04. Fractions

04-1 Definition of Fractions (1) Concepts of a Fraction: A fraction is a real number that represents a quotient or ratio of two numbers a and b a a: Numerator , a , a  b , or b a b b b: Denominator (b  0)

(a) A fraction cannot have 0 as a denominator since division by 0 is not defined. 12  , 0  (b) A fraction with 1 as the denominator is the same as the whole number that is its numerator. ex.) 1 12 1 0 3  6  (c) If the numerator and denominator of a fraction are identical, the fraction represents 1. ex.) 3 6 1

(2) Comparisons of Fractions, Ratio, Rate and Percent (a) A fraction is used to represents a part of a whole. a a  Fraction of a number: of c is  c b b a c a c  Fraction of a fraction: of is  b d b d (b) A ratio is used to compare two numbers with the same unit and is expressed as a fraction (c) A rate is a comparison of two quantities with different units. (d) A percent is a representation of a number as a part of one hundred.

[Note]  Meanings of Fraction: (i) Meaning 1: A fraction may mean division. Thus ¾ may mean 3 divided by 4, or 3  4. When a fraction means division, its numerator is the dividend and its denominator is the divisor. (ii) Meaning 2: A fraction may mean ratio. Thus ¾ may mean the ratio of 3 to 4, or 3 : 4. (iii) Meaning 3: A fraction may mean a part of a whole thing. Thus ¾ may mean three-fourths of a dollar, or 3 out of 4 dollars.

04-2 Types of Fractions Types of Fractions Examples Types of Fractions Examples (a) Decimal Fraction: A fraction 3 7 (f) Like Fractions: Fractions that 1 3 , and whose the denominator a power of 10 10 100 have the same denominator 2 2 (b) Common Fraction: A fraction 1 2 1.2 (g) Unlike Fractions: Fractions 1 2 where both numerator and , , (NOT ) and 2 3 4 that have different denominators 2 3 denominator are integers (h) Irreducible Fractions: (c) Proper Fraction: Numerator less 2 3 1 1 2 11 , , Fractions that cannot be simplified , , than denominator  (fraction)  1 5 10 2 2 3 10 any further (i) Equivalent Fractions: fractions (d) Improper Fraction: Numerator 1 2 x 9 11 3 having the same value although   greater than or equal to denominator , , 2 4 2x 5 10 2 they have different numerator and  (fraction)  1 (x  0) denominators 2 2 (j) Complex Fractions: Fractions 1 2  2  1 x  (e) Mixed Number: whole number and 5 5 5 containing at least one other 4 , , 4 a fraction. In calculation with mixed 2 1 12 fraction within it 2 3  numbers, change them into fractions.  x 5 4

[Note]  Equivalence Rules: To obtain equivalent fractions, use one of the following rules. Rule 1: The value of a fraction is not changed if its numerator and denominator are both multiplied by the same number, excluding zero. 3 30 5 5x ex.)  (3 and 4 are multiplied by 10),  (5 and 7 are multiplied by x, where x  0) 4 40 7 7x Rule 2: The value of a fraction is not changed if its numerator and denominator are divided by the same number except zero.

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2 300 3 7a 7 2 2 2 ex.)  (300 and 400 are divided by 100),  (7a and 9a are divided by a , where a  0) 400 4 9a 2 9

 Reducing a Fraction to Lowest Terms: A fraction is reduced to lowest terms when its numerator and denominator have no common factor except 1. 3 3 ex.) 4 is reduced to lowest terms, but 6 is not because 3 is a common factor of 3 and 6.

 Simplifying the Fraction: Write the fraction in the lowest terms (a) Write the numerator and denominator as a product of prime numbers. (b) Then cancel out the common factors.

04-3 Reciprocals and Their Uses (1) Reciprocal: The reciprocal of a number is 1 divided by the number.  The reciprocal of any nonzero number a is 1 a (2) Properties of Reciprocals a b (a) The fraction and are reciprocal of each other. Thus, the reciprocal of a fraction is the fraction inverted. b a  1     (b) The product of any number and its reciprocal is 1: a     1  a  b   1 ex.) 2  3  1  a   b  a  3 2 (c) To divide by a number or a fraction, multiply by its reciprocal. ex.) 8  2  8  3 3 2

[Note]  Zero has no reciprocal.  Find a reciprocal of a whole number  just put 1 over the whole number. 1 ex.) a reciprocal of 2.  2  Find a reciprocal of any fraction  switch the numerator and denominator (“invert the fraction”). 3 4 3 4 ex.) a reciprocal of .  (   1) 4 3 4 3  Find a reciprocal of a negative number  Note an unchanged sign 1 ex.) a reciprocal of   2 2 1  Find a reciprocal of 2 . 3 1 1 7  (i) convert 2 to an improper fraction: 2  3 3 3 3 (ii) turn over the numerator and denominator: 7  To solve an equation for an unknown having a fractional coefficient, multiply both members by the reciprocal 2 3 3 2 3 fraction. ex.) x  10  Multiply both sides by . Then,  x  10   x  15 3 2 2 3 2

04-4 Multiplying Fractions (1) Multiply fractions having no cancelable common factor (a) Multiply their numerators to obtain the numerator of the product (b) Multiply their denominators to obtain the denominator of the product.

(2) Multiplying fractions having a cancelable common factor (a) Factor their numerators and denominators (b) Divide out factors common to any numerator and any denominator (c) Multiply remaining factors

[Example] 3 7 37 x 5 5x (1)       5 11 511 1 2 11 3 r 3r 1 1 3 10 77 3 10 77 7x 2a 7x 2a 7x 2a 2x (2)        11        5 7 6 5 7 6 a 7  7x a 7(1  x ) a 7( 1  x ) 1  x 1 1 2 1 1

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04-5 Dividing Fractions To divide by a fraction, rewrite as multiplication using a reciprocal and then simplify the resulting fraction if possible.

[Example]  To divide any number by a fraction: (a) Find the reciprocal of the fraction. (b) Multiply the number by the reciprocal of the fraction. (c) Simplify the resulting fraction if possible. 1 (i) 3   3  4  12 4 1 3 4 (ii) 3   3   4

4 3 1

04-6 Adding or Subtracting Fractions

(1) Add or Subtracting like Fractions 12 10 8 12  10  8 14 (a) Combine their numerators and Keep the same common denominator ex.)     11 11 11 11 11 (b) Reduce resulting fraction

(2) Add or Subtracting Unlike Fractions (a) Find the Least Common Denominator (LCD) (b) Change each fraction to an equivalent fraction having the LCD as denominator (c) Combine into a single fraction in lowest terms

[Note]  Least Common Denominator (LCD): The LCD of two or more fractions is the least common multiple (LCM) of their denominators. [Example] 1 3 12 33 2  9 11 (2)       (LCM of 6 and 4  12) 6 4 62 4 3 12 12 2 7 5 1 212 7 3 5  4 1 24  21  20  1 24 2            (LCM of 3, 12, 9, and 36  36) 3 12 9 36 312 12 3 9  4 36 36 36 3

04-7 Simplifying Complex Fractions A complex fraction is a fraction containing at least one other fraction within it. 2 2 (1) Multiplication Method 12 2 4 9 2 3 (a) Find LCD of fractions in complex fraction ex.) 3  3   (LCD of and  12) 3 3 33 9 3 4 (b) Multiply both numerator and denominator by LCD 12 4 4 (2) Division Method 1  1 3  2 1 (a) Combine terms of numerator 1 5 1 6 1 ex.) 2 3  6 6  6      (b) Combine terms of denominator 1  1 3  2 5 6 6 6 5 5 (c) Divide new numerator by new denominator 2 3 6 6 6

[Example] a a 1 2  1 2  bd 1   1   18 ad  6 3  18  3  12 27 1 2 2 1 (1)  b  b   6 3    (LCD of , , , and  18) c c bc 2 1  2 1  4  54  9 49 6 3 9 2 bd  3    3   18 d d 9 2  9 2 

2 y  2 y  2 a 1  a c a d ad y y y y (2)  b          c b d b c bc 4 2  (y  2)(y  2) y  2 1  y 4 2 2 d y y 2 y

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