Combining Two Ratios

How to join two ratios that share a quantity into one ratio of three quantities, by making the shared quantity the same number of units in both.

Before you start

Equivalent ratios. Multiplying or dividing every term of a ratio by the same non-zero number gives an equivalent ratio: $2 : 3 = 8 : 12$ (every term multiplied by 4) and $15 : 25 = 3 : 5$ (every term divided by 5).

A ratio of three quantities. $a : b : c = 4 : 6 : 9$ compares three quantities at once. The ratio of any two of them can be read from it: $a : b = 4 : 6 = 2 : 3$ , $b : c = 6 : 9 = 2 : 3$ and $a : c = 4 : 9$ .

The lowest common multiple (LCM) of two whole numbers is the smallest positive whole number that is a multiple of both. The LCM of 4 and 6 is 12.

Two Ratios That Share a Quantity

Ali, Mei Ling and Ravi collect stickers. The ratio of the number of Ali's stickers to the number of Mei Ling's stickers is $2 : 3$ . The ratio of the number of Mei Ling's stickers to the number of Ravi's stickers is $4 : 5$ . What is the ratio of the number of Ali's stickers to the number of Mei Ling's stickers to the number of Ravi's stickers?

Mei Ling's stickers are in both ratios. They are the shared quantity, and they are what lets us compare Ali's stickers with Ravi's.

It is tempting to write the two ratios side by side, as $2 : 3 : 4 : 5$ or as $2 : 3 : 5$ . To see why this cannot be done, draw each ratio as a bar model, with every unit drawn the same size.

Mei Ling's bar is 3 units long in the first drawing and 4 units long in the second. But Mei Ling has only one number of stickers, so her bar ought to be the same length in both drawings. The units cannot really be the same size: 1 unit of the first ratio is not the same amount as 1 unit of the second ratio. The terms 2 and 3 count one kind of unit and the terms 4 and 5 count another, so they cannot stand together in one ratio.

Numbers make this plain. Suppose Mei Ling had 24 stickers. In the first ratio 3 units would be 24 stickers, so 1 unit would be 8 stickers. In the second ratio 4 units would be 24 stickers, so 1 unit would be 6 stickers.

The two ratios can be joined only when Mei Ling's stickers are the same number of units in both. Cut every unit of the first ratio into 4 equal small units, and every unit of the second ratio into 3 equal small units. Mei Ling's bar is then $3 \times 4 = 12$ small units in the first drawing and $4 \times 3 = 12$ small units in the second.

The heavy lines show the units of each ratio as it was given.

Compare the heavy lines in the two drawings. A unit of the first ratio (4 small units) is now longer than a unit of the second ratio (3 small units), just as 1 unit was 8 stickers in the first ratio and 6 stickers in the second. The first pair of drawings could not show this, because it drew every unit the same size.

In both drawings 1 small unit is now one twelfth of Mei Ling's stickers. So all the small units are the same size, Mei Ling's bar is the same length in both drawings, and the three bars can be put in one drawing.

The ratio of the number of Ali's stickers to the number of Mei Ling's stickers to the number of Ravi's stickers is $8 : 12 : 15$ .

Check with the numbers above. If Mei Ling had 24 stickers, Ali would have $2 \times 8 = 16$ stickers and Ravi would have $5 \times 6 = 30$ stickers, and $16 : 24 : 30 = 8 : 12 : 15$ (every term divided by 2).

Cutting every unit of a ratio into the same number of small units multiplies every term of the ratio by that number, which gives an equivalent ratio: $2 : 3 = 8 : 12$ and $4 : 5 = 12 : 15$ . The number 12 was chosen because it is the LCM of 3 and 4: the smallest number of small units that a bar of 3 units and a bar of 4 units can both be cut into.

Making the Shared Quantity the Same

The bars need not be drawn every time. Set the two ratios out as rows of a table of equivalent ratios, with one column for each quantity, so that the two terms of the shared quantity stand one under the other. Then multiply every term of each row by the number that makes those two terms the same. Here is the table for the stickers.

	Ali	Mei Ling	Ravi
Ali : Mei Ling	2	3
Mei Ling : Ravi		4	5
Ali : Mei Ling, every term multiplied by 4	8	12
Mei Ling : Ravi, every term multiplied by 3		12	15
Ali : Mei Ling : Ravi	8	12	15

Rule: combining two ratios

Find the shared quantity, the one that is in both ratios, and its term in each ratio.
Find the LCM of these two terms.
Change each ratio into an equivalent ratio in which the term of the shared quantity is the LCM, by multiplying every term of the ratio by the same number.
The shared quantity is now the same number of units in both ratios. Write the three terms as one ratio, in the order the question asks for.

Give the combined ratio in its simplest form. The ratio of any two of the three quantities can then be read from it.

Example 1: The shared quantity in the middle

If $a : b = 3 : 4$ and $b : c = 5 : 6$ , find
(a) $a : b : c$
(b) $a : c$

Solution

(a) The shared quantity is $b$ . It is 4 units in the first ratio and 5 units in the second.

The LCM of 4 and 5 is 20, so make $b$ 20 units in both ratios.

$4 \times 5 = 20$ , so multiply every term of the first ratio by 5: $a : b = 3 : 4 = 15 : 20$

$5 \times 4 = 20$ , so multiply every term of the second ratio by 4: $b : c = 5 : 6 = 20 : 24$

	$a$	$b$	$c$
$a : b$	3	4
$b : c$		5	6
$a : b$ , every term multiplied by 5	15	20
$b : c$ , every term multiplied by 4		20	24
$a : b : c$	15	20	24

15, 20 and 24 have no common factor but 1, so $15 : 20 : 24$ is in its simplest form.

(b) Read the first and third terms of the combined ratio: $a : c = 15 : 24$ . The HCF of 15 and 24 is 3, so divide both terms by 3: $a : c = 5 : 8$ .

(a) $a : b : c = 15 : 20 : 24$
(b) $a : c = 5 : 8$

Why the LCM?

Any common multiple of the two terms will do. In Example 1, with $b$ as 40 units, $a : b = 30 : 40$ and $b : c = 40 : 48$ , so $a : b : c = 30 : 40 : 48$ . This is not in its simplest form: dividing every term by the HCF, 2, gives $15 : 20 : 24$ again. The LCM gives the smallest numbers to work with.

Example 2: Shared terms that are equal, or one a multiple of the other

(a) If $p : q = 4 : 7$ and $q : r = 7 : 9$ , find $p : q : r$ .
(b) If $x : y = 5 : 3$ and $y : z = 9 : 7$ , find $x : y : z$ .

Solution

(a) The shared quantity is $q$ . It is 7 units in both ratios already, so the units of the two ratios are the same size and the three terms can be written as one ratio at once. 4, 7 and 9 have no common factor but 1.

(b) The shared quantity is $y$ . It is 3 units in the first ratio and 9 units in the second. The LCM of 3 and 9 is 9, so only the first ratio has to be changed.

$3 \times 3 = 9$ , so multiply every term of the first ratio by 3: $x : y = 5 : 3 = 15 : 9$

The second ratio stays as it is: $y : z = 9 : 7$

15 and 9 have the common factor 3, but 7 does not, so $15 : 9 : 7$ is in its simplest form.

(a) $p : q : r = 4 : 7 : 9$
(b) $x : y : z = 15 : 9 : 7$

When the Shared Quantity Is Not in the Middle

The shared quantity is not always the second quantity of one ratio and the first quantity of the other. It may come first in both ratios, or last in both, and the question may ask for the three quantities in any order. The method is the same: find the quantity that is in both ratios and make its two terms the same. In the table, write every term under its own quantity.

Example 3: The shared quantity comes first in both ratios

The ratio of Ali's age to his sister's age is $3 : 2$ . The ratio of Ali's age to his father's age is $2 : 7$ . Find the ratio of
(a) Ali's age to his sister's age to his father's age
(b) his sister's age to his father's age

Solution

(a) Ali's age is in both ratios. It is 3 units in the first ratio and 2 units in the second.

The LCM of 3 and 2 is 6, so make Ali's age 6 units in both ratios.

Multiply every term of the first ratio by 2: Ali : sister $= 3 : 2 = 6 : 4$

Multiply every term of the second ratio by 3: Ali : father $= 2 : 7 = 6 : 21$

	Ali	Sister	Father
Ali : sister	3	2
Ali : father	2		7
Ali : sister, every term multiplied by 2	6	4
Ali : father, every term multiplied by 3	6		21
Ali : sister : father	6	4	21

6, 4 and 21 have no common factor but 1, so $6 : 4 : 21$ is in its simplest form.

(b) Read the second and third terms of the combined ratio: sister : father $= 4 : 21$

Check with ages that fit: if Ali is 12, his sister is 8 and his father is 42. Then $12 : 8 = 3 : 2$ and $12 : 42 = 2 : 7$ , as given.

(a) The ratio of Ali's age to his sister's age to his father's age is $6 : 4 : 21$ .
(b) The ratio of his sister's age to his father's age is $4 : 21$ .

Example 4: The shared quantity comes last in both ratios

The ratio of Siti's height to Ravi's height is $9 : 10$ . The ratio of Hui Min's height to Ravi's height is $14 : 15$ . Find the ratio of
(a) Siti's height to Hui Min's height to Ravi's height
(b) Ravi's height to Hui Min's height to Siti's height

Solution

(a) Ravi's height is in both ratios. It is 10 units in the first ratio and 15 units in the second.

The LCM of 10 and 15 is 30, so make Ravi's height 30 units in both ratios.

Multiply every term of the first ratio by 3: Siti : Ravi $= 9 : 10 = 27 : 30$

Multiply every term of the second ratio by 2: Hui Min : Ravi $= 14 : 15 = 28 : 30$

	Siti	Hui Min	Ravi
Siti : Ravi	9		10
Hui Min : Ravi		14	15
Siti : Ravi, every term multiplied by 3	27		30
Hui Min : Ravi, every term multiplied by 2		28	30
Siti : Hui Min : Ravi	27	28	30

27, 28 and 30 have no common factor but 1, so $27 : 28 : 30$ is in its simplest form.

(b) These are the same three terms, written in the order asked for: Ravi 30 units, Hui Min 28 units, Siti 27 units.

(a) The ratio of Siti's height to Hui Min's height to Ravi's height is $27 : 28 : 30$ .
(b) The ratio of Ravi's height to Hui Min's height to Siti's height is $30 : 28 : 27$ .

Simplifying First, and Ratios with Fractions or Decimals

Combining is easiest when each given ratio has whole-number terms and is in its simplest form, so tidy each ratio first. If its terms have a common factor, divide every term by their HCF. If it has decimals, multiply every term by 10, 100 or 1000. If it has fractions, multiply every term by the LCM of the denominators, after changing any mixed number to an improper fraction. Each of these gives an equivalent ratio, so the comparison is not changed.

Whichever way you work, finish by checking the combined ratio. If its three terms have a common factor other than 1, divide every term by their HCF.

Example 5: Ratios that are not in their simplest form

If $p : q = 12 : 18$ and $q : r = 10 : 25$ , find $p : q : r$ .

Solution

Simplify first. The HCF of 12 and 18 is 6, so $p : q = 12 : 18 = 2 : 3$ . The HCF of 10 and 25 is 5, so $q : r = 10 : 25 = 2 : 5$ .

The shared quantity $q$ is 3 units in the first ratio and 2 units in the second. The LCM of 3 and 2 is 6.

Multiply every term of the first ratio by 2: $p : q = 2 : 3 = 4 : 6$

Multiply every term of the second ratio by 3: $q : r = 2 : 5 = 6 : 15$

So $p : q : r = 4 : 6 : 15$ . The terms 4 and 6 have the common factor 2, but 15 does not have it, so $4 : 6 : 15$ is in its simplest form.

Or simplify at the end. Without simplifying first, $q$ is 18 units in the first ratio and 10 units in the second, and the LCM of 18 and 10 is 90. Then $p : q = 12 : 18 = 60 : 90$ (every term multiplied by 5) and $q : r = 10 : 25 = 90 : 225$ (every term multiplied by 9), so $p : q : r = 60 : 90 : 225$ .

The HCF of 60, 90 and 225 is 15. Dividing every term by 15 gives $4 : 6 : 15$ again, but with more work.

$p : q : r = 4 : 6 : 15$

Example 6: Ratios with fractions and decimals

For a class party, Siti mixes a drink from mango juice, apple juice and soda water. The ratio of the volume of mango juice to the volume of apple juice is $\frac{3}{4} : \frac{1}{2}$ . The ratio of the volume of apple juice to the volume of soda water is $1.5 : 2.5$ . Find the ratio of
(a) the volume of mango juice to the volume of apple juice to the volume of soda water
(b) the volume of mango juice to the volume of soda water

Solution

(a) Clear the fractions. The LCM of the denominators 4 and 2 is 4, so multiply every term by 4: mango : apple $= \frac{3}{4} : \frac{1}{2} = 3 : 2$

Clear the decimals. Multiply every term by 10, then divide every term by the HCF, 5: apple : soda $= 1.5 : 2.5 = 15 : 25 = 3 : 5$

Apple juice is in both ratios. It is 2 units in the first ratio and 3 units in the second, and the LCM of 2 and 3 is 6.

Multiply every term of the first ratio by 3: mango : apple $= 3 : 2 = 9 : 6$

Multiply every term of the second ratio by 2: apple : soda $= 3 : 5 = 6 : 10$

So mango : apple : soda $= 9 : 6 : 10$ . The three terms have no common factor but 1.

(b) Read the first and third terms of the combined ratio: mango : soda $= 9 : 10$

(a) The ratio of the volume of mango juice to the volume of apple juice to the volume of soda water is $9 : 6 : 10$ .
(b) The ratio of the volume of mango juice to the volume of soda water is $9 : 10$ .

Word Problems

In a word problem, write each ratio with the names of its quantities, in the order of the words. Then look for the quantity that is named in both ratios. In the combined ratio every quantity is counted in units of the same size, so one known amount is enough to find the others.

Example 7: Three colours of markers

Siti has red, orange and green markers. The ratio of the number of red markers to the number of orange markers is $7 : 8$ . The ratio of the number of orange markers to the number of green markers is $3 : 5$ . Find the ratio of
(a) the number of red markers to the number of orange markers to the number of green markers
(b) the number of red markers to the number of green markers

Solution

(a) The orange markers are in both ratios. They are 8 units in the first ratio and 3 units in the second.

The LCM of 8 and 3 is 24, so make the orange markers 24 units in both ratios.

Multiply every term of the first ratio by 3: red : orange $= 7 : 8 = 21 : 24$

Multiply every term of the second ratio by 8: orange : green $= 3 : 5 = 24 : 40$

	Red	Orange	Green
Red : orange	7	8
Orange : green		3	5
Red : orange, every term multiplied by 3	21	24
Orange : green, every term multiplied by 8		24	40
Red : orange : green	21	24	40

21, 24 and 40 have no common factor but 1, so $21 : 24 : 40$ is in its simplest form.

(b) Read the first and third terms of the combined ratio: red : green $= 21 : 40$

(a) The ratio of the number of red markers to the number of orange markers to the number of green markers is $21 : 24 : 40$ .
(b) The ratio of the number of red markers to the number of green markers is $21 : 40$ .

Example 8: Finding an amount from the combined ratio

The ratio of Mei Ling's savings to Hui Min's savings is $5 : 6$ . The ratio of Hui Min's savings to Ravi's savings is $4 : 3$ .
(a) Find the ratio of Mei Ling's savings to Hui Min's savings to Ravi's savings.
(b) Ravi has saved $45. How much has Mei Ling saved?

Solution

(a) Hui Min's savings are in both ratios. They are 6 units in the first ratio and 4 units in the second, and the LCM of 6 and 4 is 12.

Multiply every term of the first ratio by 2: Mei Ling : Hui Min $= 5 : 6 = 10 : 12$

Multiply every term of the second ratio by 3: Hui Min : Ravi $= 4 : 3 = 12 : 9$

So Mei Ling : Hui Min : Ravi $= 10 : 12 : 9$ . The three terms have no common factor but 1.

(b) Draw the combined ratio as a bar model. Every unit is now the same size.

Ravi's savings are 9 units, so 9 units = $45 and 1 unit = $5, because $45 \div 9 = 5$ .

Mei Ling's savings are 10 units, and $10 \times 5 = 50$ .

Check: Hui Min's savings are 12 units, and $12 \times 5 = 60$ . Then $50 : 60 = 5 : 6$ and $60 : 45 = 4 : 3$ , as given.

(a) The ratio of Mei Ling's savings to Hui Min's savings to Ravi's savings is $10 : 12 : 9$ .
(b) Mei Ling has saved $50.

Common Mistakes

Writing the two ratios side by side. From $a : b = 2 : 3$ and $b : c = 4 : 5$ , both $2 : 3 : 4 : 5$ and $2 : 3 : 5$ are wrong. The first has four terms for three quantities. The second drops the 4, as though 3 units of one ratio were the same amount as 4 units of the other. Make $b$ 12 units in both ratios first: $a : b : c = 8 : 12 : 15$ .
Multiplying only the shared term. Changing $2 : 3$ into $2 : 12$ changes the ratio. Every term must be multiplied by the same number: $2 : 3 = 8 : 12$ .
Adding to make the shared terms equal. Adding 1 to each term of $2 : 3$ gives $3 : 4$ , which is not an equivalent ratio. Only multiplying or dividing every term by the same non-zero number gives an equivalent ratio.
Putting a term under the wrong quantity. From $a : b = 6 : 4$ and $a : c = 6 : 21$ , the ratio $a : b : c$ is $6 : 4 : 21$ , not $4 : 6 : 21$ . Write each term under its own quantity in the table, and give the terms in the order the question asks for.
Dividing only two of the three terms. $4 : 6 : 15$ cannot be changed to $2 : 3 : 15$ . A ratio of three quantities is simplified only by dividing all three terms by a common factor.

Practice

Work each question on paper before you open its solution.

Question 1

Find $a : b : c$ in each case.
(a) $a : b = 3 : 5$ and $b : c = 5 : 8$
(b) $a : b = 1 : 2$ and $b : c = 4 : 3$
(c) $a : b = 2 : 5$ and $b : c = 3 : 4$

ShowHide solution

(a) The shared quantity $b$ is 5 units in both ratios already, so the three terms can be written as one ratio at once. 3, 5 and 8 have no common factor but 1.

(b) $b$ is 2 units in the first ratio and 4 units in the second. The LCM of 2 and 4 is 4, so only the first ratio has to be changed. Multiply every term by 2: $a : b = 1 : 2 = 2 : 4$ . The second ratio stays as $b : c = 4 : 3$ . So $a : b : c = 2 : 4 : 3$ . 2 and 4 have the common factor 2, but 3 does not, so the ratio is in its simplest form.

(c) $b$ is 5 units in the first ratio and 3 units in the second, and the LCM of 5 and 3 is 15. Multiply every term of the first ratio by 3: $a : b = 2 : 5 = 6 : 15$ . Multiply every term of the second ratio by 5: $b : c = 3 : 4 = 15 : 20$ . So $a : b : c = 6 : 15 : 20$ . 6 and 15 share 3, and 15 and 20 share 5, but no number other than 1 divides all three.

(a) $a : b : c = 3 : 5 : 8$
(b) $a : b : c = 2 : 4 : 3$
(c) $a : b : c = 6 : 15 : 20$

Question 2

Ravi, Hui Min and Siti borrow books from the library. The ratio of the number of books Ravi borrows to the number Hui Min borrows is $3 : 2$ . The ratio of the number Hui Min borrows to the number Siti borrows is $3 : 4$ . The two bar models below show the two ratios, with every unit drawn the same size.
(a) Explain why 1 unit of the first drawing cannot stand for the same number of books as 1 unit of the second drawing.
(b) Into how many equal small units should each unit of the first drawing be cut, and each unit of the second drawing, so that Hui Min's bar is the same number of small units in both, using the fewest small units possible?
(c) Find the ratio of the number of books Ravi borrows to the number Hui Min borrows to the number Siti borrows.

ShowHide solution

(a) Hui Min's bar is 2 units in the first drawing and 3 units in the second. She borrows only one number of books, so 2 units of the first drawing are the same number of books as 3 units of the second. A unit of the first drawing must stand for more books than a unit of the second.

(b) Any common multiple of 2 and 3 would make Hui Min's bar the same in both, but the fewest small units come from the LCM. The LCM of 2 and 3 is 6, so make Hui Min's bar 6 small units in both drawings. $2 \times 3 = 6$ , so cut each unit of the first drawing into 3 small units. $3 \times 2 = 6$ , so cut each unit of the second drawing into 2 small units.

(c) Multiply every term of the first ratio by 3: Ravi : Hui Min $= 3 : 2 = 9 : 6$ . Multiply every term of the second ratio by 2: Hui Min : Siti $= 3 : 4 = 6 : 8$ .

All the small units are now the same size, so Ravi : Hui Min : Siti $= 9 : 6 : 8$ . The three terms have no common factor but 1.

(a) Hui Min's books are 2 units in one drawing and 3 units in the other, so the units are not the same size.
(b) Each unit of the first drawing into 3 small units, and each unit of the second drawing into 2 small units.
(c) The ratio of the number of books Ravi borrows to the number Hui Min borrows to the number Siti borrows is $9 : 6 : 8$ .

Question 3

It is given that $x : y = 5 : 6$ and $y : z = 9 : 4$ . Copy the table of equivalent ratios below and fill in every place marked with a question mark, making the two terms of $y$ equal to their LCM. Then write down
(a) $x : y : z$
(b) $x : z$

	$x$	$y$	$z$
$x : y$	5	6
$y : z$		9	4
$x : y$ , every term multiplied by ?	?	?
$y : z$ , every term multiplied by ?		?	?
$x : y : z$	?	?	?

ShowHide solution

The shared quantity $y$ is 6 units in the first ratio and 9 units in the second. The LCM of 6 and 9 is 18.

$6 \times 3 = 18$ , so multiply every term of the first ratio by 3. $9 \times 2 = 18$ , so multiply every term of the second ratio by 2.

	$x$	$y$	$z$
$x : y$	5	6
$y : z$		9	4
$x : y$ , every term multiplied by 3	15	18
$y : z$ , every term multiplied by 2		18	8
$x : y : z$	15	18	8

(a) 15, 18 and 8 have no common factor but 1, so $15 : 18 : 8$ is in its simplest form.

(b) Read the first and third terms of the combined ratio: $x : z = 15 : 8$

(a) $x : y : z = 15 : 18 : 8$
(b) $x : z = 15 : 8$

Question 4

If $a : b = 4 : 3$ and $a : c = 6 : 5$ , find
(a) $a : b : c$
(b) $b : c$

ShowHide solution

The shared quantity is $a$ . It is 4 units in the first ratio and 6 units in the second, and the LCM of 4 and 6 is 12.

Multiply every term of the first ratio by 3: $a : b = 4 : 3 = 12 : 9$

Multiply every term of the second ratio by 2: $a : c = 6 : 5 = 12 : 10$

(a) $a$ is 12 units, $b$ is 9 units and $c$ is 10 units. The three terms have no common factor but 1.

(b) Read the second and third terms of the combined ratio: $b : c = 9 : 10$

(a) $a : b : c = 12 : 9 : 10$
(b) $b : c = 9 : 10$

Question 5

If $p : r = 3 : 8$ and $q : r = 7 : 12$ , find
(a) $p : q : r$
(b) $p : q$

ShowHide solution

The shared quantity is $r$ . It is 8 units in the first ratio and 12 units in the second, and the LCM of 8 and 12 is 24.

Multiply every term of the first ratio by 3: $p : r = 3 : 8 = 9 : 24$

Multiply every term of the second ratio by 2: $q : r = 7 : 12 = 14 : 24$

(a) $p$ is 9 units, $q$ is 14 units and $r$ is 24 units. The three terms have no common factor but 1.

(b) Read the first and second terms of the combined ratio: $p : q = 9 : 14$

(a) $p : q : r = 9 : 14 : 24$
(b) $p : q = 9 : 14$

Question 6

If $a : b = 14 : 21$ and $b : c = 20 : 12$ , find
(a) $a : b : c$
(b) $a : c$

ShowHide solution

Simplify each ratio first. The HCF of 14 and 21 is 7, so $a : b = 14 : 21 = 2 : 3$ . The HCF of 20 and 12 is 4, so $b : c = 20 : 12 = 5 : 3$ .

The shared quantity $b$ is 3 units in the first ratio and 5 units in the second, and the LCM of 3 and 5 is 15.

Multiply every term of the first ratio by 5: $a : b = 2 : 3 = 10 : 15$

Multiply every term of the second ratio by 3: $b : c = 5 : 3 = 15 : 9$

(a) $a : b : c = 10 : 15 : 9$ . The terms 10 and 15 have the common factor 5, but 9 does not have it, so the ratio is in its simplest form.

(b) Read the first and third terms of the combined ratio: $a : c = 10 : 9$

If you did not simplify first: the LCM of 21 and 20 is 420, which gives $a : b = 280 : 420$ and $b : c = 420 : 252$ , so $a : b : c = 280 : 420 : 252$ . The HCF of the three terms is 28, and dividing every term by 28 gives $10 : 15 : 9$ again.

(a) $a : b : c = 10 : 15 : 9$
(b) $a : c = 10 : 9$

Question 7

Find $a : b : c$ in each case.
(a) $a : b = 0.4 : 1$ and $b : c = \frac{1}{3} : \frac{1}{4}$
(b) $a : b = 1\frac{1}{2} : 2$ and $a : c = 0.6 : 1.5$

ShowHide solution

(a) Multiply every term by 10, then divide every term by 2: $a : b = 0.4 : 1 = 4 : 10 = 2 : 5$

The LCM of the denominators 3 and 4 is 12, so multiply every term by 12: $b : c = \frac{1}{3} : \frac{1}{4} = 4 : 3$

The shared quantity $b$ is 5 units in the first ratio and 4 units in the second, and the LCM of 5 and 4 is 20. So $a : b = 2 : 5 = 8 : 20$ (every term multiplied by 4) and $b : c = 4 : 3 = 20 : 15$ (every term multiplied by 5).

(b) $1\frac{1}{2} = \frac{3}{2}$ . Multiply every term by 2: $a : b = \frac{3}{2} : 2 = 3 : 4$

Multiply every term by 10, then divide every term by 3: $a : c = 0.6 : 1.5 = 6 : 15 = 2 : 5$

The shared quantity $a$ is 3 units in the first ratio and 2 units in the second, and the LCM of 3 and 2 is 6. So $a : b = 3 : 4 = 6 : 8$ (every term multiplied by 2) and $a : c = 2 : 5 = 6 : 15$ (every term multiplied by 3).

In each answer the three terms have no common factor but 1.

(a) $a : b : c = 8 : 20 : 15$
(b) $a : b : c = 6 : 8 : 15$

Question 8

Ravi is given $a : b = 3 : 4$ and $b : c = 6 : 7$ . He writes: " $a : b = 3 : 12$ and $b : c = 12 : 14$ , so $a : b : c = 3 : 12 : 14$ ."
(a) What mistake has Ravi made?
(b) Find the correct ratio $a : b : c$ .

ShowHide solution

(a) Ravi is right to make $b$ 12 units in both ratios, because the LCM of 4 and 6 is 12. In the second ratio he multiplied every term by 2, which is correct: $6 : 7 = 12 : 14$ .

In the first ratio he multiplied only the 4 by 3 and left the 3 as it was. But $3 : 12$ is not equivalent to $3 : 4$ .

(b) Multiply every term of the first ratio by 3: $a : b = 3 : 4 = 9 : 12$ . With $b : c = 12 : 14$ , this gives $a : b : c = 9 : 12 : 14$ . The three terms have no common factor but 1.

(a) He multiplied only the shared term of $3 : 4$ by 3, not every term.
(b) $a : b : c = 9 : 12 : 14$

Question 9

In a school garden there are a papaya tree, a banana plant and a mango tree. The ratio of the height of the papaya tree to the height of the mango tree is $3 : 5$ . The ratio of the height of the banana plant to the height of the mango tree is $1 : 2$ . Find the ratio of
(a) the height of the papaya tree to the height of the banana plant to the height of the mango tree
(b) the height of the papaya tree to the height of the banana plant

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The height of the mango tree is in both ratios. It is 5 units in the first ratio and 2 units in the second, and the LCM of 5 and 2 is 10.

Multiply every term of the first ratio by 2: papaya : mango $= 3 : 5 = 6 : 10$

Multiply every term of the second ratio by 5: banana : mango $= 1 : 2 = 5 : 10$

(a) The papaya tree is 6 units, the banana plant is 5 units and the mango tree is 10 units. The three terms have no common factor but 1.

(b) Read the first and second terms of the combined ratio: papaya : banana $= 6 : 5$

(a) The ratio of the height of the papaya tree to the height of the banana plant to the height of the mango tree is $6 : 5 : 10$ .
(b) The ratio of the height of the papaya tree to the height of the banana plant is $6 : 5$ .

Question 10

For the pastry of her pineapple tarts, Mrs Goh uses flour, butter and sugar. The ratio of the mass of flour to the mass of butter is $2\frac{1}{2} : 1\frac{1}{2}$ . The ratio of the mass of flour to the mass of sugar is $1 : 0.3$ . Find the ratio of
(a) the mass of flour to the mass of butter to the mass of sugar
(b) the mass of butter to the mass of sugar

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$2\frac{1}{2} = \frac{5}{2}$ and $1\frac{1}{2} = \frac{3}{2}$ . Multiply every term by 2: flour : butter $= \frac{5}{2} : \frac{3}{2} = 5 : 3$

Multiply every term by 10: flour : sugar $= 1 : 0.3 = 10 : 3$

The mass of flour is in both ratios. It is 5 units in the first ratio and 10 units in the second. The LCM of 5 and 10 is 10, so only the first ratio has to be changed.

Multiply every term of the first ratio by 2: flour : butter $= 5 : 3 = 10 : 6$

(a) Flour is 10 units, butter is 6 units and sugar is 3 units. The three terms have no common factor but 1.

(b) Read the second and third terms of the combined ratio: butter : sugar $= 6 : 3$ . Divide both terms by their HCF, 3: butter : sugar $= 2 : 1$

(a) The ratio of the mass of flour to the mass of butter to the mass of sugar is $10 : 6 : 3$ .
(b) The ratio of the mass of butter to the mass of sugar is $2 : 1$ .

Question 11

The ratio of Ali's weekly pocket money to Ravi's weekly pocket money is $4 : 5$ . The ratio of Siti's weekly pocket money to Ali's weekly pocket money is $6 : 7$ . Find the ratio of
(a) Siti's pocket money to Ali's pocket money to Ravi's pocket money
(b) Ravi's pocket money to Siti's pocket money

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Ali's pocket money is in both ratios. It is the first term, 4 units, in Ali : Ravi, and the second term, 7 units, in Siti : Ali. The LCM of 4 and 7 is 28.

Multiply every term of the first ratio by 7: Ali : Ravi $= 4 : 5 = 28 : 35$

Multiply every term of the second ratio by 4: Siti : Ali $= 6 : 7 = 24 : 28$

(a) Siti's pocket money is 24 units, Ali's is 28 units and Ravi's is 35 units. The three terms have no common factor but 1.

(b) Read Ravi's term and then Siti's term: Ravi : Siti $= 35 : 24$

(a) The ratio of Siti's pocket money to Ali's pocket money to Ravi's pocket money is $24 : 28 : 35$ .
(b) The ratio of Ravi's pocket money to Siti's pocket money is $35 : 24$ .

Question 12

In a school, the ratio of the number of students in the Art Club to the number of students in the Chess Club is $5 : 2$ . The ratio of the number of students in the Art Club to the number of students in the Robotics Club is $3 : 4$ .
(a) Find the ratio of the number of students in the Art Club to the number in the Chess Club to the number in the Robotics Club.
(b) There are 12 students in the Chess Club. How many students are there in the Robotics Club?

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(a) The Art Club is in both ratios. It is 5 units in the first ratio and 3 units in the second, and the LCM of 5 and 3 is 15.

Multiply every term of the first ratio by 3: Art : Chess $= 5 : 2 = 15 : 6$

Multiply every term of the second ratio by 5: Art : Robotics $= 3 : 4 = 15 : 20$

So Art : Chess : Robotics $= 15 : 6 : 20$ . The three terms have no common factor but 1.

(b) Draw the combined ratio as a bar model.

The Chess Club is 6 units, so 6 units = 12 students and 1 unit = 2 students, because $12 \div 6 = 2$ .

The Robotics Club is 20 units, and $20 \times 2 = 40$ .

Check: the Art Club is 15 units, and $15 \times 2 = 30$ . Then $30 : 12 = 5 : 2$ and $30 : 40 = 3 : 4$ , as given.

(a) The ratio of the number of students in the Art Club to the number in the Chess Club to the number in the Robotics Club is $15 : 6 : 20$ .
(b) There are 40 students in the Robotics Club.

Key Points

Two ratios can be combined when one quantity is in both of them: the shared quantity.
The terms of two ratios cannot be written side by side unless the shared quantity is the same number of units in both.
Make the two terms of the shared quantity the same, using their LCM, by multiplying every term of each ratio by the same number.
In a table of equivalent ratios, write every term under its own quantity. The method is the same wherever the shared quantity stands.
Simplify the given ratios, and clear fractions and decimals, before combining. Check that the combined ratio is in its simplest form.
The ratio of any two of the three quantities can be read from the combined ratio, and then simplified.