Math Refresher Course

Math refresher course

Prof. Sacha Varone Haute Ecole de Gestion de Genève

Academic year 2011-2012

Contents

1 Review: numbers 3 1.1 Properties ...... 3 1.2 Notation ...... 3

2 Fractions 4 2.1 Introduction ...... 4 2.2 Multiplication and division ...... 5 2.3 Addition and subtraction ...... 5 2.4 Percents ...... 7 2.5 Ratio ...... 7

3 Powers and roots 8 3.1 Introduction ...... 8 3.2 Order of operations ...... 8 3.3 Laws ...... 9 3.4 Roots ...... 9

4 Equations 11 4.1 Introduction ...... 11 4.2 Properties: addition and multiplication ...... 11 4.3 Linear equation with one variable ...... 12

5 Systems of Linear Equations 14 5.1 Introduction ...... 14 5.2 Method of substitution ...... 14 5.3 Method of elimination ...... 15 5.4 Graph representation ...... 17

6 Statistics 18

1 6.1 Measures of central tendency ...... 18 6.2 Measures of dispersion ...... 18

7 Exercises 20 7.1 Numbers ...... 21 7.2 Fractions ...... 22 7.3 Powers and Roots ...... 24 7.4 Equations ...... 25 7.5 Systems ...... 26 7.6 Statistitics ...... 28 Remark Syllabus and content may change before or during the course.

2 1 Review: numbers

Numbers can be divided into several sets:

1. Natural numbers: 0, 1, 2, 3,...

2. Integer numbers: ..., −2, −1, 0, 1, 2,...

2 27 3. Rational numbers: 3 , − 333 ,... p those numbers are of the form q where q is diﬀerent from 0 and p,q are integer numbers √ 4. Real numbers: 2, π, −2.333 ...,...

1.1 Properties

Property (a, b real numbers) Illustration (−1) · a = −a (−1)6.3 = −6.3 −(−a) = a −(−2) = 2 (−a)b = −(ab) = a(−b) (−2)5 = −(2 · 5) = 2(−5) (−a)(−b) = ab (−2)(−6) = 2 · 6

Deﬁnition The absolute value of a number a is the value of a without its sign. The notation is |a| and its value is a if a > 0 −a if a < 0

1.2 Notation

Let’s consider two real numbers a and b, with b diﬀerent from zero

1. Some calculators produce results such as ”4.543E13”. This means 4.543 · 1013, which is 4543 followed by 10 zeros.

−1 1 2. b means b 3. a − b means a + (−b)

a 4. a : b means b

3 2 Fractions

2.1 Introduction

DeﬁnitionA numeric fraction is a quotient of two numbers. The top number is called the numerator and the bottom number is referred to as the denominator. The denominator cannot equal 0. a where b 6= 0 b

2 10 Example 5 or 100

1 A number could be written in the form 5 4 (an interest rate for example). In order to transform it into a fraction, we simply use the denominator of the fraction, 1 5·4+1 21 and use 5 ∗ 4 + 1 as the numerator. Hence 5 4 = 4 = 4 a bc + a c = b b

Deﬁnition A fraction is simpliﬁed if the numerator and denominator do not have any common factors other than 1. Property a · c a = where b, c 6= 0 b · c b

To simplify a fraction, we use a factorisation into prime numbers. A prime number is a whole number that has only two distinct factors, 1 and itself. Examples of prime numbers are 2, 3, 5, 7, 11, ... To simplify a fraction, we can rewrite the denominator and the numerator using a product of prime numbers, and then use the property.

12 Example Simplify 18 . The prime factorization of the numerator 12 would be 12 = (2)(2)(3) . The prime factorization of the denominator 18 would be 18 = (2)(3)(3) Therefore we have 12 2 · 2 · 3 2 = = 18 2 · 3 · 3 3

4 Deﬁnition A fraction is irreducible if it can not be further simpliﬁed.

Example 2 3 is irreducible 2 1 4 is not irreducible since it can be simpliﬁed as 2

2.2 Multiplication and division

Deﬁnition

• Multiplying fractions consists in multiplying the numerators together and multiplying the denominators together. a c ac = b d bd • Two numbers (and hence fractions) are reciprocal of each other if their product is 1 1 a · = 1 where a 6= 0 a

• Dividing a fraction A by another fraction B consists in a multiplying A by the reciprocal of B a c a d ad : = · = b d b c bc

2 4 Example 3 : 5 =? 2 4 2 5 First rewrite the division with a multiplication by the reciprocal: 3 : 5 = 3 · 4 2 5 1 5 Then try to simplify the product: 3 · 2·2 = 3 · 2 1·5 5 Finally do the product: 3·2 = 6

2.3 Addition and subtraction

If both fractions have the same denominator, then a b a + b a b a − b + = − = c c c c c c

5 3 5+3 8 Example 7 + 7 = 7 = 7

5 If the fractions to be added or subtracted do not have a common denominator, then

1. multiply each numerator by the denominator of the other fraction

2. add the two results. This is the new numerator.

3. multiply the denominators together. This is the new denominator

4. simplify the obtained fraction.

a c ad + bc + = b d bd

2 4 Example 3 + 5 =?

1.2 · 5 = 10 and 4 · 3 = 12

2. the new numerator is 10 + 12 = 22

3. the new denominator is 3 · 5 = 15

22 2·11 22 4. the fraction is 15 = 3·5 = 15

6 2.4 Percents

Percent = ”per hundred”. Notation: x%

10 Example 10% = 100 = .10

How to write a percent as a decimal number? Drop the percent sign and move your decimal two places to the left.

Example 75% (=75.00%) becomes .75 163% becomes 1.63 0.2% → 0.002

How to write a decimal number as a percent? Move your decimal two places to the right and then put the % sign at the end of the number. Example .62 → 62% 8 → 800% 0.123 → 12.3%

2.5 Ratio

DeﬁnitionA ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. It is often written as two numbers separated by a colon (:).

Example

• Most movie theater screens have an aspect ratio of 16:9, which means that the screen is 16/9 as wide as it is high.

• In probability, the ratio of the probability of something happening to the probability of it not happening is called the odds of the thing happening.

7 3 Powers and roots

3.1 Introduction

Deﬁnition The power of a number, noted an is the product of a by itself, n times. a is called the base and n is called the exponent. We refer to an as a to the nth power (or simply a to the n) an = a · a · a · ... · a | {z } n times

Example

• 24 = 2 · 2 · 2 · 2 = 16 • (−3)2 = (−3) · (−3) = 9

Property

Property (a, b real numbers) Illustration

a1 = a a 6= 0 (−2.3)1 = (−2.3) 1a = 1 1123 = 1 a0 = 1 a 6= 0 20 = 1 0k = 0 k 6= 0 025 = 0 −k 1 −2 1 1 a = ak a 6= 0 3 = 32 = 9 1 √ √ a q = q a 160.5 = 16 = 4

3.2 Order of operations

Please Parenthesis or grouping symbols Excuse Exponents (and radicals) My Dear Multiplication/Division left to right Aunt Sally Addition/Subtraction left to right

Example −32 + 23 · (−3)2 = −3 · 3 + 2 · 2 · 2 · (−3) · (−3) = −9 + 8 · 9 = 63

8 3.3 Laws

The followings laws apply for real numbers a and b, and integer numbers m and n

Law Illustration aman = am+n 3231 = 33 = 27 (am)n = amn (23)2 = 26 = 64 (ab)n = anbn (10)2 = (2 · 5)2 = 2252 = 4 · 25 = 100

a n an 3 2 32 9 ( b ) = bn where b 6= 0 ( 2 ) = 22 = 4

am m−n 24 1 an = a 23 = 2 = 2

3.4 Roots

1 √ Roots can be considered as special cases of powers: a q = q a

Deﬁnition Let n be a positive√ integer greater than 1, and let a be a real number. The principal nth root n a of a real number a is deﬁned as follows:

√ 1. If a = 0, then n a = 0. √ 2. If a > 0, then n a is the positive real number b such that bn = a.

3. If a < 0, then √ (a) if n is odd, then n a is the negative real number b such that bn = a. √ (b) if n is even, then n a is not a real number.

Warning: if a 6= 0 and b 6= 0

√ √ √ • a + b 6= √a + b √ √ √ Example: 7 + 2 = 9 6= 7 + 2 √ √ √ 2 2 2 2 • a + b 6=√ a + b √ √ √ Example: 22 + 32 = 13 6= 4 + 9 = 5 √ 2 • x = |x| √ Example: 9 = +3

9 Remark It is a good idea to rationalize the denominator of a fraction: √ √ 1 1 a a √ = √ · √ = a a a a

10 4 Equations

4.1 Introduction

Deﬁnition An equation is a statement that two quantities or expressions are equal.

Example A car is moving during t = 2 hours at the speed of r=50 km per hour. What is the distance x traveled after t = 2 hours? x = t · r that is to say x = 2 · 50 = 100

DeﬁnitionA solution or also called root of an equation in x is a number b that yields a true statement when substituted for x.

It is also said that b satisﬁes an equation in x if b is a solution of the equation.

Example 100 is a solution to the equation x = 2 · 50 +1 or -1 are both solutions to the equation x2 − 1 = 0

Deﬁnition Two equations are said equivalent if they have exactly the same solutions.

Example x2 − 1 = 0 and (x + 1)(x − 1) = 0 are equivalent since they both have +1 and -1 as solutions (and no other solutions).

Deﬁnition Solving an equation means to ﬁnd all solutions of the equation.

4.2 Properties: addition and multiplication

If a = b, then a + c = b + c If a = b, then a − c = b − c

Example

• Solve x − 5 = 0 x − 5 + 5 = 0 + 5 x = 5

• Solve 2x + 1 − 3x = 8 − 2x

11 2x − 3x + 1 = −2x + 8 −x + 1 = −2x + 8 −x + 1 − 1 = −2x + 8 − 1 −x = −2x + 7 −x + 2x = −2x + 7 + 2x x = 7

Multiplication and division properties

If a = b, then a · c = b · c If a = b, then a/c = b/c where c is not equal to 0.

Example

• Solve 3x = 12 3x 12 3 = 3 That is x = 4

• Solve 2x − 1 = 0 2x = 1 addition of +1 at each side 1 1 2x · 2 = 1 · 2 1 That is x = 2

4.3 Linear equation with one variable

DeﬁnitionA linear equation in x is an equation that can be written in the form ax+b = 0 where a and b are constant.

Example 5x − 3 = 0 is a linear equation x2 − 1 = 0 is not a linear equation. 2x − 3x + 1 = 3 is a linear equation since it is equivalent to −x − 2 = 0

12 Method for solving linear equations

1. Simplify each side, if needed.

2. Use Add./Sub. properties to move the variable term to one side and all other terms to the other side.

3. Use Mult./Div. properties to remove any values that are in front of the variable.

4. Check your answer.

Example Solve 5 − 3x + 5 − x = 2

1. 10 − 4x = 2

2. 10 − 4x − 10 = 2 − 10 that is −4x = −8

3. −4x/(−4) = (−8)/(−4) that is x = 2

4.5 − 3 · 2 + 5 − 2 = 2

It is possible that the equation has no solution.

Example Solve 4x − 2 = 4x 4x − 2 − 4x = 4x − 4x that is −2 = 0 which is impossible. Therefore the equation 4x − 2 = 4x has no solution.

It is possible that all numbers can be solutions to an equation.

Example Solve 2x − 2 = 2(x − 1) 2x − 2 = 2x − 2 2x − 2 − 2x = 2x − 2 − 2x that is 2 = 2 Therefore all values for x are solutions.

13 5 Systems of Linear Equations

5.1 Introduction

Several linear equations involving the same variables form a system of linear equations. Solving a system of linear equations means to ﬁnd all solutions that satisfy all equations involved in the system of equations. To ﬁnd the solutions of a system, we manipulate the equations until obtaining an equivalent system, i.e. a system with the same solutions.

Theorem Given a system of equations, an equivalent system results if

• two equations are interchanged

• an equation is multiplied or divided by a non zero constant

• a constant multiple of one equation is added to another equation

Example The following systems are equivalent

2x + y= 0 x − y =12

2x + y = 0 ⇔ Multiplication of the second equation by -2 −2x + 2y=−24 2x + y= 0 ⇔ Addition of the ﬁrst equation with the second equation 3y =−24

5.2 Method of substitution

1. Isolate one variable in one equation

2. Replace the variable by its expression in all other equations

3. Solve the remaining system (which contains one less unknown)

4. Find the value of the isolated variable by giving values to the variables in its expression.

14 5. Finally, check your solution.

Example Solve the system 2x + y= 0 x − y =12

The system is equivalent to the following one, in which y has been added to the second equation (both sides)

2x + y= 0 x =12 + y

Substitute x by its expression 12 + y

2(12 + y) + y= 0 x =12 + y

Solve the remaining system, which is a linear equation with one unknown y. 2(12 + y) + y = 0 ⇔ 24 + 2y + y = 0 ⇔ 3y = −24 ⇔ y = −8

Finally x = 12 + y = 12 + (−8) = 4

5.3 Method of elimination

The principle is to keep one equation with all the variables, and to eliminate the same variable to all other equations by a linear combination of equations.

1. Choose a variable to eliminate

2. Keep one equation with all variables.

3. For each equation of the remaining subsystem, add a multiple of the kept equation so that the coeﬃcient of the chosen variable becomes 0

4. Solve the remaining subsystem recursively.

Then you can ﬁnd all the solution by substitution. Do not forget to check your answer!

15 Example Solve the system 2x + y= 0 x − y =12

1. Eliminate the x variable

2. Keep the second equation x − y = 12

3. We can write

x − y =12 (−2)(x − y)=(−2)12 (−2)(x − y) = (−2)12 ⇔ ⇔ ⇔ 2x + y= 0 2x + y = 0 2x + y − 2x + 2y=0 + (−24) x − y= 12 3y =−24

4. The remaining subsystem gives y = −24/3 = −8

Therefore y = −8 and we can use the equation x − y = 12 to ﬁnd the value for x: x − (−8) = 12 ⇔ x = 4

16 5.4 Graph representation

A system of equations with 2 variables can be represented in the x, y plane. The inter- section of the lines representing the equations corresponds to the solution(s) of the system.

Example Solve graphically the system

2x + y= 0 x − y =12

Note that the system is equivalent to

y= −2x y=x − 12

-2*x -4 x-12

-6

-8

-10

-12

-1 0 1 2 3 4 5 6 7

17 6 Statistics

A set of data has to be summarized. The measures most used for one variable are measures of central tendency, and measures of dispersion. A population is the set of (or the measurements obtained from) all objects or individuals of interest. A sample is a subset of the population.

6.1 Measures of central tendency

Deﬁnition The mean of n data values xi, i = 1, . . . , n is its arithmetic average. n 1 X x¯ = x n i i=1

Notation:x ¯ for a sample, µ for a population.

1 Example The mean of 3, 10 and 20 is 3 (3 + 10 + 20) = 11

Deﬁnition In an ordered array, the median of n data values xi, i = 1, . . . , n is the ”middle” value. If n is odd, the median is the middle number. If n is even, the median is the average of the two middle numbers.

Example The median of 1,2,3,4,5 is 3 The median of 1,2,3,4 is 2.5 The median of 10,5,20 is 10

Deﬁnition The mode of n data values xi, i = 1, . . . , n is the value that occurs most often.

Example The mode of 1,2,2,4,4,4,4,5,6,7,7 is 4

6.2 Measures of dispersion

Measures of dispersion give information on the spread or variability of the data values.

Deﬁnition The range of n data values xi, i = 1, . . . , n is the diﬀerence between the largest and the smallest observations

18 Example The range of 10, 1, 1, 20, 100, 5, 5, 5, 5 is (100-1)=99

Deﬁnition The variance of n data values xi, i = 1, . . . , n is the average of squared deviations of values from the mean

• Sample variance n P 2 (xi − x¯) s2 = i=1 n − 1 • Population variance n P 2 (xi − x¯) σ2 = i=1 n

Example The variance of the sample data 10, 20, 30. The mean is (10 + 20 + 30)/3 = 20 2 (10−20)2+(20−20)2+(30−20)2 The variance is s = 3−1 = 100

Deﬁnition The standard deviation of n data values xi, i = 1, . . . , n is the square root of its variance

• Sample standard deviation v u n u P 2 u (xi − x¯) s = t i=1 n − 1

• Population standard deviation v u n u P(x − x¯)2 t i σ = i=1 n

√ Example The standard deviation of the sample data 10, 20, 30 is s = 100 = 10

19 7 Exercises

20 7.1 Numbers

Exercise 7.1 To which set belong those nunmbers?

Natural Integer Rational Real -18.2 1 3 π −e 3E2 -3

Exercise 7.2 If x < 0 and y > 0, determine the sign of the real number

1. xy 4. x : y 7. x : y + x 2. x ∗ x ∗ y 5. y − x 3. (−x)(−y) 6. y(y − x) 8. y ∗ x ∗ y ∗ x ∗ y ∗ x

Exercise 7.3 Rewrite the number without using the absolute value symbol and simplify the result.

1. | − 1 − 5| 3. (−5)|6 − 3| 5. −|3 − 6| − | − 1 − 2|

|3| 2. | − 2| − | − 3| 4. |−2| 6. | − 2||3|(−1)

Exercise 7.4 Discuss the sign of the expression (rewrite without using the absolute value symbol) |5 − x|

Exercise 7.5 Rewrite using the scientiﬁc notation:

1. 17000000 3. 2.6 5. 0.0000000001

2. 0.0000234 4. 256 6. 30002

21 7.2 Fractions

Exercise 7.6 Write the following numbers under the form of a irreducible fraction

1. 13.6 3. 1.8E1 5. -5

2. 0.00012 4. -20.1 6. 80%

Exercise 7.7 Represent under the form of pie charts the following fractions:

• 2/5

• 7/6

Exercise 7.8 Add the correct symbol <, > or = between the following fractions: 12/8 13/8 −5/9 5/(−9) 10/15 2/3 (−3)/(−6) 6/3 −4/5 −10/5

Exercise 7.9 Propose a geometric interpretation of a product of two fractions (for ex- 1 1 ample 2 · 3 )

Exercise 7.10 Give the inverse of the following fractions:

• 3/2

• 3/x

• x/(x+y)

• (-12)/x

22 Exercise 7.11 Do the addition 1. 1/2 + 1/3 3. 2/3 − 5/6 + 1/4 5. 7/5 − (−2)/10

2. 0.8 − 3/5 4. −3/2 + 2/3 6. 2/x + 1/3

Exercise 7.12 Do the product 1. 1/2 · 3/4 · 4/5 4. 3/x · (3/x + 3/2)

2. 3/(−4) · (−2)/6 · 1/3 5. x · (4/7) · x/2

3. 1/2 · (4/5 − 2/10) 6. (2/9)−1(1/9)−1

Exercise 7.13 Simplify

• 200:25 • 100:40 • (2x):(2x+2) • (2x-y+x):(x-y)

2 10 1 • ( 3 − 30 ): 6 2 10 1 • ( 3 ):( 30 ):( 6 )

Exercise 7.14 Bargain! Walking along the streets of Geneva, you see the perfect trousers for you. The price is 150Frs but the shop makes a reduction of 30%. What is the real price that you will have to pay?

Exercise 7.15 You want to buy 100 shares at the price of 66Frs. You know that trading costs you 0.6% of your investment (i.e. the price of the 100 ac- tions) in bank A. In bank B, you only pay a ﬁxed cost of 100Frs, not depending on the amount of your investment.

1. Which bank oﬀers you the best deal? 2. How many shares do you have to buy such that the best deal is done with bank B?

Exercise 7.16 An abo for Unireso (Geneva area) costs today 70 Sfr. Some years ago, its cost was 60 Sfr. What is the increase of the price in percents?

Exercise 7.17 A share costs 15$ at the morning. In the end of the day, its value has decrease by 25%. In the end of the next day, its value has increased by 25%. What is the new price of the share in the end of the next day?

23 7.3 Powers and Roots

Exercise 7.18 Use the scientiﬁc notation. A telecommunication data center gets 500 Go (Gigaoctet, Giga=109) of raw data per day. Raw data is represented by a succession of lines in ﬁle, with each line providing information about the caller id, the duration of the call, the called id, .... One line of data occupies about 10 octet. How many lines are recorded each day?

Exercise 7.19 Do the calculation

1. 52 · 23 6. (3x3) : (2x2)

2. 33 + 3 7. x−1 : x−2

5 3 4 −2 3. (−3) · (23 − 6)2 8. (10 · 100 · 0.001 ) : (0.1 · 100 ) 3 −5 9. 8x y 4. [3 − 23]2 − 52/25 4x−1y2

2 −3 2 3 −1 2 x 5. x y · x · y 10. 3y

Exercise 7.20 Let a = 3, b=-4, c=-3/4. Calculate −b − (b2 − 4ac)0.5 x = 2a

Exercise 7.21 Rationalize each denominator

1. √1 3

q 2 2. 7

p3 y 3. x2

Exercise 7.22 from [?] On a clear day, the distance d in miles that can be√ seen from the top of a tall building of height h (in feet) can be approximate by d = 1.2 h. Approximate the distance that can be seen from the top of the Chicago Sears Tower, which is 1454 feet tall.

24 7.4 Equations

Exercise 7.23 Solve the following equations

1. x − 2 = 1 − x 7. x2 − 9 = 0

2. 2x − 3 − x = x − 3 8. (x + 1)(x − 2)2 = 0

3x 6 3. 3(x − 1) − 5x = 2x + 2 9. x−2 = 1 + x−2

1 4 4. |x + 4| = 10 10. 2x−1 = 8x−4

9x 3 5. 2|3 − x| = 1 11. 3x−1 = 2 + 3x−1

√ 2 3 −2x+7 6. 20 − x = 4 12. 2x+1 − 2x−1 = 4x2−1

Exercise 7.24 A car travels a distance d = t · y, where t is a number of hours, y is the speed in km/h and d is the distance in km. Given the distance and the time, what is the mean speed of the car ?

Exercise 7.25 The marks in mathematics of a student are 3, 4.5, 4 and there is still one exam to do. What mark should he receive to get a mean of 4 ?

Exercise 7.26 In a shop, every price has been decreased by 20%. If you ﬁnd a shirt at the price of 28 Sfr, what was the original price ?

Exercise 7.27 100’000$ have to be invested in two hedge funds A and B. A has an annual interest rate of 15%, but with some high risk, and B has an interst rate of 10%. Is there a way to invest the monney so that the annual interest is 12’000$ ?

Exercise 7.28 A task can be made in 3 hours by Anna, and in 4 hours by John. If both John and Anna are working on that task, how long does it take to make it ?

Exercise 7.29 A theater manager wants to know how many entries are necessary to make proﬁt. The price paid by the customers is 50 Frs. There is a ﬁxed cost for the show: 2000Frs as well as a variable cost of 3 Frs for each entry.

Exercise 7.30 Claire gets 30$ and Brice gets 30$. How much Brice should give to Claire so that Claire has twice as much as Brice?

Exercise 7.31 The sum of three consecutive integers is 375. What are those integers?

Exercise 7.32 In a zoo, the visit costs 30$ for grown-ups and 18$ for children. In the end of a particular day, 630 people have visited the zoo, and the beneﬁt was 14’220$. How many children and grown-ups were there?

25 7.5 Systems

Exercise 7.33 Use the substitution method to solve the following systems of equations.

1. x − y = 4 x + 2y=10

2. 3x + y = 10 −x + 2y=−8

3. 2x + 4y= 12 x − 2y =−4

4. x + y2 =6 x + 2y=3

5.   x − 2y + 3z=28 2x − 3y − z=35  x + y + 2z = 5

Exercise 7.34 Use the method of elimination to solve the following systems of equations.

1. x + 3y=−1 2x − y= 5

2. −3x = y 2x + 4y + 2=8x − 6y

3. 5x − y =−1 −10x + 2y= 2

Exercise 7.35 Use a graph representation to represent the situations in the previous exercise.

Exercise 7.36 Solve the following systems of equations

1. 3x + 6y=5 2x + 4y=2

26 2. 5x − 10y=−5 4x − 8y =−4

3. 5x − y =−1 −10x + 2y= 2

Exercise 7.37 The admission price to a cinema is 11 Frs for students and 16 Frs for non-students. If 300 tickets were sold for a total of Frs 4’300, how many of each kind were purchased?

27 7.6 Statistitics

Exercise 7.38 An ice-cream seller has written during several days the number of sold ice-creams per day and the temperature of that day.

Number of ice-creams 115 149 212 163 64 73 95 65 194 220 Temperature 22 25 31 26 16 17 20 17 29 32

He wants to deduce the number of sold ice-creams in function of the temperature by using a simple linear regression: Estimated number of sold ice-creams = 10.16 x Temperature - 103.72

1. Compute the mean and the standard deviation of the temperature variable.

2. Represent this situation (data + regression line) with a scatter plot.

3. He can sell 15 ice-creams with a 1 liter ice-cream bag. If the temperature forecasting for tomorrow is 24 degree, which quantity of 1 liter ice-cream bag should be taken?