Concordia University Department of Education

EDUC 387 TEACHING MATHEMATICS II COURSE NOTES

PART II. RATIONAL NUMBERS. RATIO. PERCENTS. (REVISED VERSION)

by ANNA SIERPINSKA

Professor in The Department of Mathematics and Statistics

© 2015 Anna Sierpinska

Table of contents

INTRODUCTION 1

CHAPTER I. RATIONAL NUMBERS 2

I.1 Thinking of numbers in terms of number systems 2

I.2 Appreciation of the differences between the theories of rational numbers and fractions of

quantities 10

I.3 Brief outline of the formal theory of rational numbers 12

CHAPTER II. RATIO 22

II.1 Ratios and fractions 22

II.2 Definition of ratio 24

CHAPTER III. PERCENTS 31

III.1 Interpreting statements about percents of quantities 31

III.2 Three basic types of percent problems and their solutions 33

III.3 Increasing/decreasing a quantity by a certain percent 34

Problem set 38

REFERENCES 44

List of Figures

Figure 1. Confusing the language of quantities with the language of rational

numbers. ..............................................................................................................................12

Figure 2. A rectangle with side lengths in the ratio 3 : 2. .......................................25

Figure 3. A map of Canada. ......................................................................................27

Figure 4. The puzzle fits into a square whose side is 8 cm. .....................................29

Figure 5. "Fibonacci rectangles."..............................................................................30

Figure 6. A student’s solution of a price reduction problem. .................................37

Figure 7. A receipt for two books bought at the Université de Montréal bookstore

in 2013. .................................................................................................................................40

Figure 8. Scotch tape package advertised as "33 % more". .....................................41

Figure 9. Some data from Table 2 represented in the form of a bar diagram. ......42

1

Introduction

This is the second of the three parts of the text for the Teaching Mathematics II

course for prospective elementary school teachers. In the first part, we studied fractions

of quantities.

In this second part, in Chapter I, we abstract the notion of fraction from the

context of quantities and conceptualize it as part of an abstract number system – the

rational numbers system – embedded in an even larger system of real numbers. One aim

here is to situate the theory of fractions of quantities within a broader mathematical

horizon. Another is that we need the notion of real number to understand the notion of

ratio, which we introduce in Chapter II, and distinguish it from fraction. The notion of

percent, studied in Chapter III, is closely related to ratio, and therefore also requires a

broadening of the concept of number.

Note: This is a slightly revised – mainly corrected – version of course notes under the

same title published in 2014.

2

Chapter I. Rational numbers

In this chapter, we make a brief excursion into a theory of abstract fractions. In

the expression

Quantity 𝐴 is the fraction 𝑎

𝑏 of the quantity 𝐵

studied in Part I, we will now ignore (abstract from) everything but the fraction:

Quantity 𝐴 is the fraction 𝑎

𝑏 of the quantity 𝐵

In the theory of abstract fractions, a fraction is treated as any other number. But

“number” in this theory means something a little different from the old “whole number”

from early elementary school years, used for counting objects. Indeed, to begin thinking

of fractions as numbers we must start thinking differently about numbers and

operations on them.

I.1 Thinking of numbers in terms of number systems

In formal mathematical theory, numbers are conceived of as elements of systems

that are “closed under certain operations.”

Natural numbers

For example, the “whole numbers” of elementary school are thought of as a

system of numbers (called “natural numbers” and labelled ℕ) obtained from the number

1 by addition. So if we have the number 1 and are allowed to add, then the system

contains the number

1 + 1 , labeled “2”;

1 + 1 + 1 , labeled “3”,

etc.

3

The system of natural numbers ℕ is closed under addition: when we add any two

natural numbers, the result is another natural number.

But ℕ is not closed under subtraction, where

“to subtract a number 𝑏 from a number 𝑎”

means

“to find a number 𝑐 such that 𝑐 + 𝑏 = 𝑎.”

The number 𝑐 is denoted by the symbol 𝑎 – 𝑏, and called the “difference.”

For example, 5 − 3 = 2 because 2 + 3 = 5.

The difference 𝑎 – 𝑏 of two natural numbers is sometimes a natural number

(e.g., 5 – 3 is a natural number), and sometimes not (e.g., 3 – 5 is not a natural

number).

Why is 3 − 5 not a natural number?

Because, if it were, then there would be a natural number 𝑐 such that 𝑐 + 5 = 3.

But if 𝑐 is a natural number, then 𝑐 + 5 is always greater than 5, so it can never be

equal to 3.

Integers

A larger system of numbers must be constructed if we want to allow not only

unconstrained addition but also unconstrained subtraction. So, if we want the system to

contain all possible differences of natural numbers, then the system must include the

differences:

1 − 1, 2 − 2, 3 − 3, 4 − 4, etc., labelled “0”;

1 − 2, 2 − 3, 3 − 4, 4 − 5, etc., labelled “−1”;

4

1 − 3, 2 − 4, 3 − 5, 4 − 6, etc., labelled “−2”;

etc.,

as well as the differences

2 − 1; 3 − 2; 4 − 3; 5 − 4; … labelled “1” or “+1”;

3 − 1; 4 − 2; 5 − 3; 6 − 4; … labelled “2” or “+2”;

etc.

The smallest number system containing ℕ and closed under addition and

subtraction is called “integers” and is labelled ℤ.

Let us note here, that “containing ℕ” used in the above sentence, is an abuse of

language, because the integer +2 is not exactly the same as the natural number 2. The

“sameness” refers only to a structural similarity between positive integers and natural

numbers, not to their nature. The natural number 2 represents the number of objects in

a certain type of sets; so it refers to a state of affairs. The integer +2, on the other hand,

refers to a change in the state of affairs; an increase of the number of elements in a set by

2. In general, an integer can be interpreted as a measure of the decrease or the increase

in the number of objects in a set.

Just as positive integers are not exactly the same as natural numbers, so the

operation of addition in integers is not exactly the same as the operation of addition in

natural numbers. For example, (+3) + (+2) = +5 just as in natural numbers 3 + 2 =

5. But we need to modify our notion of addition to account for sums such as, e.g.,

(−3) + (+2) = −1.

Both ℕ and ℤ are closed under multiplication: the product of any two natural

numbers is a natural number; the product of any two integers is an integer. But, again,

the operation of multiplication in ℤ cannot refer to repeated addition which made sense

5

in the natural numbers. It must be redefined to apply to multiplying two negative

numbers, for example, (−1) × (−1). The fact that this product is equal to 1 cannot be

explained by repeated addition, which works only if at least one of the factors is positive.

But if we want the distributive law to be valid in integers, and multiplication of any

number by 0 to yield 0, and +1 multiplied by any number to be equal to that number,

then (−1) times (−1) must be equal to +1. It is enough to show that

(−1) × (−1) + (−1) = 0 (*)

because this means that (−1) × (−1) is “the opposite” of (−1) (these two numbers

“cancel each other out”). But the only opposite to (−1) is (+1); so (−1) × (−1) must

be the number +1

Here is a demonstration of the equality (*):

(−1) × (−1) + (−1)

= (−1) × (−1) + (−1) × (+1)

= (−1) × ((−1) + (+1))

= (−1) × 0 = 0

This demonstration is given here only to give the reader a taste of the nature and

formal theory of integers. A more detailed development is beyond the scope of this

booklet1.

Rational numbers

The system of integers is closed under the extended operation of multiplication,

signaled above, but it is not closed under division. Division is understood here as the

inverse of multiplication: to divide a number 𝑎 by a number 𝑏 is to find a number 𝑐

1 Interested readers are referred to the article (Coltharp, 1966), and to (Courant & Robbins, 1953, pp. 54-5). The book by Courant and Robbins is a classic of popular books about more advanced mathematics, addressed to general public.

6

such that 𝑐 multiplied by 𝑏 yields 𝑎: 𝑐 × 𝑏 = 𝑎. The number 𝑐 is denoted by the

symbol 𝑎 ÷ 𝑏, and called the “quotient.”

The quotient 𝑎 ÷ 𝑏 of two integers is an integer only if 𝑎 is divisible by 𝑏

without remainder; in other words – if “𝑏 is a factor of 𝑎.” For example, 12 ÷ 4 is an

integer because 4 is a factor of 12; but 12 ÷ 5 is not an integer because there is no

integer that, multiplied by 5, yields 12.

Again, a larger system of numbers has to be constructed to allow for

unconstrained division. This leads to the system of “rational numbers” (labelled ℚ),

where an almost unconstrained division is possible; “almost” – because the system does

not allow for division by 0.

The system of rational numbers contains the quotient of not only 1 ÷ 1, but also

1 ÷ 2, 1 ÷ 3, 1 ÷ 4, 1 ÷ 5, …

labelled

1

2,

1

3,

1

4,

1

5, …, respectively.

These symbols are called “fractions.” The symbol 1

3 denotes a number which,

multiplied by 3, gives 1.

The system contains also the numbers

2 ÷ 1, 2 ÷ 2, 2 ÷ 3, 2 ÷ 4, 2 ÷ 5, …

labelled

2

1,

2

3,

2

4,

2

5 , …, respectively,

as well as the numbers

−1 ÷ 2, − 1 ÷ 3, − 1 ÷ 4, − 1 ÷ 5, …

labelled

7

−1

2, −

1

3, −

1

4, −

1

5, …, respectively;

and so on, for any quotient 𝑎 ÷ 𝑏, where 𝑎, and 𝑏 are any integers (positive or

negative) and 𝑏 is not equal to 0. We can write the elements of this system succinctly

as the set:

ℚ = { 𝑎

𝑏: 𝑎, 𝑏 integers and 𝑏 ≠ 0 }

The reason why we have not included quotients such as 𝑎 ÷ 0 in the system of

rational numbers is quite simple. For suppose we did claim that 𝑎 ÷ 0 is equal to some

rational number 𝑐. Then, by the definition of quotient, we would have to conclude that

𝑐 × 0 = 𝑎. But 𝑐 × 0 = 0 for any number 𝑐. So 𝑎 would have to be 0 for any 𝑎,

meaning that there would be no other numbers except 0 in the system of rational

numbers. Such system would be useless.

Rational numbers are often also called “fractions”, as mentioned above, and we

will use this term in this sense whenever it will be clear from the context that we are

talking about rational numbers and not fractions of quantities. Sometimes, to stress that

we are talking about rational numbers and not about fractions of quantities, we will use

the expression “abstract fractions.”

In some publications, the term “fraction” refers only to the symbolic expression of

the form 𝑎

𝑏 where 𝑎 and 𝑏 are integers, and 𝑏 ≠ 0 (The National Council of Teachers

of Mathematics, 2010, pp. 9, 15). This form is distinguished from the “decimal form” in

which rational numbers can also be written. The quoted NCTM publication says that 29

20

“is a fraction”, but 1.45 is not, although they both represent the same rational number.

This multiplicity of senses in which the term “fraction” is used, in mathematics, in

school, and out-of-school contexts, is, in itself, a source of misunderstanding and

confusion in learners. But there is not much we can do about it. We can only be aware of

8

the problem and take the time to clarify what we and our interlocutors mean whenever

the possibility of confusion arises.

The system of rational numbers requires yet another extension of the arithmetical

operations known in the previous systems. We will define these operations later in this

chapter (section I.3).

Real numbers

The system of rational numbers, although closed under all four arithmetical

operations, was still not sufficient for the purposes of developing mathematical theories

and their applications. In particular, it does not allow to express, exactly, the number of

times that the circumference of a circle is greater than its diameter. This number is the

number 𝜋, which cannot be calculated as the quotient of one integer by another. So

mathematicians constructed the “real numbers.” One may ask if the extension of

rational numbers to real numbers can also be explained by a desire to waive the

constraints of some operation and, if yes, what this operation would be. The answer to

this question is positive in a certain sense, but its exposition would be well beyond the

scope of this booklet. It requires knowledge of Mathematical Analysis. A general idea

will be presented in a lecture, in an intuitive way.

For the purposes of this course, our understanding of real numbers will be

restricted to the notion that each real number has a decimal expansion, and that any

“decimal expression” with a finite integer part (the part to the left of the decimal point)

represents a real number. The part to the right of the decimal can be finite or infinite,

periodic or not. If it is finite or periodic, the decimal represents a rational number. If it is

0, the decimal represents an integer.

For example, the decimal

0.666 … ..

9

represents the rational number 2

3. It is a periodic decimal, with the period being the digit

6. The decimal

2.367367367367 … ..

represents the rational number 2367

999.

The decimal

0.999 …

represents the integer 1. (Explain why)

The decimal

6.999 …

represents the integer 7. A decimal needn’t have 0 to the right of the decimal point to

represent an integer.

On the other hand, the decimal

1.01001000100001000001 ….

where the number of zeros between the 1s is increasing by one, is not a rational number.

Exercise I.1 Decimal representations of fractions

Explain why a decimal representation of a fraction is either finite or periodic.

Exercise I.2 Representing a periodic decimal as a fraction

Represent 0. 123̅̅ ̅̅ ̅ in the form of a fraction 𝑎

𝑏 .

What is the general principle in converting periodic decimals into fractions?

Exercise I.3 The number 𝜋

True or false?

10

a. 𝜋 = 3.14

b. 𝜋 = 3.1415926536

c. 𝜋 =22

7

d. 𝜋 is a rational number

e. 𝜋 is a real number

f. 𝜋 ≈ 3.14

Exercise I.4 Identifying rational numbers

Which of the following symbols represent rational numbers?

a. 23

10, 2.3, 3,

3

1,

6

2, − 3, −

2

3

b. 0.5, 1

2,

−6

7,

22

7

c. 3.141141141141 … . , 0.66666666 ….

d. √2

e. √4

I.2 Appreciation of the differences between the theories of rational

numbers and fractions of quantities

It is important for teachers to be aware of the differences between the theories of

fractions of quantities and rational numbers. Many children’s difficulties with fractions

could be avoided or lessened if teachers, textbooks, and other resources did not mix

elements of the two theories in the same sentence or problem.

Here is a simple example of such confusion. It can be found on the popular

Canadian website “IXL” which offers many electronically graded exercises and problems

for every elementary school grade2. One set of “Practice problems” for Grade 5 is titled

“Fraction of a number”, and has the following question:

2 http://ca.ixl.com/math/grade-5/fractions-of-a-number

11

“What number is 1

10 of 10?”

In the theory of fractions of quantities, expressions of the form “𝑎

𝑏 of [… ]” were

common. In this theory, after “1

10 of”, one expects a name of a quantity or its measure

in terms of a denominate number (e.g., 10 𝑘𝑔, or 10 𝑐𝑚2, or 10 marbles), and not an

abstract number such as 10. So the question is not a legitimate question in the theory of

fractions of quantities. If “of” is used in the question, then it would make more sense to

ask, e.g., “The fraction 1

10 of 10 𝑙 of water represents what volume of water?”

In the theory of rational numbers, on the other hand, there is no concept of

“fraction of a quantity.” A fraction is an entity in its own right; it is a number. We don’t

speak of fractions of something; “of” is not an arithmetical operation. Therefore, an

expression such as “1

10 of 10” is not grammatically correct in this theory. Here, it would

make more sense to ask, “Calculate 1

10 × 10.”

In Table 1 we give contrasting examples of statements in the language of abstract

numbers and in the language of quantities.

The language of abstract numbers The language of quantities

1

7× 98

1

7 of 98 baby wipes

2

5+ 100

The total distance is 2

5 𝑘𝑚 + 100 𝑚

1

2÷ 5 If

1

2 𝑔𝑎𝑙𝑙𝑜𝑛 of water is distributed

equally among 5 children then each

child gets 1

2 𝑔𝑎𝑙𝑙𝑜𝑛 ÷ 5 of water.

Table 1. Contrasting examples of statements about abstract numbers and quantities.

12

Another example of confusion between the language of fractions of quantities and

the language of rational numbers was given in Part I, Problem II.2. This problem quotes

an exercise given in (Lamon, 2012, p. 122). We repeat the quote in Figure 1.

Figure 1. Confusing the language of quantities with the language of rational numbers.

An additional source of confusion in this problem can be the fact that it is not clear

what quantity the rectangle with the stars is meant to represent, and two-thirds of what

quantity is to be calculated. We can perhaps guess from the question "2

3 is how many

stars?” that the quantity is “number of stars”, rather than, for example, the area of the

rectangle. But too much in this problem depends on being familiar with the implicit

graphical conventions used in representing school problems about fractions, and

correctly guessing the intended meanings behind them.

I.3 Brief outline of the formal theory of rational numbers

The theory of rational numbers defines a rational number as an ordered pair of

integers. Some pairs will be considered equal; equal pairs will be said to represent the

same rational number. Equal pairs will be also called “equivalent.” The theory also

defines what it means to add, subtract, multiply and divide rational numbers, and shows

how the result of these operations does not change if we replace a given pair by an

equivalent one.

These definitions agree with the rules of adding or subtracting fractions of

quantities, calculating a fraction of a fraction of a quantity, and finding the quantity of

13

which a given fraction is the given quantity. We could say that the theory of rational

numbers has been inspired by the practice of working with fractions of quantities.

We give some details of the theory in the subsections below.

I.3.1 Definition of a rational number

Formally, a rational number is defined as an ordered pair of integers, usually

written using a symbol of the form 𝑎

𝑏 , called “fraction”, with the assumption that the

second element of the pair is not zero, and the assumption that two such pairs, 𝑎

𝑏 and

𝑐

𝑑

represent the same rational number (“are equivalent”) if 𝑎 × 𝑑 = 𝑏 × 𝑐. In this case, we

write, 𝑎

𝑏=

𝑐

𝑑.

For example, we can say that 2

3=

4

6 because 2 × 6 = 3 × 4. For similar reasons,

the fractions 6

9,

8

12,

10

15,

12

18 represent the same rational number as

2

3.

Exercise I.3.1.1 Producing and verifying examples of equivalent fractions

a. Give other examples of equivalent fractions.

b. Do the fractions 23

5 and

46

11 represent the same rational number?

c. Give three fractions representing the same rational number as 28

63. How many more such

equivalent fractions can you give?

Exercise I.3.1.2 Constructing equivalent fractions

a. Find an integer 𝑛 such that 𝑛

7=

8

4.

b. Find an integer 𝑑 such that 7

𝑑= 455.

c. Invent questions that would be of the same types as the two questions above. Are your

questions more difficult or easier? Justify.

14

Exercise I.3.1.3 Contrasting relations between rational numbers and fractions of

quantities

In the theory of abstract fractions it is true that 1

3=

3

9.

Is it always the case that 1

3 of a quantity is the same as

3

9 of a quantity? Justify if true; give a

counterexample if false.

I.3.2 Order in rational numbers

We have seen in Part I of the course notes, Chapter VIII, that whether or not a

fraction 𝑎

𝑏 of one quantity represents a bigger, smaller or the same quantity as a

fraction 𝑐

𝑑 of another quantity depends as much on the numbers 𝑎, 𝑏, 𝑐 and 𝑑 as on

the sizes of the reference quantities. In the theory of rational numbers, the order

between fractions 𝑎

𝑏 and

𝑐

𝑑 depends solely on the numbers 𝑎, 𝑏, 𝑐 and 𝑑. For

example, one-half is always less than three-quarters: 1

2<

3

4 . The order is defined

formally, as follows.

We say that the fraction 𝑎

𝑏 is less than the fraction

𝑐

𝑑 ,

written 𝑎

𝑏<

𝑐

𝑑 , whenever 𝑎 × 𝑑 < 𝑏 × 𝑐.

For example, 1

2<

3

4 because 1 × 4 < 2 × 3. Note that if, instead of

1

2 , we take

any fraction equivalent to it, it will still be smaller than 3

4. Let us take

750

1500, which

represents the same rational number as 1

2. It is still true that

750

1500<

3

4 (Verify).

Exercise I.3.2.1 Ordering given numbers

a. Order the numbers below from the largest to the smallest:

21

5; 1;

32

13;

303

100;

9

4;

5

2; 2

1

3

b. Order the numbers below from the smallest to the largest:

1; 6

5;

1

2;

3

7;

5

6;

8

7;

7

8;

8

9

15

Exercise I.3.2.2 Constructing rational numbers given inequality constraints

a. Find the largest integer 𝑛 such that 𝑛

7<

5

4.

b. Find the smallest integer 𝑑 such that 7

𝑑<

4

5.

c. Find the smallest integer 𝑑 such that 7

𝑑< 3.

I.3.3 Addition of rational numbers

In the theory of fractions of quantities, we had to be very careful about the kinds

of quantities we were adding, because addition did not always make sense. In the theory

of rational numbers, any two rational numbers can be added, producing another rational

number. Here is the definition of addition of two rational numbers:

𝑎

𝑏+

𝑐

𝑑≝

𝑎 × 𝑑 + 𝑏 × 𝑐

𝑏 × 𝑑

where the multiplication and addition signs on the right-hand side of the equation refer

to operations on integers, assumed to have been defined and known before.

Note that, formally, this definition is the same as the rule in the general solution

to the problem of addition of two fractions of the same quantity in Part I of the course

notes, Section IV.3:

𝑎

𝑏 𝑜𝑓 𝑄 +

𝑐

𝑑 𝑜𝑓 𝑄 =

𝑎 × 𝑑 + 𝑏 × 𝑐

𝑏 × 𝑑 𝑜𝑓 𝑄

This definition implies that, if the fractions have the same denominators, then a

simpler rule applies:

𝑎

𝑏+

𝑐

𝑏=

𝑎 + 𝑐

𝑏

16

Exercise I.3.3.1 Why can we just add the numerators if the denominators are the same?

Prove the rule of addition of fractions with the same denominators, stated above.

The theory of rational numbers allows to add any two fractions and does not

concern itself with whether these fractions of the same quantity or not. But it is

concerned with the fact that any given rational number can be represented by many

different fraction symbols. Will the result of addition change if a different, but equivalent

fraction is used to represent it?

For example,

3

5+

7

10=

3×10+5×7

5×10=

65

50=

13

10.

If we take, say, 9

15 instead of

3

5, and

14

20 instead of

7

10 to represent the same

rational numbers, will the resulting fraction be equivalent to 13

10 ?

9

15+

14

20=

390

300

The answer is positive, since 390

300=

13

10.

This is just an example, but, in the theory of rational numbers, it is shown that the

sum of any two rational numbers does not depend on the choice of the fractions that

represent them. This theorem allows taking shortcuts in addition. For example, instead

of applying the definition to calculate the sum in the example above, we could have

replaced 3

5 by the equivalent fraction

6

10 and applied the rule of adding fractions with

the same denominators.

Exercise I.3.3.2 Why is the definition of addition of fractions so complicated?

Why do you think addition of fractions was not defined by the rule: “add the numerators and add

the denominators”: 𝑎

𝑏+

𝑐

𝑑=

𝑎+𝑐

𝑏+𝑑 ?

17

Exercise I.3.3.3 A computational exercise

Compute the sums below using at least two different strategies. Explain why the result does not

depend on the choice of the strategy.

a. 4

9+

7

12

b. 2

7+ 5

c. 53

4+ 7

5

6

Exercise I.3.3.4 Defining subtraction

a. Define the operation of subtraction of two rational numbers, calling the result “the

difference.”

b. Does the difference of two rational numbers depend on the choice of the fractions used to

represent them? Justify your answer.

c. Give examples and show different strategies of computing the difference of two fractions.

d. Write a sequence of exercises for practicing subtraction, ordered from the easiest to the most

difficult. Explain what makes each exercise more difficult than the preceding one.

I.3.4 Multiplication of rational numbers

If two rational numbers are represented by fractions 𝑎

𝑏 and

𝑐

𝑑 , the result of the

multiplication of these rational numbers is the rational number represented by the

fraction 𝑎×𝑐

𝑏×𝑑, called their “product.” Symbolically:

𝑎

𝑏×

𝑐

𝑑≝

𝑎 × 𝑐

𝑏 × 𝑑

where the multiplication signs on the right-hand side of the equation refer to

multiplication of integers, assumed to have been defined and known before.

Note that this definition has been inspired by the rule in the general solution of

the fraction of a fraction of a quantity problem in Part I of the course notes, Section V.2:

18

If 𝑋 is 𝑎

𝑏 of 𝑌,

and 𝑌 is 𝑐

𝑑 of 𝑍,

then 𝑋 is the fraction 𝑎×𝑐

𝑏×𝑑 of 𝑍.

The theory of rational numbers is concerned with the independence of the result

of multiplication from the choice of their representatives in the form of fractions. We

leave the explanation of this independence as an exercise.

Exercise I.3.4.1

a. Calculate 4

9×

7

12.

b. If we replace the fractions we have multiplied in part (a) by equivalent fractions, is the product

we get a different fraction from the one we got before? Justify.

c. Try to prove, in general terms, the independence of the result of multiplication of rational

numbers from the choice of their representatives in the form of fractions.

d What is the difference between 2 ×7

9 and 2

7

9 ?

Exercise I.3.4.2

Multiplication of abstract fractions is commutative, i.e., the result does not depend on the order

of the terms. Can we say that multiplication of fractions of quantities is commutative?

I.3.5 Division of rational numbers

To divide a rational number 𝑎

𝑏 by a rational number

𝑐

𝑑 is to find a rational

number 𝑥

𝑦 such that

𝑎

𝑏=

𝑐

𝑑×

𝑥

𝑦:

𝑎

𝑏÷

𝑐

𝑑=

𝑥

𝑦 if and only if

𝑎

𝑏=

𝑐

𝑑×

𝑥

𝑦

This is how we understand division of numbers: as an operation that becomes

necessary in answering questions about multiplication where the product is given but one

of the factors is not: “𝐴 times WHAT equals 𝐶?”; or “WHAT times 𝐵 equals 𝐶?” T

19

We will derive the rule for division of abstract fractions by analyzing the meaning

of the definition above.

In this definition, the equality 𝑎

𝑏=

𝑐

𝑑×

𝑥

𝑦 is to be understood as equivalence of

the abstract fractions, and the multiplication sign – as multiplication of such fractions, as

defined in section I.3.4. So this equality can be written as

𝑎

𝑏=

𝑐 × 𝑥

𝑑 × 𝑦

which, according to the definition of equivalent fractions, means that

𝑎 × 𝑑 × 𝑦 = 𝑏 × 𝑐 × 𝑥

If we look at this equality this way, using associativity of multiplication of integers,:

(𝑎 × 𝑑) × 𝑦 = (𝑏 × 𝑐) × 𝑥

then it can be interpreted as equivalence of a different pair of fractions:

𝑥

𝑦=

𝑎 × 𝑑

𝑏 × 𝑐

The right-hand side of this equation can be interpreted as a product of the dividend and

the inverse of the divisor:

𝑥

𝑦=

𝑎

𝑏×

𝑑

𝑐

So this is how we divide a fraction by a fraction:

𝑎

𝑏÷

𝑐

𝑑=

𝑎

𝑏×

𝑑

𝑐

which explains why “to divide” is often defined as “to multiply by the inverse.”

We have already seen a similar rule in the context of fractions of quantities, in

Part I of the course notes, Section VI.3, in the form of the theorem:

20

If 𝐴 is a quantity and 𝑚 and 𝑛 are non-zero whole numbers then

𝐴 ÷𝑚

𝑛=

𝑛

𝑚× 𝐴

Exercise I.3.5.1 Why the rule for dividing fractions cannot be simpler?

Give a good reason why the rule for dividing fractions cannot be: divide the numerators and

divide the denominators, i.e., 𝑎

𝑏÷

𝑐

𝑑=

𝑎÷𝑐

𝑏÷𝑑 ?

Exercise I.3.5.2 Computational exercises

Calculate, treating all numbers as rational numbers and all operations as operations on rational

numbers.

a. 21

42÷

7

8

b. 3

4÷

4

3

c. 3

4×

4

3

d. Divide 1 by one-half.

e. Divide 1 by 2.

f. 3

4÷ 2

g. 3

4÷

1

2

h. 2 ÷3

4

i. 21

4÷

3

4

j. 12 ÷ 5

k. (1+ 1

4) ÷

10

4

l. 720 ÷ 21

2

m. 1

8+

1

3×

1

4+

1

8+ (

7

8÷

7

4) +

1

6

Exercise I.3.5.3 Constructing a computational exercise

Construct an exercise on calculations with abstract fractions that would look quite complicated

but the result would be a simple whole number, for example, 2.

21

Exercise I.3.5.4 Fraction bar as division sign

The fraction bar is sometimes used in the sense of division. For example, the expression 3

3

5

8

means the same as 33

5÷ 8. Calculate

24

3+5

6

.

Exercise I.3.5.5 Independence of division of rational numbers from the choice of their

representations in the form of fraction symbols.

a. Verify that the result of dividing 5

6 by

1

2 is the same rational number as the result of dividing

10

12 by

3

6 .

b. Show that the result of dividing 𝑎

𝑏 by

𝑐

𝑑 is the same rational number as the result of

dividing 𝑎×𝑘

𝑏×𝑘 by

𝑐×𝑚

𝑑×𝑚 for any integers 𝑘, 𝑚 .

Exercise I.3.5.6 Assessing a student’s response to a problem about fractions of quantities

The teacher gave the following problem to a class of 6th graders:

On a certain day, a small stall for tourists in Mexico attracted 40 customers and each one bought an item. Since it was such a

swelteringly hot day, 1

2 of the customers bought straw sun hats;

1

4

of the customers bought handmade souvenir cloth dolls, 1

8 of the

customers bought disposable cameras, and 1

8 bought T-shirts. How

many people bought each item? Explain how you derived your answer.3

Asked how they solved the problem, one of the students says:

“I divide 40 customers in half and get 20 sun hat buyers. Then I do

40 ÷1

4 = 10, so 10 customers bought dolls.”

a. How would you assess this response? b. How would you explain it? c. What would you say to the student?

3 The problem is adapted from Phillips, T. (2013). Around the world in twelve problems. A problem book for the use of elementary school teachers. Montreal: Concordia University. The book was presented to A. Sierpinska in June 2013 in partial fulfillment of the requirements for the course EDUC 387 Teaching Mathematics II.

22

Chapter II. Ratio

In this chapter we study the meaning of statements such as

In this hospital, the ratio of nurses to doctors is 2 ∶ 1 (“two to one”).

This statement does not say exactly how many nurses or doctors are in the

hospital. It only describes a multiplicative relationship between two quantities. It says

that for every doctor, there are two nurses; or that for every 2 nurses, there is one

doctor. Equivalently, it says,

In this hospital, there are twice as many nurses than doctors.

or,

In this hospital, the number of doctors is 1

2 of the number of nurses.

There appears to be a similarity, therefore, between the notions of fraction and

ratio, but, as we will see, there are important differences, too. We look into this matter

in the next section.

II.1 Ratios and fractions

In the example above the ratio was given by a pair of numbers. But it can also be

given as a single number, as in these examples:

In Canada, the male to female ratio at birth is 1.056; in the population over 65 years old, the ratio is 0.78.4

The ratio of the circumference of a circle to its diameter is the number 𝜋.

The first of the above statements means that, for every 1000 baby girl births in

Canada, 1056 baby boys are born. Among the population 65 or older, however, for every

1000 women there are only 780 men. This statement can be translated into one about

4 Source: http://en.wikipedia.org/wiki/List_of_countries_by_sex_ratio

23

fractions, although not directly, since a conversion of the decimals into some fractional

representation is necessary. For example, about the male to female ratio at birth, we

could say,

The number of baby girl births is the fraction 125

132 of the number of

baby boy births.

(Verify if this is indeed correct.)

There is no such possibility with the second statement, about a relationship

between the circumference and the diameter of a circle. The statement says that, if the

diameter of a circle is, say, 2 𝑚, then the length of the circumference is 𝜋 × 2 𝑚. This

length is somewhere between 6.28 𝑚 and 6.30 𝑚; more precisely, it is between

6.282 𝑚 and 6.284 𝑚. We can continue increasing the degree of precision further,

using consecutive digits of the decimal expansion of 𝜋, but we cannot translate this

statement into one saying that the diameter is such and such fraction of the

circumference. This is because 𝜋 is not a rational number5.

There is a subtlety in the above explanation that is easily overlooked. When we

say, above, that the circumference is 𝜋 × 2 𝑚, the multiplication sign in this expression

refers to multiplication of a quantity by a real number which is not a rational number. We

know what it means to multiply a quantity by a rational number – 𝑎

𝑏× 𝑄 ≝

𝑎

𝑏 𝑜𝑓 𝑄 – but

we have not defined what it means to multiply a quantity by a real number that is not

rational. The precise definition of this operation exceeds the scope of this course.

Interested readers are referred to an intuitive explanation of the real number system in

5 The proof that the ratio of the circumference of the circle to the diameter of the circle cannot be represented as a ratio of integers, i.e., that 𝜋 is not a rational number, is not elementary. It requires knowledge of Calculus which is not a prerequisite for this course. Interested readers can search for proofs of irrationality of the number 𝜋 on the internet. One source is, for example, a Wikipedia article at http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational .

24

the popular book “What is Mathematics?” by Courant and Robbins (Courant & Robbins,

1953, pp. 58-72).

In the next section, we define formally what it means for one quantity to be in

such and such ratio to another quantity of the same kind.

II.2 Definition of ratio

The ratio of one quantity to another can be expressed as an ordered pair of real

numbers, or as a single real number.

Let 𝐴 and 𝐵 be two quantities of the same kind, and let 𝑎 and 𝑏 be positive

real numbers.

DEFINITION OF RATIO AS A PAIR OF NUMBERS (DoR-I)

We say that the ratio of 𝐴 to 𝐵 is 𝑎 ∶ 𝑏 if 𝑏 × 𝐴 = 𝑎 × 𝐵.

DEFINITION OF RATIO AS A SINGLE NUMBER (DoR-II)

We say that the ratio of 𝐴 to 𝐵 is the number 𝑟, if 𝐴 = 𝑟 × 𝐵.

For example, if we say that, in a hospital department, there are 63 nurses and 14

doctors, then the common kind of quantity we are talking about here is the number of

people.

We can say that the ratio of nurses to doctors in this department is 9 ∶ 2. We

could also say that the ratio of nurses to doctors is 63 ∶ 14, or that it is 18 ∶ 4 (Verify!),

but the representation 9 ∶ 2 is the simplest and gives a better idea of the relationship.

We can also say that the nurses to doctors ratio is the single number 4.5, because

the number of nurses is 4.5 times the number of doctors.

Ratios are extensively used in statistics – we have seen an example above – but

also in drawing plans and maps to scale, and in many other contexts. In everyday life,

ratios are omnipresent in the context of scaling cooking recipes.

25

A deeper understanding of ratios will be developed by solving the exercises below.

Problem set II

Problem II.1 Making a rectangular shape out of a piece of wire

Suppose you have 16 𝑚 of wire and you want to make a rectangular shape out of it. You want

the lengths of the sides of this rectangle to be in the ratio 3 ∶ 2, so that the rectangle looks like

the one in Figure 2.

Figure 2. A rectangle with side lengths in the ratio 3 : 2.

a. What should be the lengths of the sides?

b. Express the ratio of the sides as a single number.

c. Can the statement about the ratio of the sides be translated into a statement about the length

of one side being a fraction of the length of another? If yes, give this fraction.

Problem II.2 What is the ratio?

Suppose that a 253

5 𝑦𝑎𝑟𝑑𝑠-long piece of string has been cut into a piece of 12 𝑦𝑎𝑟𝑑𝑠 and a

piece of 133

5 𝑦𝑎𝑟𝑑𝑠. In what ratio has it been cut? Express the ratio as

(a) a pair of numbers;

(b) a single number.

Compare your answer with those of your peers. If your answers are different, does it mean that

some of these answers are necessarily incorrect?

26

Problem II.3 Dividing a segment in a given ratio

Mark two points, 𝐴 and 𝐵, on a sheet of blank paper. Find a point 𝐶 on the segment 𝐴𝐵 so

that the ratio 𝐴𝐶 ∶ 𝐶𝐵 is

a. 1 : 1 b. 1 : 2 c. 3 : 4 d. 2 : 3

Problem II.4 Dividing a given distance in a given ratio

The distance between city 𝐴 and city 𝐵 is 240 𝑘𝑚. City 𝐶 is between 𝐴 and 𝐵 and it is

situated so that the ratio of its distance to city 𝐴 to its distance to city 𝐵 is 7 ∶ 8. How far is it

from city 𝐶 to city 𝐴?

Problem II.5 Drawing the number line

A line segment of 18 𝑐𝑚 has been drawn horizontally on a sheet of paper. It is supposed to

represent the number line. The left end of the segment was marked “0”. A point marked “11

2”

divides the segment in the ratio 1 ∶ 2.

a. What number marks the right end of the segment?

b. In what ratio does the point marked “21

2 “ divide the segment?

c. Explain how you found the answers and how you made sure they were correct.

d. Compare this problem with Problem VIII.3 in Part I of the course notes.

Problem II.6 The boys to girls ratio in a class

Suppose 𝐵 is the number of boys in a class, and 𝐺 is the number of girls in that class.

a. Suppose 𝐵 = 5 students and 𝐺 = 35 students.

i. What is the ratio 𝐵 ∶ 𝐺? What is the ratio 𝐺 ∶ 𝐵? Express these ratios both as pairs of

numbers and single numbers. Are both ratios rational numbers?

ii. Can we say that 𝐵 is a fraction of 𝐺? If yes, what fraction is it?

iii. Can we say that 𝐺 is a fraction of 𝐵? If yes, what fraction is it?

iv. 𝐵 is what fraction of the number of all students in the class?

b. Same questions with 𝐵 = 32 students and 𝐺 = 10 students.

c. Suppose that in a particular class the ratio of boys to girls is 1 : 1.5. It is also known that the

number of girls is more than 20 but less than 30. How many girls and how many boys could there

be in this class?

27

Problem II.7. Estimating a distance

Look at the map of Canada in Figure 3. Based on this map, estimate the distance, in straight line,

from Montreal to Sept-Iles. Information about the scale in the bottom-right corner of the map

will be very useful in this task. Compare your result with information about the driving distance

between the two cities, which you can find on the internet. Analyze your solution: what

arithmetical operations have you used? Have you used proportional reasoning? If yes, explain

where and how.

Figure 3. A map of Canada.

28

Problem II.8 Scaling a recipe

If 3

4 cups of sugar and

1

2 teaspoons of salt are required in a pumpkin pie recipe to make 4

servings, how many cups of sugar and teaspoons of salt will be needed to make this pie for 10

servings?

NOTE: Amounts of ingredients in recipes are assumed to be proportional to the number of

servings.

Have you used ratios in solving the problem? Could you have used them? Explain how.

Problem II.9 Ratios among recipe ingredients

To make shortbread cookies, one needs butter, sugar, flour and cornstarch. The ratios between

the volumes of ingredients must be exactly as given below (otherwise it is not shortbread that

obtains but some other type of pastry). To simplify the notation, we will use abbreviations:

volume of butter will be denoted by “B”; of sugar – “S”; of flour – “F”, and of cornstarch – “C”.

B : S is 2 : 1

B : F is 2 : 3

F : C is 6 : 1

For 2 dozen shortbread cookies, one needs 1 cup of butter. How much (in cups) of the other

ingredients does one need?

Problem II.10 Design a recipe problem

Construct a problem about using a recipe that would require conversion of units and

using ratios and fractions. Solve the problem.

Problem II.11 Paper formats

Find the ratio of the longer to shorter side of some of the known paper formats: letter, legal, A4,

B4, A3. How would you characterize the differences between these shapes and how is this

difference reflected in the ratios? Draw these paper formats to scale on a sheet of paper.

Problem II.12 Ratios in the context of elections

In a certain election the participation was such that for every 18 registered voters who did vote

there were 7 registered voters who did not.

a. Suppose there were 6 million registered voters. How many voted and how many did not?

29

b. Suppose 532,000 registered voters had not voted. If the ratio of the registered voters who

voted to those who did not was still 18 ∶ 7, how many registered voters would have voted?

Problem II.13 Ratios and fractions of quantities in the context of parliamentary

representation

Four parties (P1, P2, P3 and P4) have their representatives in the parliament of a certain country.

Party P1 occupies 1

4 of the seats.

Party P4 has 1

11 of the number of seats occupied by P1.

Parties P2 and P3 together have 8

11 of all seats.

The ratio of the number of seats occupied by P2 to the number of seats occupied by P3

is 25 ∶ 3.

P3 has 24 seats.

a. How many seats are there in that parliament? Justify your answer.

b. How many seats each party has? Justify your answer.

Problem II.14 Enlarging a puzzle

Given a puzzle as in Figure 4. The puzzle fits into an 8 𝑐𝑚 × 8 𝑐𝑚 square.

Scale the puzzle so that it fits into a square with side

a. 10 𝑐𝑚;

b. 5 𝑐𝑚.

Figure 4. The puzzle fits into a square whose side is 8 cm.

30

Problem II.15 The Golden Rectangle

Suppose we have 1 𝑚 of wire and we want to use this wire to make a rectangular frame. We

would like the lengths 𝑎 and 𝑏 of the sides of the rectangle to satisfy the equation

𝑎 ∶ 𝑏 = (𝑎 + 𝑏) ∶ 𝑎

Find exact or approximate values of the lengths of the sides 𝑎 and 𝑏 of the rectangle, and their

ratio.

NOTE: A rectangle whose sides satisfy the equation is called the “Golden rectangle.”

Problem II.16 A sequence of Fibonacci rectangles

A Fibonacci sequence is an infinite sequence of numbers that starts with the numbers 1, 1 and

every next number is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

Construct a sequence of rectangles, whose width and length are two consecutive terms of the

Fibonacci sequence in centimeters. Thus the first rectangle is a square with side 1 𝑐𝑚; the

second is a 2 𝑐𝑚 × 1 𝑐𝑚 rectangle; the third is a 3 𝑐𝑚 × 2 𝑐𝑚 rectangle, etc. (Figure 5) We

call this sequence “Fibonacci rectangles.”

Figure 5. "Fibonacci rectangles."

Figure 5 shows only four Fibonacci rectangles. Draw 3 more. Notice that the rectangles become

more and more similar in shape. Verify your impression by calculating the ratios of the length to

the width of the Fibonacci rectangles and representing them by decimal numbers. What do you

notice about these numbers?

31

Chapter III. Percents

Fractions and ratios are two ways of representing multiplicative relationships

between quantities. Percents are yet another way. In this chapter, we first analyze the

meaning of statements of the type,

Quantity 𝐴 is 𝑝 % of quantity 𝐵.

and then we identify and solve several types of problems about percents.

III.1 Interpreting statements about percents of quantities

Let us look at the following example. We are told that in the class we will be

teaching next year, 75% of the children will be girls. We cannot say how many girls

exactly there will be, but this information tells us that if there were a 100 children in the

class, then 75 of them would be girls, and that the ratio of the number of girls (𝐺) to the

number of all children (𝐴) would always be 75 ∶ 100, whatever the class size:

𝐺 ∶ 𝐴 = 75 ∶ 100

By DoR-I, this means that 100 × 𝐺 = 75 × 𝐴. Since 75 and 100 are whole

numbers, then, by DoF-I (Part I, Section II.3) we can say that

𝐺 =75

100× 𝐴

So if the number of children in the class is 24, the number of girls is the fraction

75

100 of 24 children, or, equivalently,

3

4 of the 24 children, which is 18 children.

It may be tempting to define “𝐴 is 𝑝% of 𝐵” by “𝐴 is the fraction 𝑝

100 of 𝐵”,

based on this example. But in this example, the percentage was given as a whole number

(75) of “percents”, so we could easily translate the information about the percent of the

class into a statement about a fraction of the class.

32

What if 𝑝 is not a whole number? Then 𝑝

100 is not necessarily a fraction. It could

be any non-negative rational number. In principle, it could be any non-negative real

number, although irrational numbers of percents are seldom, if ever, used in practice.

If 𝑝 = 0.9, for example, as in “0.9% saline solution”, then we should not say “the

volume of NaCl is the fraction 0.9

100 of the volume of the solution”, because

0.9

100 is not,

technically, a fraction, although it is a rational number and can be represented in the

proper fractional form 9

1000.

Note, however, that in our example about 75% of the class being girls, we could

also represent the fraction 75

100 is decimal form, and write

𝐺 = 0.75 × 𝐴

Since 0.75 is also a result of division of 75 by 100, this suggests defining “𝐴 is

𝑝% of 𝐵” by 𝐴 = (𝑝 ÷ 100) × 𝐵. This is probably how you have always calculated a

percent of a quantity: you would divide the number of “percents” by 100 and multiply

the result by the quantity.

DEFINITION OF A PERCENT OF A QUANTITY (DoP)

Let 𝐴 and 𝐵 be quantities of the same kind, and let 𝑝 be any non-negative real number.

We say that 𝐴 is 𝑝 % of 𝐵 if 𝐴 = (𝑝 ÷ 100) × 𝐵.

Exercise III.1 A vinegar solution

Windows are best washed with a vinegar solution such that if we take 1

4 cup vinegar, then we

should take 1

2 teaspoon dishwashing detergent, and 2 cups of water. Assume that 1 cup =

250 𝑚𝑙 and 1 tsp = 5 𝑚𝑙. The volume of vinegar in the solution is what percent of the volume

of the solution as a whole? Approximate your answer to an integer number of percents.

33

III.2 Three basic types of percent problems and their solutions

Let us look back at the statement:

Quantity 𝐴 is 𝑝 % of quantity 𝐵.

There are three variables in this statement: quantity 𝐴, number 𝑝 and quantity

𝐵. In a problem about percents, the value of one of these variables can be unknown and

the values of the other two – given. This gives three basic types of problems about

percents.

Percent Problem Type 1. WHAT QUANTITY 𝑨 is the given 𝒑 % of the given

quantity 𝑩?

The solution of this problem requires only a direct application of the definition

DoP.

Percent Problem Type 2. A given quantity 𝑨 is the given 𝒑 % of WHAT

QUANTITY 𝑩?

Since, by DoP, 𝐴 = (𝑝 ÷ 100) × 𝐵, to calculate 𝐵, we need to “invert” the

multiplication operation: we divide 𝐴 by 𝑝 ÷ 100. This is equivalent to dividing 𝐴 by

𝑝 and multiplying the result by 100. (Explain why)

Percent Problem Type 3. A given quantity 𝑨 is WHAT PERCENT of the given

quantity 𝑩?

Here, too, we need to invert the multiplication operation, but now, it is the

variable 𝑝 that is unknown. Directly from DoP, we get:

𝑝 ÷ 100 = 𝐴 ÷ 𝐵

But we need the number 𝑝 alone. It is enough to invert the division operation:

𝑝 = (𝐴 ÷ 𝐵) × 100

34

Exercise III.2 Composing simple practice problems about percents

Write two problems of each of the three types presented above. Make one of the pair more

difficult than the other. Solve the problems. Explain what accounts for the greater difficulty in

each case.

III.3 Increasing/decreasing a quantity by a certain percent

The basic percent problems are often embedded in the context of increase or

decrease of some quantity by a certain percent: increase/decrease of production,

salaries, prices, taxes, etc. We look at some examples of such contexts.

Example III.3.1 Salary increase

According to information published on one Canadian website6, the average salary

increase in Canadian industry in 2014 was expected to be 2.6 %. In some sectors of

industry the increase was projected to be 4 % (Oil and gas), in some – 2.1 % (Forestry

and paper), but, on the average, across all sectors, it was expected to be at 2.6 %.

What does it mean to have one’s salary increased by 2.6 % ?

This means that the increase, i.e., the difference between the new salary (𝑁𝑆)

and the former salary (𝐹𝑆), is 𝑝 % of the former salary:

(𝑁𝑆 − 𝐹𝑆) is 2.6 % of 𝐹𝑆

or

𝑁𝑆 − 𝐹𝑆 = (2.6 ÷ 100) × 𝐹𝑆 = 0.026 × 𝐹𝑆

If 𝐹𝑆 is known, then the calculation of 2.6 % of 𝐹𝑆 is a Percent Problem of

Type 1. The result is the difference between the former and the new salaries.

If 𝐹𝑆 is known, 𝑁𝑆 can be easily found:

6 “PLANT” (PLANT, 2013) : http://www.plant.ca/sustainability/canadians-average-salary-increase-for-2014-is-2-6-114126/

35

𝑁𝑆 = 𝐹𝑆 + 0.026 × 𝐹𝑆 = 1.026 × 𝐹𝑆

So we could say that

𝑁𝑆 is 102.6 % of 𝐹𝑆.

Exercise III.3 Industry workers’ salary increases

a. In 2013, an oil industry worker’s annual salary was 125 000 $. In 2014, his salary was

increased by 4 %. What was his new annual salary?

b. After a 2.6 % increase, a construction worker’s annual salary became 58,300 $. What was his

annual salary before the increase?

Example III.3.2 Price reduction

In August, a department store in Montreal announced a sale of its barbecue grills

(BBQ): the regular prices were to be reduced by 15 %. But looking for details on the

store’s website, a customer found that a BBQ whose regular price was 499.99 $ will now

cost 429.99 $. The customer found this odd because this was not exactly a 15 %

reduction. According to her calculations, the reduction was much closer to 14 % than

15 %. Was she right?

To answer this question, we need to know what quantity should be the 15 % of

what other quantity.

If 𝑅𝑒𝑔𝑃 is the regular price, and 𝑅𝑒𝑑𝑃 is the reduced price, then

𝑅𝑒𝑔𝑃 − 𝑅𝑒𝑑𝑃 = 70 $ should be 15 % of 𝑅𝑒𝑔𝑃 = 499.99 $

This is a basic type of percent problem, type 3: we need to find what percent 𝑝 of

499.99 $ is 70 $.

𝑝 = (70 $ ÷ 499.99 $) × 100 ≅ 14.00028

So the reduction is about 14 %, not 15 %.

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Basic problems about price reduction can be categorized into three types,

depending on what is unknown: the regular price, the reduced price or the percent of

reduction.

Exercise III.3.1 Composing problems about price reduction

Write 3 problems about price reduction: one where the regular price is unknown; one where the

reduced price is unknown and one where the percent of reduction is unknown. Solve the

problems.

Exercise III.3.2 Assessment of a student’s solution to a price reduction problem

A student was solving the problem:

“The regular price of a large bookcase was 283.50 $. It is now being sold at a reduced price, 250 $. What percent reduction is it?”

She solved the problem two ways, and was surprised she did not get the same answer. She

became confused and asked the teacher for help. Her solution is reproduced in Figure 6. Write

an explanation for the student.

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Figure 6. A student’s solution of a price reduction problem.

The rest of this chapter is a Problem set about situations where one quantity is a

certain percent of another quantity.

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Problem set

Problem III.1

In a European country, a customer bought a laptop whose price was displayed as 283.50 €. At

the cash, the VAT tax (Value Added Tax) was added, and she was charged 339.20 €. What is the

VAT on electronic equipment (in percents) in this country?

Problem III.2 Price of gas

Make up three problems about percents, each of a different type, based on the following

information, posted August 11, 2014, on the website of NorthJersey.com:

Survey: US gas prices down 6 cents a gallon. The average U.S. price of gasoline has dipped 6 cents a gallon in the past two weeks and prices in California have fallen 9 cents in the same period. According to Lundberg Survey released Sunday, the average nationwide for a gallon of regular is now $3.52. The average price for midgrade gasoline is $3.83 and premium is $3.88. (NorthJersey.com)

Problem III.3 Increase of the mass salary of management in Quebec universities

On the 19th of January 2013, the Quebec newspaper Le Devoir published an article by Jessica

Nadeau about the increase of the mass salary of the management in Quebec universities over the

previous 10 years.

… in 1997-98, the mass salary of rectors, vice-rectors and other management staff in Quebec universities was about 129 million. Twelve years later, in 2008-09, the sum of salaries reached 328 million, an increase of 154%. (Jessica Nadeau, Le Devoir, 19 janvier 2013)

a. How has the increase of 154% been calculated?

b. Does it mean that every administrator’s salary has increased by 154%?

c. Suppose an administrator’s annual salary was $100,000 in 1997-98 and has been increased by

154% by 2008-09. What was the amount of the salary in 2008-09?

Problem III.4 Taxes in Quebec

In Canada, customers pay two taxes: “GST” or Goods and Services Tax, which is a federal tax, and

a provincial sales tax. In Quebec, customers pay the GST and the QST or the Quebec Sales Tax.

In French, the taxes are called TPS (Taxes sur Produits et Services) and TVQ (Taxe des Ventes du

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Québec), respectively. Receipts at the store often use the acronym TVP instead of TVQ, for

Taxe des Ventes Provinciale.

In 2012, customers in Quebec were paying 5 % GST and 9.5 % QST. Here is how the

taxes were calculated. If the amount before taxes was 𝐴 $, then the amount to pay was

calculated as follows:

First, the GST was added to 𝐴 $: 𝐴 $ + 5 % of 𝐴 $ (call it 𝐵 $)

Then, the QST was added to 𝐵 $: 𝐵 $ + 9.5 % of 𝐵 $

This was the amount to pay.

In 2013, the QST was increased, but the tax started to be calculated differently. The

customers in Quebec were paying 5 % GST and 9.975 % QST, and here is how the taxes were

calculated. If the amount before taxes was 𝐴 $, then the amount to pay was calculated as

follows:

First, the GST of 𝐴 $ was calculated: 5% 𝑜𝑓 𝐴 $ (call it 𝐶 $)

Then, the QST of 𝐴 $ was calculated: 9.975 % of 𝐴 $ (call it 𝐷 $)

The amount to pay was 𝐶 $ + 𝐷 $.

Answer the following questions:

a. Were Quebeckers paying higher sales taxes in 2013 than they were in 2012? Explain.

b. Look at the 2013 receipt from the Université de Montréal bookstore reproduced in Figure 7.

Two books were bought:

- “20 Formules Pédagogiques” (printed, with an ISBN): selling price $22.10

- “PLU 6035 – H13 - Pratique de l’enseignement supérieur. Recueil des textes” (photocopied, no

ISBN): selling price $ 30.65

How were the TPS and TVP calculated?

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Figure 7. A receipt for two books bought at the Université de Montréal bookstore in 2013.

Problem III.5 Advertisement of the type: “33% more”

Scotch tape is sometimes sold in packages of three. The package advertises that you get “33 %

more”. (Figure 8) On the package, “7.62 𝑚” is crossed out and replaced by “10.1 𝑚”; also

“22.86 𝑚” is crossed out and replaced by “30.3 𝑚. ”

a. To what does the “33 % more” refer? Justify your answer.

b. Do you think that this is an honest advertisement? Justify.

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Figure 8. Scotch tape package advertised as "33 % more".

Problem III.6 Statistics

Numbers of students enrolled in courses offered for credit at Concordia in 2012/13 are given in

Table 2.7 The data are also represented in the form of a bar diagram in Figure 9.

Compose at least 3 problems about one quantity being a percent of another quantity requiring

the use of these data.

Department or faculty

Undergraduate Master’s PhD Total

Arts and Science 15 958 1 569 503 18 030

Education 824 308 40 1 172

Maths &Stats 581 70 26 677

Table 2. Numbers of students enrolled in courses offered for credit in a university

7 Source: http://www.concordia.ca/about/fast-facts.html

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Figure 9. Some data from Table 2 represented in the form of a bar diagram.

Problem III.7 Investments

Jack invested some of his savings in a GIC (Guaranteed Investment Certificate) for 3.5 years, at the

interest rate of 1.5 %. This means that if he does not withdraw his money during this time, after

the 3.5 years, his savings will be increased by 1.5 % of the sum he deposited at the beginning.

Suppose he deposited 40,000 $ at the beginning. How much money will he have in his GIC at

maturity?

Problem III.8 Reduction/enlargement in photocopying - 1

A rectangular 6” x 9” picture is placed on a letter-sized sheet of paper so that its sides are parallel

to the edges of the sheet of paper. The general rule is that when this sheet of paper is

photocopied with a reduction (or enlargement) set at 𝑝 % then the lengths of the sides of the

picture are 𝑝 % of the original lengths. Suppose you photocopied the sheet with the picture with

a reduction set at 75 % and then you lost the original.

You want to recover the original picture by enlarging the 75 % photocopy. You decided to set the

enlargement at 133 %. Will you recover the original exactly? If yes, justify it. If not, what

fraction of the original length of the picture (9”) will be the difference between of length of the

original picture and the length of the enlarged copy?

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Problem III.9 Reduction/enlargement in photocopying – 2

A rectangular picture is placed on a letter-sized sheet of paper so that its sides are parallel to the

edges of the sheet of paper. When this sheet of paper is photocopied with a reduction set at

“80 %”, then the lengths of the sides of the picture are 80 % of the original lengths. Suppose you

photocopied the sheet with the picture with a reduction set at 80 % and then you lost the

original.

a. You want to recover the original size by enlarging the 80 % photocopy. What percent

enlargement will you use? What can you do to verify if your answer is correct?

b. You not only lost the original but realized that you were not supposed to reduce the original by

20 % but to enlarge it by 20 %. What enlargement will you use in copying the 80 % copy to

obtain the expected enlargement of the original?8 How do you know your answer is correct?

c. If the original width and length of the picture were 9.9 𝑐𝑚 and 12.8 𝑐𝑚, respectively, what

are the width and length of the picture on a

(i) 80 % copy ?

(ii) 120 % copy ?

d. Is the area of the picture on an 80 % copy also reduced by 20 % relative to the original? If

not, by what percent is it reduced?

e. Generalize: The area of the picture on a 𝑝 % copy is reduced BY WHAT PERCENT relative to the

original? Justify your answer.

Problem III.10 Beliefs about price reductions

One day, a department store announces a sale: 25 % off the prices of all electric appliances. But

the buyers are not exactly rushing to buy them. So a week later the store advertises that they are

taking 20 % off the already reduced prices. Now, the store is crowded with buyers. A quick poll

reveals that almost 50% of the buyers believe that they are getting a reduction of 45 % off the

original price. Are these people right? Justify your answer.

8 This problem has been inspired by an example in (Lamon, 2012, p. 197).

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References

Coltharp, F. (1966). Introducing integers as ordered pairs. School Science and

Mathematics, 66(3), 277-282.

Courant, R., & Robbins, H. (1953). What is mathematics? London: Oxford University Press.

Lamon, S. (2012). Teaching fractions and ratios for understanding. New York: Routledge.

PLANT. (2013, August 20). Canadians' average salary increase for 2014 is 2.6%. Retrieved

from PLANT: http://www.plant.ca/sustainability/canadians-average-salary-

increase-for-2014-is-2-6-114126/

The National Council of Teachers of Mathematics. (2010). Developing essential

understanding of rational numbers. Grades 3-5. Reston, Virginia: National Council

of Teachers of Mathematics.