?

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13. Sally paid $9 400 for the rates last year. At the beginning of this year, she moves to a new flat and its rateable value is $142 800. What is the percentage change in the rates payable this year as compared to last year? (Give the answer correct to the nearest 1% .)

14. The average monthly income of Gigi is $25 000. If she has a total allowance of $120 000, find her salaries tax payable.

15. (a) Find the salaries tax payable in each of the following situations: (i) Net chargeable income = $40 000 (ii) Net chargeable income = $80 000 (iii) Net chargeable income = $120 000 (b) Joe has to pay the salaries tax of $10 100. Using the results of (a) , calculate his net chargeable income.

16. Benny’s salaries tax payable is $13 500. Find his net chargeable income.

17. Glen has a total allowance of $130 000. He has to pay $1 500 in salaries tax. (a) Find his net chargeable income. (b) Find his average monthly income.

18. (a) The net chargeable income of Gary is $123 000. Find his salaries tax payable. (b) The net chargeable income of Raymond is one third of that of Gary. Is the salaries tax payable of Raymond one third of that of Gary? Explain your answer. If not, find the difference in their salaries tax payable.

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Consolidation Exercise 3F 1. $18 000 2. $37 500 3. $961 250 4. $360 000 5. $576 000 6. $1 760 000 7. $640 8. $3 530 9. $6 120 10. yes 11. $3 320 12. $117 000 13. −24% 14. $18 600 15. (a) (i) $800 (ii) $3 600 (iii) $8 400 (b) $130 000 16. $150 000 17. (a) $50 000 (b) $15 000 18. (a) $8 910 (b) no, $8 040

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