Equivalent Ratios and Simplest Form

Why multiplying or dividing every term of a ratio by the same number gives an equivalent ratio, how to write a ratio in its simplest form, and how to find a missing term.

Before you start

A ratio compares two quantities of the same kind. If a bracelet has 2 glass beads and 3 wooden beads, the ratio of the number of glass beads to the number of wooden beads is $2 : 3$ , read "2 to 3". The numbers 2 and 3 are the terms of the ratio. Their order follows the order of the words, and a ratio has no unit of measurement.

The highest common factor (HCF) of two numbers is the largest whole number that is a factor of both. By prime factorisation, $12 = 2^{2} \times 3$ and $18 = 2 \times 3^{2}$ , so the HCF of 12 and 18 is $2 \times 3 = 6$ .

A fraction keeps its value when its numerator and its denominator are multiplied or divided by the same non-zero number: $\frac{2}{3} = \frac{4}{6} = \frac{8}{12}$ . These are equivalent fractions.

Equivalent Ratios

Siti makes bracelets. Every bracelet has 2 glass beads and 3 wooden beads. The table shows the beads she needs for 1, 2, 3, 4 and 5 bracelets.

Number of bracelets	Glass beads	Wooden beads	Ratio of glass beads to wooden beads
1	2	3	$2 : 3$
2	4	6	$4 : 6$
3	6	9	$6 : 9$
4	8	12	$8 : 12$
5	10	15	$10 : 15$

The numbers of beads change from row to row, but the comparison does not: in every row there are 2 glass beads for every 3 wooden beads. Ratios that make the same comparison are called equivalent ratios, and we join them with an equals sign: $2 : 3 = 4 : 6 = 6 : 9 = 8 : 12 = 10 : 15$ . Each of them is $2 : 3$ with both terms multiplied by the number of bracelets.

Why the ratio does not change: the bar model

Take the beads for 4 bracelets: 8 glass beads and 12 wooden beads. In a bar model each quantity is a bar made of equal units. Here are the same beads drawn three times, each time with units of a different size.

With 1 bead in a unit, the bars have 8 units and 12 units. The ratio is $8 : 12$ .

With 2 beads in a unit, the bars have 4 units and 6 units. The ratio is $4 : 6$ .

With 4 beads in a unit, the bars have 2 units and 3 units. The ratio is $2 : 3$ .

The bars are the same length in all three drawings, because they stand for the same beads. Only the size of a unit has changed. When each unit is made 2 times as large, both bars have half as many units, so both terms are divided by 2. When each unit is made 4 times as large, both terms are divided by 4. Going the other way, cutting every unit into 2 or 4 equal smaller units multiplies both terms by 2 or by 4. The comparison between the two bars is the same every time, so $8 : 12 = 4 : 6 = 2 : 3$ .

Why the ratio does not change: the fraction

The ratio $8 : 12$ also says that the number of glass beads is $\frac{8}{12}$ of the number of wooden beads. In the same way, $4 : 6$ says that it is $\frac{4}{6}$ of the number of wooden beads, and $2 : 3$ says that it is $\frac{2}{3}$ of the number of wooden beads. These are equivalent fractions, $\frac{8}{12} = \frac{4}{6} = \frac{2}{3}$ , so the three ratios say one and the same thing. Multiplying or dividing both terms of a ratio by the same number is the same step as multiplying or dividing the numerator and the denominator of a fraction by the same number.

Rule: equivalent ratios

Multiplying or dividing every term of a ratio by the same non-zero number gives an equivalent ratio.

Multiplying: $2 : 3 = 2 \times 4 : 3 \times 4 = 8 : 12$

Dividing: $12 : 18 = 12 \div 6 : 18 \div 6 = 2 : 3$

The number cannot be 0. Multiplying by 0 would give $0 : 0$ , which compares nothing, and no number can be divided by 0.

Only multiplying and dividing do this. If Siti puts 1 more glass bead and 1 more wooden bead on a bracelet, it has 3 glass beads and 4 wooden beads, and $3 : 4$ is not equivalent to $2 : 3$ : the number of glass beads is now $\frac{3}{4}$ of the number of wooden beads, and $\frac{3}{4}$ is not equal to $\frac{2}{3}$ .

Example 1: Writing equivalent ratios

(a) Write three ratios equivalent to $3 : 7$ by multiplying both terms by 2, by 5 and by 10.
(b) Write three ratios equivalent to $36 : 60$ by dividing both terms by 2, by 4 and by 12.

Solution

(a) Multiply both terms by the same number each time. Any whole number greater than 1 gives an equivalent ratio with whole-number terms; this question asks for 2, 5 and 10.

$3 \times 2 : 7 \times 2 = 6 : 14$

$3 \times 5 : 7 \times 5 = 15 : 35$

$3 \times 10 : 7 \times 10 = 30 : 70$

(b) Divide both terms by the same common factor of 36 and 60 each time. Their common factors greater than 1 are 2, 3, 4, 6 and 12, so any of these would do; this question asks for 2, 4 and 12.

$36 \div 2 : 60 \div 2 = 18 : 30$

$36 \div 4 : 60 \div 4 = 9 : 15$

$36 \div 12 : 60 \div 12 = 3 : 5$

(a) $6 : 14$ , $15 : 35$ and $30 : 70$ are equivalent to $3 : 7$ .
(b) $18 : 30$ , $9 : 15$ and $3 : 5$ are equivalent to $36 : 60$ .

The Simplest Form of a Ratio

Of all the ratios equivalent to $8 : 12$ , the one with the smallest whole-number terms is $2 : 3$ . It cannot be divided any further, because 2 and 3 have no common factor other than 1. It is the easiest of them to read and to compare with other ratios, so an answer that is a ratio is given in this form unless the question says otherwise.

Definition: simplest form

A ratio is in its simplest form when its terms are whole numbers that have no common factor other than 1. This is the same as saying that the HCF of its terms is 1.

$2 : 3$ , $5 : 3$ and $1 : 4$ are in their simplest form. $8 : 12$ is not, because 4 is a common factor of 8 and 12.

A ratio has only one simplest form. In this lesson every term is a whole number and the quantities compared are already in the same unit of measurement.

How to write a ratio in its simplest form

In one step: find the HCF of the terms and divide every term by it.
In several steps: divide every term by any common factor greater than 1, do the same to the new ratio, and go on until the terms have no common factor other than 1.

Both ways end at the same ratio. Dividing by the HCF is shorter when you can see the HCF or find it by prime factorisation. Dividing in steps needs no HCF, but you must not stop too soon. Either way, the terms stay in the same order.

Example 2: Two ways to the simplest form

Write $20 : 12$ in its simplest form.

Solution

The two ways are set out side by side.

In one step, by the HCF	In several steps, by common factors
Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 12: 1, 2, 3, 4, 6, 12 The HCF of 20 and 12 is 4, so divide both terms by 4: $20 : 12 = 20 \div 4 : 12 \div 4 = 5 : 3$	20 and 12 are both even, so divide both terms by 2: $20 : 12 = 20 \div 2 : 12 \div 2 = 10 : 6$ 10 and 6 are both even, so divide both terms by 2 again: $10 : 6 = 10 \div 2 : 6 \div 2 = 5 : 3$ 5 and 3 have no common factor other than 1, so stop.

In the right-hand column, the two divisions by 2 together make one division by $2 \times 2 = 4$ , which is the HCF.

On a bar model, dividing both terms by 4 joins the units in fours: bars of 20 units and 12 units become bars of 5 units and 3 units, and the bars are the same length as before.

Check: 5 and 3 are different prime numbers, so their only common factor is 1.

In its simplest form, $20 : 12 = 5 : 3$ .

Example 3: A ratio from a situation

A school library buys 52 English books and 76 Chinese books. Find, in its simplest form, the ratio of
(a) the number of English books to the number of Chinese books,
(b) the number of Chinese books to the number of English books.

Solution

(a) English books : Chinese books $= 52 : 76$

By prime factorisation, $52 = 2^{2} \times 13$ and $76 = 2^{2} \times 19$ . The only prime factor in both is 2, and its lowest power is $2^{2}$ , so the HCF is $2^{2} = 4$ .

$52 : 76 = 52 \div 4 : 76 \div 4 = 13 : 19$

In steps, the answer is the same: both terms are even, so $52 : 76 = 26 : 38$ ; both are even again, so $26 : 38 = 13 : 19$ .

13 and 19 are different prime numbers, so their only common factor is 1. The ratio is in its simplest form.

(b) The order of the words is turned round, so the terms are turned round: Chinese books : English books $= 76 : 52 = 76 \div 4 : 52 \div 4 = 19 : 13$

$13 : 19$ and $19 : 13$ are different ratios. Writing a ratio in its simplest form never changes the order of its terms.

(a) The ratio of the number of English books to the number of Chinese books is $13 : 19$ .
(b) The ratio of the number of Chinese books to the number of English books is $19 : 13$ .

Example 4: Larger terms

Write $90 : 126$ in its simplest form.

Solution

In one step. $90 = 2 \times 3^{2} \times 5$ and $126 = 2 \times 3^{2} \times 7$ , so the HCF is $2 \times 3^{2} = 18$ .

$90 : 126 = 90 \div 18 : 126 \div 18 = 5 : 7$

In several steps. Both terms are even, so divide by 2: $90 : 126 = 45 : 63$

The digits of 45 add up to $4 + 5 = 9$ and the digits of 63 add up to $6 + 3 = 9$ . Both sums are divisible by 3, so 45 and 63 are divisible by 3: $45 : 63 = 15 : 21$

15 and 21 are divisible by 3 again: $15 : 21 = 5 : 7$

The numbers divided by (2, 3 and 3) multiply to $2 \times 3 \times 3 = 18$ . Dividing in steps is dividing by the HCF a part at a time.

Check: 5 and 7 are different prime numbers, so their only common factor is 1.

In its simplest form, $90 : 126 = 5 : 7$ .

Making Sure a Ratio Is in Its Simplest Form

A ratio is in its simplest form only when the HCF of its terms is 1. The look of the terms can mislead, so test them.

How to test

Write each term as a product of prime factors. If no prime number appears in both, the only common factor is 1 and the ratio is in its simplest form. If a prime number appears in both, it is a common factor and the ratio is not in its simplest form.

Quick signs that a ratio is not in its simplest form: both terms are even (2 is a common factor); both terms end in 0 or 5 (5 is a common factor); the sum of the digits of each term is divisible by 3 (3 is a common factor); the smaller term is greater than 1 and divides the larger term exactly (the smaller term is a common factor).

These do not show that a ratio is in its simplest form: both terms are odd; one of the terms is a prime number.

The terms of a ratio in its simplest form do not have to be prime numbers. They only have to share no factor other than 1.

Example 5: Is it in its simplest form?

Which of these ratios are in their simplest form? Write the others in their simplest form.
(a) $9 : 16$
(b) $15 : 21$
(c) $17 : 51$
(d) $35 : 48$

Solution

(a) $9 = 3^{2}$ and $16 = 2^{4}$ . No prime number appears in both, so the only common factor is 1. Neither term is a prime number, yet the ratio is in its simplest form.

(b) Both terms are odd, but $15 = 3 \times 5$ and $21 = 3 \times 7$ have the common factor 3: $15 : 21 = 15 \div 3 : 21 \div 3 = 5 : 7$

(c) 17 is a prime number, but $51 = 3 \times 17$ is a multiple of 17, so 17 is a common factor: $17 : 51 = 17 \div 17 : 51 \div 17 = 1 : 3$ . The first term is now 1, and it is written: the ratio is $1 : 3$ , not "3".

(d) $35 = 5 \times 7$ and $48 = 2^{4} \times 3$ . No prime number appears in both, so the only common factor is 1.

(a) $9 : 16$ is in its simplest form.
(b) $15 : 21 = 5 : 7$
(c) $17 : 51 = 1 : 3$
(d) $35 : 48$ is in its simplest form.

Deciding Whether Two Ratios Are Equivalent

Equivalent ratios can look quite different. In the table of bracelets, $6 : 9$ (3 bracelets) and $10 : 15$ (5 bracelets) are equivalent, yet they have no term in common, and no whole number multiplies 6 and 9 to give 10 and 15. Their simplest forms show that they are equivalent: $6 : 9 = 2 : 3$ on dividing both terms by 3, and $10 : 15 = 2 : 3$ on dividing both terms by 5.

Test for equivalent ratios

Write each ratio in its simplest form. Two ratios are equivalent if their simplest forms are the same, with the terms in the same order. If their simplest forms are different, the ratios are not equivalent.

A shortcut, when you can see it: if both terms of one ratio, multiplied by the same number, give the terms of the other in the same order, the ratios are equivalent by the rule for equivalent ratios.

The fractions agree with the test: $\frac{6}{9} = \frac{2}{3}$ and $\frac{10}{15} = \frac{2}{3}$ .

Example 6: Equivalent or not?

Are the two ratios equivalent?
(a) $3 : 4$ and $9 : 12$
(b) $20 : 8$ and $45 : 18$
(c) $12 : 16$ and $15 : 21$
(d) $8 : 12$ and $12 : 8$

Solution

(a) $3 \times 3 = 9$ and $4 \times 3 = 12$ : both terms of $3 : 4$ are multiplied by 3. The test says the same: $3 : 4$ is in its simplest form, and $9 : 12 = 3 : 4$ on dividing both terms by 3.

(b) $20 : 8 = 5 : 2$ on dividing both terms by 4. $45 : 18 = 5 : 2$ on dividing both terms by 9. The simplest forms are the same.

(c) $12 : 16 = 3 : 4$ on dividing both terms by 4. $15 : 21 = 5 : 7$ on dividing both terms by 3. The simplest forms are different.

(d) $8 : 12 = 2 : 3$ and $12 : 8 = 3 : 2$ . The numbers are the same but their order is not, so the simplest forms are different.

(a) $3 : 4$ and $9 : 12$ are equivalent.
(b) $20 : 8$ and $45 : 18$ are equivalent.
(c) $12 : 16$ and $15 : 21$ are not equivalent.
(d) $8 : 12$ and $12 : 8$ are not equivalent.

Finding a Missing Term

When two ratios are known to be equivalent and one term is missing, the rule for equivalent ratios finds it. Suppose $5 : 8$ and $20 : x$ are equivalent, where the letter $x$ stands for a term we do not know yet. We write $5 : 8 = 20 : x$ . Both first terms are known, and $20 \div 5 = 4$ , so the first term has been multiplied by 4. Every term must be multiplied by the same number, so the second term is multiplied by 4 too: $x = 8 \times 4 = 32$ .

Ratio	First term	Second term
$5 : 8$	5	8
Both terms	$\times 4$	$\times 4$
$20 : x$	20	$x = 32$

How to find a missing term

Find the two known terms in the same position: both first terms, or both second terms.
Divide the larger of them by the smaller. The answer is the multiplier between the two ratios.
Go from the complete ratio to the other one: multiply its other term by the multiplier if the terms are getting larger, and divide by it if they are getting smaller.

If the multiplier is not a whole number, write the complete ratio in its simplest form first and start again from there. Check by writing your finished ratio in its simplest form.

Example 7: Missing terms

Find the missing term $x$ in each pair of equivalent ratios.
(a) $4 : 9 = 28 : x$
(b) $x : 66 = 15 : 22$
(c) $56 : 40 = 7 : x$
(d) $9 : 12 = 15 : x$

Solution

(a) The first terms are both known. $28 \div 4 = 7$ , so going from $4 : 9$ to $28 : x$ every term is multiplied by 7: $x = 9 \times 7 = 63$ . Check: $28 : 63 = 4 : 9$ on dividing both terms by 7.

(b) The second terms are both known. $66 \div 22 = 3$ , so going from $15 : 22$ to $x : 66$ every term is multiplied by 3: $x = 15 \times 3 = 45$ . Check: $45 : 66 = 15 : 22$ on dividing both terms by 3.

(c) The first terms are both known. $56 \div 7 = 8$ , so going from $56 : 40$ to $7 : x$ every term is divided by 8: $x = 40 \div 8 = 5$ . Check: $7 \times 8 : 5 \times 8 = 56 : 40$ .

(d) The first terms are 9 and 15, and 15 is not a whole number of times 9. Write the complete ratio in its simplest form first: $9 : 12 = 3 : 4$ on dividing both terms by 3. Now $3 : 4 = 15 : x$ , and $15 \div 3 = 5$ , so $x = 4 \times 5 = 20$ . Check: $15 : 20 = 3 : 4$ on dividing both terms by 5.

(a) $x = 63$
(b) $x = 45$
(c) $x = 5$
(d) $x = 20$

Example 8: A recipe

Mrs Goh bakes butter cookies. In her recipe, the ratio of the mass of sugar to the mass of flour is $2 : 5$ . She uses 80 g of sugar. How many grams of flour does she need?

Solution

Sugar : flour $= 2 : 5$ . With 80 g of sugar the ratio is $80 : x$ , where $x$ stands for the number of grams of flour. The recipe is the same, so the two ratios are equivalent: $2 : 5 = 80 : x$

The first terms are both known. $80 \div 2 = 40$ , so every term of $2 : 5$ is multiplied by 40.

$x = 5 \times 40 = 200$

On the bar model, the multiplier is the size of 1 unit: the 2 units of sugar are 80 g, so 1 unit is 40 g.

Check: $80 : 200 = 80 \div 40 : 200 \div 40 = 2 : 5$

Mrs Goh needs 200 g of flour.

Common Mistakes

Dividing only one term, or dividing the terms by different numbers. $12 : 18$ is not $6 : 18$ , and it is not $12 \div 4 : 18 \div 6 = 3 : 3$ . Every term is divided by the same number: $12 : 18 = 12 \div 6 : 18 \div 6 = 2 : 3$ .
Adding or subtracting the same number. $2 : 3$ is not equivalent to $4 : 5$ , although 2 has been added to both terms: $\frac{2}{3}$ is not equal to $\frac{4}{5}$ . Only multiplying and dividing give an equivalent ratio.
Stopping too soon. $24 : 36 = 12 : 18$ is true, but 12 and 18 still have the common factor 6. The simplest form is $2 : 3$ .
Turning the terms round. $20 : 12 = 5 : 3$ , not $3 : 5$ . The smaller term does not have to come first.
Dropping a term of 1. $12 : 4 = 3 : 1$ . A ratio always has both its terms, so the answer is $3 : 1$ and not "3".

Practice

Work each question on paper before you open its solution.

Question 1

Copy and complete the table so that every row is a ratio equivalent to $3 : 5$ .

First term	Second term
3	5
6	?
?	20
15	?
?	50

ShowHide solution

Row 2: $6 = 3 \times 2$ , so the second term is $5 \times 2 = 10$ .

Row 3: $20 = 5 \times 4$ , so the first term is $3 \times 4 = 12$ .

Row 4: $15 = 3 \times 5$ , so the second term is $5 \times 5 = 25$ .

Row 5: $50 = 5 \times 10$ , so the first term is $3 \times 10 = 30$ .

The rows are $3 : 5$ , $6 : 10$ , $12 : 20$ , $15 : 25$ and $30 : 50$ .

Question 2

The bar model shows the numbers of boys and girls in a school choir. Every unit stands for the same number of students.

(a) Write the ratio of the number of boys to the number of girls from the units as they are drawn.
(b) Write this ratio in its simplest form.
(c) How can the units be joined so that the bar model shows the simplest form?

ShowHide solution

(a) The bar for the boys has 6 units and the bar for the girls has 10 units, so boys : girls $= 6 : 10$ .

(b) The HCF of 6 and 10 is 2. $6 : 10 = 6 \div 2 : 10 \div 2 = 3 : 5$ . 3 and 5 are different prime numbers, so this is the simplest form.

(c) Join the units in twos. Each new unit is 2 of the old units, so the boys have 3 units and the girls have 5 units. The bars are the same length as before.

(a) $6 : 10$
(b) $3 : 5$
(c) Join the units in twos, to give 3 units for the boys and 5 units for the girls.

Question 3

Write each ratio in its simplest form.
(a) $15 : 25$
(b) $18 : 24$
(c) $28 : 7$
(d) $54 : 36$

ShowHide solution

(a) The HCF of 15 and 25 is 5. $15 : 25 = 15 \div 5 : 25 \div 5 = 3 : 5$

(b) The HCF of 18 and 24 is 6. $18 : 24 = 18 \div 6 : 24 \div 6 = 3 : 4$

(d) The HCF of 54 and 36 is 18. $54 : 36 = 54 \div 18 : 36 \div 18 = 3 : 2$ . In steps: dividing both terms by 2 gives $27 : 18$ , and dividing both by 9 gives $3 : 2$ .

(a) $15 : 25 = 3 : 5$
(b) $18 : 24 = 3 : 4$
(c) $28 : 7 = 4 : 1$
(d) $54 : 36 = 3 : 2$

Question 4

Write each ratio in its simplest form in two ways: in several steps, dividing by one common factor at a time, and in one step, dividing by the HCF found by prime factorisation.
(a) $84 : 108$
(b) $105 : 168$
(c) $198 : 132$

ShowHide solution

(a) In steps: dividing both terms by 2 gives $42 : 54$ ; dividing by 2 again gives $21 : 27$ ; dividing by 3 gives $7 : 9$ .

In one step: $84 = 2^{2} \times 3 \times 7$ and $108 = 2^{2} \times 3^{3}$ , so the HCF is $2^{2} \times 3 = 12$ . $84 \div 12 : 108 \div 12 = 7 : 9$

(b) In steps: the digits of 105 add up to $1 + 0 + 5 = 6$ and the digits of 168 add up to $1 + 6 + 8 = 15$ . Both sums are divisible by 3, so both terms are divisible by 3: $105 : 168 = 35 : 56$ . Both 35 and 56 are multiples of 7: $35 : 56 = 5 : 8$ .

In one step: $105 = 3 \times 5 \times 7$ and $168 = 2^{3} \times 3 \times 7$ , so the HCF is $3 \times 7 = 21$ . $105 \div 21 : 168 \div 21 = 5 : 8$

In one step: $198 = 2 \times 3^{2} \times 11$ and $132 = 2^{2} \times 3 \times 11$ , so the HCF is $2 \times 3 \times 11 = 66$ . $198 \div 66 : 132 \div 66 = 3 : 2$

(a) $84 : 108 = 7 : 9$
(b) $105 : 168 = 5 : 8$
(c) $198 : 132 = 3 : 2$

Question 5

Which of these ratios are in their simplest form? Write the others in their simplest form.
(a) $8 : 15$
(b) $33 : 55$
(c) $13 : 91$
(d) $57 : 76$

ShowHide solution

(a) $8 = 2^{3}$ and $15 = 3 \times 5$ . No prime number appears in both, so the only common factor is 1.

(b) Both terms are odd, but $33 = 3 \times 11$ and $55 = 5 \times 11$ have the common factor 11: $33 : 55 = 33 \div 11 : 55 \div 11 = 3 : 5$

(d) $57 = 3 \times 19$ and $76 = 2^{2} \times 19$ have the common factor 19: $57 : 76 = 57 \div 19 : 76 \div 19 = 3 : 4$

(a) $8 : 15$ is in its simplest form.
(b) $33 : 55 = 3 : 5$
(c) $13 : 91 = 1 : 7$
(d) $57 : 76 = 3 : 4$

Question 6

Are the two ratios equivalent? Show how you know.
(a) $4 : 5$ and $28 : 35$
(b) $8 : 12$ and $14 : 21$
(c) $9 : 12$ and $15 : 18$
(d) $5 : 8$ and $8 : 5$

ShowHide solution

(a) $4 \times 7 = 28$ and $5 \times 7 = 35$ : both terms of $4 : 5$ are multiplied by 7.

(b) $8 : 12 = 2 : 3$ on dividing both terms by 4, and $14 : 21 = 2 : 3$ on dividing both terms by 7. The simplest forms are the same.

(c) $9 : 12 = 3 : 4$ on dividing both terms by 3, and $15 : 18 = 5 : 6$ on dividing both terms by 3. The simplest forms are different.

(d) Both ratios are in their simplest form, and their terms are in a different order. $5 : 8$ says that the first quantity is $\frac{5}{8}$ of the second; $8 : 5$ says that it is $\frac{8}{5}$ of the second.

(a) $4 : 5$ and $28 : 35$ are equivalent.
(b) $8 : 12$ and $14 : 21$ are equivalent.
(c) $9 : 12$ and $15 : 18$ are not equivalent.
(d) $5 : 8$ and $8 : 5$ are not equivalent.

Question 7

Find the missing term $x$ in each pair of equivalent ratios.
(a) $5 : 6 = x : 48$
(b) $7 : x = 63 : 36$
(c) $x : 13 = 48 : 52$
(d) $15 : 25 = 21 : x$

ShowHide solution

(a) The second terms are both known. $48 \div 6 = 8$ , so every term of $5 : 6$ is multiplied by 8: $x = 5 \times 8 = 40$

(b) The first terms are both known. $63 \div 7 = 9$ , so every term of $63 : 36$ is divided by 9: $x = 36 \div 9 = 4$

(d) The first terms are 15 and 21, and 21 is not a whole number of times 15. In its simplest form, $15 : 25 = 3 : 5$ on dividing both terms by 5. Now $3 : 5 = 21 : x$ , and $21 \div 3 = 7$ , so $x = 5 \times 7 = 35$ . Check: $21 : 35 = 3 : 5$ on dividing both terms by 7.

(a) $x = 40$
(b) $x = 4$
(c) $x = 12$
(d) $x = 35$

Question 8

In Secondary 1 at a school there are 126 boys and 98 girls. Find, in its simplest form, the ratio of
(a) the number of boys to the number of girls,
(b) the number of girls to the number of boys,
(c) the number of girls to the total number of Secondary 1 students.

ShowHide solution

(a) Boys : girls $= 126 : 98$ . By prime factorisation, $126 = 2 \times 3^{2} \times 7$ and $98 = 2 \times 7^{2}$ , so the HCF is $2 \times 7 = 14$ . $126 \div 14 : 98 \div 14 = 9 : 7$

(b) The order is turned round: girls : boys $= 98 : 126 = 7 : 9$

(c) Total number of students $= 126 + 98 = 224$ , so girls : total $= 98 : 224$ . $224 = 2^{5} \times 7$ , so the HCF of 98 and 224 is $2 \times 7 = 14$ . $98 \div 14 : 224 \div 14 = 7 : 16$

Check: $9 = 3^{2}$ and 7 share no prime factor, and 7 and $16 = 2^{4}$ share none, so each ratio is in its simplest form.

(a) The ratio of the number of boys to the number of girls is $9 : 7$ .
(b) The ratio of the number of girls to the number of boys is $7 : 9$ .
(c) The ratio of the number of girls to the total number of students is $7 : 16$ .

Question 9

Three friends make orange drinks from the same syrup. Ali mixes 120 ml of syrup with 300 ml of water. Siti mixes 200 ml of syrup with 500 ml of water. Hui Min mixes 150 ml of syrup with 400 ml of water. Two drinks taste the same when they have syrup and water in the same ratio. Which drinks taste the same?

ShowHide solution

All the amounts are in millilitres, so write each ratio of syrup to water and put it in its simplest form.

Ali: $120 : 300 = 120 \div 60 : 300 \div 60 = 2 : 5$

Siti: $200 : 500 = 200 \div 100 : 500 \div 100 = 2 : 5$

Hui Min: $150 : 400 = 150 \div 50 : 400 \div 50 = 3 : 8$

Ali's ratio and Siti's ratio have the same simplest form, so they are equivalent. Hui Min's ratio has a different simplest form.

Ali's drink and Siti's drink taste the same. Hui Min's drink does not taste the same as theirs.

Question 10

For a school camp, the ratio of the number of teachers to the number of students must be $2 : 15$ . There are 135 students going to the camp. How many teachers must go?

ShowHide solution

Teachers : students $= 2 : 15$ . With 135 students the ratio is $x : 135$ , where $x$ stands for the number of teachers. The two ratios must be equivalent: $2 : 15 = x : 135$

The second terms are both known. $135 \div 15 = 9$ , so every term of $2 : 15$ is multiplied by 9.

$x = 2 \times 9 = 18$

Check: $18 : 135 = 18 \div 9 : 135 \div 9 = 2 : 15$

18 teachers must go to the camp.

Question 11

To make a shade of green, Mr Tan mixes 8 litres of blue paint with 12 litres of yellow paint. For a second batch of the same shade he has 15 litres of yellow paint. How many litres of blue paint does he need?

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Blue : yellow $= 8 : 12$ . For the second batch the ratio is $x : 15$ , where $x$ stands for the number of litres of blue paint. The shade is the same, so the ratios are equivalent: $8 : 12 = x : 15$

The second terms are 12 and 15, and 15 is not a whole number of times 12. Write the complete ratio in its simplest form first: $8 : 12 = 8 \div 4 : 12 \div 4 = 2 : 3$

Now $2 : 3 = x : 15$ . $15 \div 3 = 5$ , so every term of $2 : 3$ is multiplied by 5: $x = 2 \times 5 = 10$

Check: $10 : 15 = 2 : 3$ on dividing both terms by 5, and $8 : 12 = 2 : 3$ as well.

Mr Tan needs 10 litres of blue paint.

Question 12

Three students were asked to write $48 : 72$ in its simplest form. Ali wrote $24 : 36$ , Siti wrote $4 : 6$ and Ravi wrote $3 : 2$ .
(a) What is wrong with each answer?
(b) Write $48 : 72$ in its simplest form.
(c) Mei Ling says that $5 : 9 = 7 : 11$ "because 2 has been added to both terms". Is she right?

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(a) Ali divided both terms by 2. $24 : 36$ is equivalent to $48 : 72$ , but 24 and 36 still have common factors greater than 1 (their HCF is 12), so it is not the simplest form.

Siti divided both terms by 12. $4 : 6$ is equivalent to $48 : 72$ , but 4 and 6 still have the common factor 2.

Ravi turned the terms round. $3 : 2$ is the simplest form of $72 : 48$ , not of $48 : 72$ .

(b) $48 = 2^{4} \times 3$ and $72 = 2^{3} \times 3^{2}$ , so the HCF is $2^{3} \times 3 = 24$ . $48 : 72 = 48 \div 24 : 72 \div 24 = 2 : 3$

(c) $5 : 9$ and $7 : 11$ are both in their simplest form, and they are different, so the ratios are not equivalent. The fractions agree: $\frac{5}{9} = \frac{55}{99}$ and $\frac{7}{11} = \frac{63}{99}$ , which are not equal. Adding the same number to both terms changes a ratio.

(a) Ali and Siti stopped too soon, and Ravi turned the terms round.
(b) $48 : 72 = 2 : 3$
(c) No. $5 : 9$ and $7 : 11$ are not equivalent.

Key Points

Multiplying or dividing every term of a ratio by the same non-zero number gives an equivalent ratio. Adding or subtracting the same number does not.
On a bar model, equivalent ratios are the same bars cut into units of a different size. As fractions, they are equivalent fractions: $8 : 12 = 2 : 3$ because $\frac{8}{12} = \frac{2}{3}$ .
A ratio is in its simplest form when its terms are whole numbers with no common factor other than 1.
To write a ratio in its simplest form, divide every term by the HCF of the terms, or divide by common factors one after another until none is left but 1.
To be sure, look for a prime factor that is in both terms. Two odd terms, or one prime term, prove nothing.
Two ratios are equivalent if their simplest forms are the same, with the terms in the same order.
To find a missing term, find the multiplier between the two known terms in the same position and use it on the other term. If it is not a whole number, write the complete ratio in its simplest form first.