Ratios with Fractions and Decimals

How to write a ratio of decimals, fractions or mixed numbers in its simplest form, by first making every term a whole number.

Before you start

A ratio compares two quantities of the same kind, measured in the same unit of measurement. The terms follow the order of the words: if a rope is 3 m long and a pole is 5 m long, the ratio of the length of the rope to the length of the pole is $3 : 5$ .

Multiplying or dividing every term of a ratio by the same non-zero number gives an equivalent ratio: $3 : 5 = 3 \times 4 : 5 \times 4 = 12 : 20$ . The comparison is the same because the fraction is the same. The ratio $3 : 5$ says that the first quantity is $\frac{3}{5}$ of the second, and $\frac{12}{20} = \frac{3}{5}$ .

A ratio is in its simplest form when its terms are whole numbers with no common factor other than 1. To reach it, divide every term by their highest common factor (HCF): the HCF of 12 and 18 is 6, so $12 : 18 = 12 \div 6 : 18 \div 6 = 2 : 3$ .

This lesson also uses the lowest common multiple (LCM): the LCM of 4 and 6 is 12, the smallest number that both 4 and 6 divide exactly. A mixed number is changed to an improper fraction like this: $2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}$ .

Making Every Term a Whole Number

The terms of a ratio are not always whole numbers. Siti has a ribbon 1.5 m long and Mei Ling has a ribbon 2.5 m long. Both lengths are in metres, so the ratio of the length of Siti's ribbon to the length of Mei Ling's ribbon is $1.5 : 2.5$ . This ratio is correct, but it is not in its simplest form, because its terms are not whole numbers.

Cut both ribbons into pieces 0.5 m long. Siti's ribbon makes 3 pieces and Mei Ling's ribbon makes 5 pieces of the same size, so the ratio is $3 : 5$ . In the bar model, each unit stands for one piece.

Counting in half-metres doubles both numbers: $1.5 \times 2 : 2.5 \times 2 = 3 : 5$ . The ribbons have not changed. Only the size of the piece they are counted in has changed, so the comparison is the same. This is why multiplying every term by the same non-zero number gives an equivalent ratio when the terms are decimals or fractions too.

Method

To write a ratio of decimals, fractions or mixed numbers in its simplest form:

Check that the quantities are in the same unit of measurement.
Multiply every term by the same non-zero number, chosen so that every term becomes a whole number.
Divide every term by their HCF.

The rest of this lesson shows how to choose the number in step 2.

Ratios with Decimals

Multiplying a decimal by 10 moves every digit one place to the left, so it clears one decimal place: $0.7 \times 10 = 7$ . Multiplying by 100 clears two decimal places, and multiplying by 1000 clears three: $0.25 \times 100 = 25$ and $0.204 \times 1000 = 204$ .

Rule: a ratio of decimals

Find the term with the most decimal places. Multiply every term by 10 if that term has 1 decimal place, by 100 if it has 2, and by 1000 if it has 3. Then write the ratio in its simplest form.

Ratio	Most decimal places	Multiply every term by	Whole-number terms	Simplest form
$0.3 : 1.2$	1	10	$3 : 12$	$1 : 4$
$0.3 : 0.12$	2	100	$30 : 12$	$5 : 2$
$0.3 : 0.125$	3	1000	$300 : 125$	$12 : 5$

A term with fewer decimal places is multiplied by the same number as the others. In the second row, $0.3 \times 100 = 30$ , not 3.

Example 1: Ratios of decimals

Express each ratio in its simplest form.
(a) $2.5 : 5.5$
(b) $1.6 : 4$
(c) $0.45 : 0.6$

Solution

(a) Each term has 1 decimal place, so multiply every term by 10: $2.5 \times 10 : 5.5 \times 10 = 25 : 55$

The HCF of 25 and 55 is 5: $25 \div 5 : 55 \div 5 = 5 : 11$

(b) 1.6 has 1 decimal place, so multiply every term by 10. The whole number 4 is a term of the ratio, so it is multiplied by 10 too: $1.6 \times 10 : 4 \times 10 = 16 : 40$

The HCF of 16 and 40 is 8: $16 \div 8 : 40 \div 8 = 2 : 5$

(c) 0.45 has 2 decimal places and 0.6 has 1. The most is 2, so multiply every term by 100: $0.45 \times 100 : 0.6 \times 100 = 45 : 60$

The HCF of 45 and 60 is 15: $45 \div 15 : 60 \div 15 = 3 : 4$

(a) $2.5 : 5.5 = 5 : 11$
(b) $1.6 : 4 = 2 : 5$
(c) $0.45 : 0.6 = 3 : 4$

Example 2: Three decimal places

Express $0.18 : 0.204$ in its simplest form.

Solution

0.18 has 2 decimal places and 0.204 has 3. The most is 3, so multiply every term by 1000: $0.18 \times 1000 : 0.204 \times 1000 = 180 : 204$

Multiplying by 100 would not be enough. It gives $18 : 20.4$ , and 20.4 is still a decimal.

Find the HCF of 180 and 204 by prime factorisation: $180 = 2^{2} \times 3^{2} \times 5$ and $204 = 2^{2} \times 3 \times 17$ . Take each prime factor that is common to both, to the lower of its two powers: the HCF is $2^{2} \times 3 = 12$ .

$180 \div 12 : 204 \div 12 = 15 : 17$

Dividing by common factors in several steps gives the same ratio: dividing by 2, by 2 again and then by 3, $180 : 204 = 90 : 102 = 45 : 51 = 15 : 17$ .

15 and 17 have no common factor other than 1, so the ratio is in its simplest form.

$0.18 : 0.204 = 15 : 17$

Example 3: Two masses

A durian has a mass of 1.8 kg and a mango has a mass of 0.45 kg. Find the ratio of the mass of the durian to the mass of the mango.

Solution

Both masses are in kilograms. The durian is named first, so its mass is the first term: $1.8 : 0.45$

0.45 has 2 decimal places, which is the most, so multiply every term by 100: $1.8 \times 100 : 0.45 \times 100 = 180 : 45$

The HCF of 180 and 45 is 45: $180 \div 45 : 45 \div 45 = 4 : 1$

Check: $4 \times 0.45 = 1.8$ , so the durian is 4 times as heavy as the mango.

Multiplying by 1000 works too. It is the same as changing both masses from kilograms to grams: 1800 g to 450 g is $1800 : 450 = 4 : 1$ . Any number that makes every term a whole number leads to the same simplest form. Multiplying by 100 rather than 1000 keeps the terms smaller.

The ratio of the mass of the durian to the mass of the mango is $4 : 1$ .

Ratios with Fractions

Fractions with the same denominator are counted in pieces of the same size. $\frac{3}{8}$ is 3 eighths and $\frac{5}{8}$ is 5 eighths, so $\frac{3}{8} : \frac{5}{8}$ compares 3 pieces with 5 pieces of the same size: it is $3 : 5$ . Multiplying every term by the denominator, 8, gives the same ratio: $\frac{3}{8} \times 8 : \frac{5}{8} \times 8 = 3 : 5$ .

Example 4: The same denominator

Express each ratio in its simplest form.
(a) $\frac{2}{7} : \frac{5}{7}$
(b) $\frac{4}{9} : \frac{2}{9}$

Solution

(a) Multiply every term by the denominator, 7: $\frac{2}{7} \times 7 : \frac{5}{7} \times 7 = 2 : 5$

2 and 5 have no common factor other than 1, so the ratio is in its simplest form.

(b) Multiply every term by the denominator, 9: $\frac{4}{9} \times 9 : \frac{2}{9} \times 9 = 4 : 2$

The HCF of 4 and 2 is 2: $4 \div 2 : 2 \div 2 = 2 : 1$

When the denominators are the same, the ratio of the fractions is the ratio of their numerators. It may still have to be simplified, as in (b).

(a) $\frac{2}{7} : \frac{5}{7} = 2 : 5$
(b) $\frac{4}{9} : \frac{2}{9} = 2 : 1$

Fractions with different denominators are counted in pieces of different sizes, so their numerators cannot be compared as they stand. First find a piece that fits both fractions.

Example 5: Two times

Hui Min spends $\frac{2}{3}$ of an hour on her maths homework and $\frac{1}{2}$ of an hour on her science homework. Find the ratio of the time she spends on maths to the time she spends on science.

Solution

Both times are in hours, and maths is named first: $\frac{2}{3} : \frac{1}{2}$

Thirds and halves are pieces of different sizes. The LCM of the denominators 3 and 2 is 6, so an hour can be cut into sixths, which fit both: $\frac{2}{3} = \frac{4}{6}$ and $\frac{1}{2} = \frac{3}{6}$ . The maths time is 4 sixths of an hour and the science time is 3 sixths of an hour.

4 pieces to 3 pieces of the same size is $4 : 3$ . Multiplying every term by 6 gives this in one step: $\frac{2}{3} \times 6 : \frac{1}{2} \times 6 = 4 : 3$

Check in minutes: $\frac{2}{3}$ of an hour is 40 minutes and $\frac{1}{2}$ of an hour is 30 minutes, and $40 : 30 = 4 : 3$ .

The ratio of the time she spends on maths to the time she spends on science is $4 : 3$ .

Rule: a ratio of fractions

Multiply every term by the LCM of the denominators. Every denominator divides the LCM exactly, so every term becomes a whole number. Then write the ratio in its simplest form.

When the denominators are the same, their LCM is that denominator.

Example 6: Different denominators

Express each ratio in its simplest form.
(a) $\frac{2}{3} : \frac{5}{6}$
(b) $\frac{3}{4} : \frac{5}{6}$
(c) $\frac{3}{4} : \frac{9}{10}$

Solution

(a) 6 is a multiple of 3, so the LCM of 3 and 6 is 6. Multiply every term by 6: $\frac{2}{3} \times 6 : \frac{5}{6} \times 6 = 4 : 5$

(b) The LCM of 4 and 6 is 12. Multiply every term by 12: $\frac{3}{4} \times 12 : \frac{5}{6} \times 12 = 9 : 10$

The terms are whole numbers, but 15 and 18 have the common factor 3. The HCF of 15 and 18 is 3: $15 \div 3 : 18 \div 3 = 5 : 6$

(a) $\frac{2}{3} : \frac{5}{6} = 4 : 5$
(b) $\frac{3}{4} : \frac{5}{6} = 9 : 10$
(c) $\frac{3}{4} : \frac{9}{10} = 5 : 6$

There is a second way. The ratio $4 : 3$ says that the first quantity is $\frac{4}{3}$ of the second, and $\frac{4}{3}$ is $4 \div 3$ , the first term divided by the second. So a ratio of two terms can also be found by dividing the first term by the second. Write the answer as a fraction in its simplest form: its numerator is the first term of the ratio and its denominator is the second term.

Example 7: Dividing the first term by the second

Find each ratio by dividing the first term by the second.
(a) $\frac{2}{3} : \frac{1}{2}$
(b) $\frac{3}{4} : \frac{9}{10}$

Solution

(a) To divide by a fraction, multiply by its reciprocal: $\frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3}$

The first term is $\frac{4}{3}$ of the second, so the ratio is $4 : 3$ , as in Example 5. Leave $\frac{4}{3}$ as an improper fraction: its numerator and its denominator are the two terms.

(b) $\frac{3}{4} \div \frac{9}{10} = \frac{3}{4} \times \frac{10}{9} = \frac{30}{36} = \frac{5}{6}$

The first term is $\frac{5}{6}$ of the second, so the ratio is $5 : 6$ , as in Example 6 (c).

If the division gives a whole number, write it over 1. In Example 3, $1.8 \div 0.45 = 4 = \frac{4}{1}$ , which is the ratio $4 : 1$ .

(a) $\frac{2}{3} : \frac{1}{2} = 4 : 3$
(b) $\frac{3}{4} : \frac{9}{10} = 5 : 6$

A Fraction and a Whole Number

When one term is a fraction and the other is a whole number, multiply every term by the denominator of the fraction. The whole number has no denominator to clear, but it is a term of the ratio, so it is multiplied by the same number. It may help to write the whole number as a fraction with denominator 1: $2 = \frac{2}{1}$ .

Example 8: A fraction and a whole number

Express each ratio in its simplest form.
(a) $\frac{4}{5} : 1$
(b) $\frac{3}{5} : 2$
(c) $6 : \frac{3}{4}$

Solution

(a) Multiply every term by 5: $\frac{4}{5} \times 5 : 1 \times 5 = 4 : 5$

(b) Multiply every term by 5: $\frac{3}{5} \times 5 : 2 \times 5 = 3 : 10$

The ratio is not $3 : 2$ . That would multiply the first term by 5 and leave the second as it was.

The HCF of 24 and 3 is 3: $24 \div 3 : 3 \div 3 = 8 : 1$

Check: $8 \times \frac{3}{4} = 6$ , so the first term is 8 times the second.

(a) $\frac{4}{5} : 1 = 4 : 5$
(b) $\frac{3}{5} : 2 = 3 : 10$
(c) $6 : \frac{3}{4} = 8 : 1$

Ratios with Mixed Numbers

A mixed number is a whole number and a fraction added together: $1\frac{1}{2} = 1 + \frac{1}{2}$ . Its two parts cannot be handled separately in a ratio. Change every mixed number to an improper fraction first. The ratio is then a ratio of fractions.

Example 9: Mixed numbers

Express each ratio in its simplest form.
(a) $1\frac{3}{5} : 2\frac{4}{5}$
(b) $1\frac{1}{2} : 2\frac{1}{4}$
(c) $2\frac{1}{4} : 1\frac{2}{3}$

Solution

(a) $1\frac{3}{5} = \frac{8}{5}$ and $2\frac{4}{5} = \frac{14}{5}$ , so the ratio is $\frac{8}{5} : \frac{14}{5}$ .

The denominators are the same. Multiply every term by 5: $\frac{8}{5} \times 5 : \frac{14}{5} \times 5 = 8 : 14$

The HCF of 8 and 14 is 2: $8 \div 2 : 14 \div 2 = 4 : 7$

(b) $1\frac{1}{2} = \frac{3}{2}$ and $2\frac{1}{4} = \frac{9}{4}$ , so the ratio is $\frac{3}{2} : \frac{9}{4}$ .

The LCM of 2 and 4 is 4. Multiply every term by 4: $\frac{3}{2} \times 4 : \frac{9}{4} \times 4 = 6 : 9$

The HCF of 6 and 9 is 3: $6 \div 3 : 9 \div 3 = 2 : 3$

The LCM of 4 and 3 is 12. Multiply every term by 12: $\frac{9}{4} \times 12 : \frac{5}{3} \times 12 = 27 : 20$

$27 = 3^{3}$ and $20 = 2^{2} \times 5$ have no common prime factor, so the ratio is in its simplest form.

(a) $1\frac{3}{5} : 2\frac{4}{5} = 4 : 7$
(b) $1\frac{1}{2} : 2\frac{1}{4} = 2 : 3$
(c) $2\frac{1}{4} : 1\frac{2}{3} = 27 : 20$

A Fraction and a Decimal

When one term is a fraction and the other is a decimal, write both terms the same way first: both as decimals or both as fractions. Then use the rule for that kind of ratio. Use decimals only when the fraction is a terminating decimal, such as $\frac{3}{4} = 0.75$ . A fraction such as $\frac{2}{3} = 0.666\ldots$ is a recurring decimal, and a rounded value such as 0.67 would give a wrong ratio, so change the decimal to a fraction instead. The same choice is open when the other term is a whole number: in Example 8 (a), $\frac{4}{5} : 1 = 0.8 : 1 = 8 : 10 = 4 : 5$ .

Example 10: A fraction and a decimal

Express each ratio in its simplest form.
(a) $0.6 : \frac{3}{4}$
(b) $0.5 : \frac{2}{3}$
(c) $0.75 : 1\frac{1}{2}$

Solution

(a) As decimals: $\frac{3}{4} = 0.75$ , so the ratio is $0.6 : 0.75$ . Multiply every term by 100: $60 : 75$ . The HCF of 60 and 75 is 15: $60 \div 15 : 75 \div 15 = 4 : 5$

As fractions: $0.6 = \frac{6}{10} = \frac{3}{5}$ , so the ratio is $\frac{3}{5} : \frac{3}{4}$ . The LCM of 5 and 4 is 20: $\frac{3}{5} \times 20 : \frac{3}{4} \times 20 = 12 : 15$ . The HCF of 12 and 15 is 3: $12 \div 3 : 15 \div 3 = 4 : 5$ . Both ways give the same ratio.

(b) $\frac{2}{3}$ is a recurring decimal, so use fractions: $0.5 = \frac{1}{2}$ , and the ratio is $\frac{1}{2} : \frac{2}{3}$ .

The LCM of 2 and 3 is 6. Multiply every term by 6: $\frac{1}{2} \times 6 : \frac{2}{3} \times 6 = 3 : 4$

The HCF of 75 and 150 is 75: $75 \div 75 : 150 \div 75 = 1 : 2$

(a) $0.6 : \frac{3}{4} = 4 : 5$
(b) $0.5 : \frac{2}{3} = 3 : 4$
(c) $0.75 : 1\frac{1}{2} = 1 : 2$

Ratios in Word Problems

In a word problem, check first that the quantities are in the same unit of measurement, and write the terms in the order of the words. Then make every term a whole number and simplify.

Example 11: Two distances

Ravi jogs 2.4 km and Ali jogs $1\frac{4}{5}$ km. Find the ratio of
(a) the distance Ravi jogs to the distance Ali jogs,
(b) the distance Ali jogs to the total distance the two boys jog.

Solution

Both distances are in kilometres. One is a decimal and the other is a mixed number, so write both as decimals: $1\frac{4}{5} = 1.8$

(a) Ravi is named first: $2.4 : 1.8$ . Multiply every term by 10: $24 : 18$ . The HCF of 24 and 18 is 6: $24 \div 6 : 18 \div 6 = 4 : 3$

(b) The total distance is $2.4 + 1.8 = 4.2$ km. Ali is named first: $1.8 : 4.2$ . Multiply every term by 10: $18 : 42$ . The HCF of 18 and 42 is 6: $18 \div 6 : 42 \div 6 = 3 : 7$

Check with a bar model. From (a), Ravi's distance is 4 units and Ali's is 3 units, so the total is $4 + 3 = 7$ units, and Ali's distance to the total is $3 : 7$ .

(a) The ratio of the distance Ravi jogs to the distance Ali jogs is $4 : 3$ .
(b) The ratio of the distance Ali jogs to the total distance is $3 : 7$ .

Common Mistakes

Multiplying the terms by different numbers. $0.4 : 0.12$ is not $4 : 12$ . That multiplies the first term by 10 and the second by 100. Multiply both by 100: $40 : 12 = 10 : 3$ . In the same way, $\frac{3}{5} : 2$ is not $3 : 2$ , because the 2 has not been multiplied by 5. It is $3 : 10$ .
Clearing the decimals and forgetting to simplify. $1.6 : 2.4 = 16 : 24$ is not finished, because 16 and 24 have the common factor 8. In its simplest form the ratio is $2 : 3$ .
Multiplying by the product of the denominators and stopping there. For $\frac{3}{4} : \frac{5}{6}$ , multiplying every term by $4 \times 6 = 24$ gives $18 : 20$ . This is an equivalent ratio, but it is not in its simplest form: dividing by 2 gives $9 : 10$ . Multiplying by the LCM, 12, gives $9 : 10$ straight away. Whichever number you multiply by, look for a common factor afterwards.
Splitting a mixed number. In $1\frac{1}{2} : 2\frac{1}{4}$ , comparing the whole numbers gives $1 : 2$ and comparing the fraction parts gives $\frac{1}{2} : \frac{1}{4} = 2 : 1$ . Neither is right. Change to improper fractions first: $\frac{3}{2} : \frac{9}{4} = 6 : 9 = 2 : 3$ .

Practice

Work each question on paper before you open its solution.

Question 1

Express each ratio in its simplest form.
(a) $0.3 : 0.9$
(b) $1.2 : 2.8$
(c) $0.25 : 0.35$
(d) $4.5 : 6$

ShowHide solution

(a) Each term has 1 decimal place. Multiply every term by 10: $3 : 9$ . The HCF of 3 and 9 is 3: $3 \div 3 : 9 \div 3 = 1 : 3$

(b) Multiply every term by 10: $12 : 28$ . The HCF of 12 and 28 is 4: $12 \div 4 : 28 \div 4 = 3 : 7$

(c) Each term has 2 decimal places. Multiply every term by 100: $25 : 35$ . The HCF of 25 and 35 is 5: $25 \div 5 : 35 \div 5 = 5 : 7$

(d) 4.5 has 1 decimal place. Multiply every term by 10, the whole number 6 as well: $45 : 60$ . The HCF of 45 and 60 is 15: $45 \div 15 : 60 \div 15 = 3 : 4$

(a) $0.3 : 0.9 = 1 : 3$
(b) $1.2 : 2.8 = 3 : 7$
(c) $0.25 : 0.35 = 5 : 7$
(d) $4.5 : 6 = 3 : 4$

Question 2

Express each ratio in its simplest form.
(a) $0.9 : 0.24$
(b) $0.28 : 0.182$
(c) $1.05 : 0.7$
(d) $0.125 : 2$

ShowHide solution

(a) 0.24 has 2 decimal places, the most. Multiply every term by 100: $0.9 \times 100 : 0.24 \times 100 = 90 : 24$ . The HCF of 90 and 24 is 6: $90 \div 6 : 24 \div 6 = 15 : 4$

(b) 0.182 has 3 decimal places, the most. Multiply every term by 1000: $0.28 \times 1000 : 0.182 \times 1000 = 280 : 182$

$280 = 2^{3} \times 5 \times 7$ and $182 = 2 \times 7 \times 13$ , so the HCF is $2 \times 7 = 14$ : $280 \div 14 : 182 \div 14 = 20 : 13$

(c) 1.05 has 2 decimal places, the most. Multiply every term by 100: $1.05 \times 100 : 0.7 \times 100 = 105 : 70$ . The HCF of 105 and 70 is 35: $105 \div 35 : 70 \div 35 = 3 : 2$

(d) 0.125 has 3 decimal places. Multiply every term by 1000, the whole number 2 as well: $125 : 2000$ . The HCF of 125 and 2000 is 125: $125 \div 125 : 2000 \div 125 = 1 : 16$

(a) $0.9 : 0.24 = 15 : 4$
(b) $0.28 : 0.182 = 20 : 13$
(c) $1.05 : 0.7 = 3 : 2$
(d) $0.125 : 2 = 1 : 16$

Question 3

Express each ratio in its simplest form.
(a) $\frac{5}{12} : \frac{7}{12}$
(b) $\frac{9}{10} : \frac{3}{10}$
(c) $\frac{4}{15} : \frac{14}{15}$

ShowHide solution

(a) The denominators are the same. Multiply every term by 12: $5 : 7$ . 5 and 7 have no common factor other than 1.

(b) Multiply every term by 10: $9 : 3$ . The HCF of 9 and 3 is 3: $9 \div 3 : 3 \div 3 = 3 : 1$

(a) $\frac{5}{12} : \frac{7}{12} = 5 : 7$
(b) $\frac{9}{10} : \frac{3}{10} = 3 : 1$
(c) $\frac{4}{15} : \frac{14}{15} = 2 : 7$

Question 4

Express each ratio in its simplest form.
(a) $\frac{1}{2} : \frac{4}{5}$
(b) $\frac{3}{5} : \frac{7}{10}$
(c) $\frac{5}{6} : \frac{3}{8}$
(d) $\frac{4}{9} : \frac{2}{3}$

ShowHide solution

(a) The LCM of 2 and 5 is 10. Multiply every term by 10: $\frac{1}{2} \times 10 : \frac{4}{5} \times 10 = 5 : 8$ . 5 and 8 have no common factor other than 1.

(b) The LCM of 5 and 10 is 10. Multiply every term by 10: $\frac{3}{5} \times 10 : \frac{7}{10} \times 10 = 6 : 7$

(c) The LCM of 6 and 8 is 24. Multiply every term by 24: $\frac{5}{6} \times 24 : \frac{3}{8} \times 24 = 20 : 9$ . 20 and 9 have no common factor other than 1.

(d) The LCM of 9 and 3 is 9. Multiply every term by 9: $\frac{4}{9} \times 9 : \frac{2}{3} \times 9 = 4 : 6$ . The HCF of 4 and 6 is 2: $4 \div 2 : 6 \div 2 = 2 : 3$

(a) $\frac{1}{2} : \frac{4}{5} = 5 : 8$
(b) $\frac{3}{5} : \frac{7}{10} = 6 : 7$
(c) $\frac{5}{6} : \frac{3}{8} = 20 : 9$
(d) $\frac{4}{9} : \frac{2}{3} = 2 : 3$

Question 5

Find the ratio $\frac{5}{8} : \frac{3}{4}$ in two ways:
(a) by multiplying every term by the LCM of the denominators,
(b) by dividing the first term by the second.

ShowHide solution

(a) 8 is a multiple of 4, so the LCM of 8 and 4 is 8. Multiply every term by 8: $\frac{5}{8} \times 8 : \frac{3}{4} \times 8 = 5 : 6$

(b) $\frac{5}{8} \div \frac{3}{4} = \frac{5}{8} \times \frac{4}{3} = \frac{20}{24} = \frac{5}{6}$

The first term is $\frac{5}{6}$ of the second, so the ratio is $5 : 6$ . The two ways agree.

$\frac{5}{8} : \frac{3}{4} = 5 : 6$

Question 6

Express each ratio in its simplest form.
(a) $\frac{2}{3} : 4$
(b) $3 : \frac{3}{4}$
(c) $1 : \frac{7}{10}$

ShowHide solution

(a) Multiply every term by 3, the whole number 4 as well: $\frac{2}{3} \times 3 : 4 \times 3 = 2 : 12$ . The HCF of 2 and 12 is 2: $2 \div 2 : 12 \div 2 = 1 : 6$

(b) Multiply every term by 4: $3 \times 4 : \frac{3}{4} \times 4 = 12 : 3$ . The HCF of 12 and 3 is 3: $12 \div 3 : 3 \div 3 = 4 : 1$

(c) Multiply every term by 10: $1 \times 10 : \frac{7}{10} \times 10 = 10 : 7$ . 10 and 7 have no common factor other than 1.

(a) $\frac{2}{3} : 4 = 1 : 6$
(b) $3 : \frac{3}{4} = 4 : 1$
(c) $1 : \frac{7}{10} = 10 : 7$

Question 7

Express each ratio in its simplest form.
(a) $2\frac{1}{2} : 3\frac{1}{2}$
(b) $1\frac{1}{3} : 2\frac{2}{9}$
(c) $3\frac{3}{4} : 1\frac{1}{6}$
(d) $4 : 1\frac{3}{5}$

ShowHide solution

(a) $2\frac{1}{2} = \frac{5}{2}$ and $3\frac{1}{2} = \frac{7}{2}$ . Multiply every term by 2: $\frac{5}{2} \times 2 : \frac{7}{2} \times 2 = 5 : 7$

(b) $1\frac{1}{3} = \frac{4}{3}$ and $2\frac{2}{9} = \frac{20}{9}$ . The LCM of 3 and 9 is 9: $\frac{4}{3} \times 9 : \frac{20}{9} \times 9 = 12 : 20$ . The HCF of 12 and 20 is 4: $12 \div 4 : 20 \div 4 = 3 : 5$

(c) $3\frac{3}{4} = \frac{15}{4}$ and $1\frac{1}{6} = \frac{7}{6}$ . The LCM of 4 and 6 is 12: $\frac{15}{4} \times 12 : \frac{7}{6} \times 12 = 45 : 14$ . $45 = 3^{2} \times 5$ and $14 = 2 \times 7$ have no common prime factor.

(d) $1\frac{3}{5} = \frac{8}{5}$ . Multiply every term by 5, the whole number 4 as well: $4 \times 5 : \frac{8}{5} \times 5 = 20 : 8$ . The HCF of 20 and 8 is 4: $20 \div 4 : 8 \div 4 = 5 : 2$

(a) $2\frac{1}{2} : 3\frac{1}{2} = 5 : 7$
(b) $1\frac{1}{3} : 2\frac{2}{9} = 3 : 5$
(c) $3\frac{3}{4} : 1\frac{1}{6} = 45 : 14$
(d) $4 : 1\frac{3}{5} = 5 : 2$

Question 8

Express each ratio in its simplest form.
(a) $0.4 : \frac{3}{5}$
(b) $\frac{5}{6} : 0.25$
(c) $1.2 : 2\frac{2}{5}$
(d) $1\frac{1}{3} : 0.8$

ShowHide solution

(a) $\frac{3}{5} = 0.6$ , so the ratio is $0.4 : 0.6$ . Multiply every term by 10: $4 : 6$ . The HCF of 4 and 6 is 2: $4 \div 2 : 6 \div 2 = 2 : 3$

(b) $\frac{5}{6}$ is a recurring decimal, so use fractions: $0.25 = \frac{1}{4}$ , and the ratio is $\frac{5}{6} : \frac{1}{4}$ . The LCM of 6 and 4 is 12: $\frac{5}{6} \times 12 : \frac{1}{4} \times 12 = 10 : 3$

(c) $2\frac{2}{5} = 2.4$ , so the ratio is $1.2 : 2.4$ . Multiply every term by 10: $12 : 24$ . The HCF of 12 and 24 is 12: $12 \div 12 : 24 \div 12 = 1 : 2$

(d) $1\frac{1}{3} = \frac{4}{3}$ is a recurring decimal, so use fractions: $0.8 = \frac{8}{10} = \frac{4}{5}$ , and the ratio is $\frac{4}{3} : \frac{4}{5}$ . The LCM of 3 and 5 is 15: $\frac{4}{3} \times 15 : \frac{4}{5} \times 15 = 20 : 12$ . The HCF of 20 and 12 is 4: $20 \div 4 : 12 \div 4 = 5 : 3$

(a) $0.4 : \frac{3}{5} = 2 : 3$
(b) $\frac{5}{6} : 0.25 = 10 : 3$
(c) $1.2 : 2\frac{2}{5} = 1 : 2$
(d) $1\frac{1}{3} : 0.8 = 5 : 3$

Question 9

At her drinks stall, Mrs Goh makes bandung by mixing 0.35 litres of rose syrup with 1.4 litres of milk. Find the ratio of
(a) the volume of rose syrup to the volume of milk,
(b) the volume of milk to the volume of bandung she makes.

ShowHide solution

Both volumes are in litres.

(a) The rose syrup is named first: $0.35 : 1.4$ . 0.35 has 2 decimal places, the most, so multiply every term by 100: $0.35 \times 100 : 1.4 \times 100 = 35 : 140$

The HCF of 35 and 140 is 35: $35 \div 35 : 140 \div 35 = 1 : 4$

(b) The volume of bandung is $0.35 + 1.4 = 1.75$ litres. The milk is named first: $1.4 : 1.75$ . Multiply every term by 100: $140 : 175$

The HCF of 140 and 175 is 35: $140 \div 35 : 175 \div 35 = 4 : 5$

(a) The ratio of the volume of rose syrup to the volume of milk is $1 : 4$ .
(b) The ratio of the volume of milk to the volume of bandung is $4 : 5$ .

Question 10

Mei Ling spends $\frac{2}{5}$ of her pocket money on food and $\frac{1}{4}$ of it on transport. She saves the rest. Find the ratio of
(a) the money she spends on food to the money she spends on transport,
(b) the money she spends on food to the money she saves.

ShowHide solution

The fractions are all fractions of the same amount, her pocket money, so they can be compared by a ratio.

(a) Food is named first: $\frac{2}{5} : \frac{1}{4}$ . The LCM of 5 and 4 is 20. Multiply every term by 20: $\frac{2}{5} \times 20 : \frac{1}{4} \times 20 = 8 : 5$

(b) The fraction she saves is $1 - \frac{2}{5} - \frac{1}{4} = \frac{20}{20} - \frac{8}{20} - \frac{5}{20} = \frac{7}{20}$

Food is named first: $\frac{2}{5} : \frac{7}{20}$ . The LCM of 5 and 20 is 20. Multiply every term by 20: $\frac{2}{5} \times 20 : \frac{7}{20} \times 20 = 8 : 7$

Check: in twentieths of her pocket money, food is 8, transport is 5 and savings are 7, and $8 + 5 + 7 = 20$ .

(a) The ratio of the money spent on food to the money spent on transport is $8 : 5$ .
(b) The ratio of the money spent on food to the money saved is $8 : 7$ .

Question 11

Three friends each complete the same jigsaw puzzle. Ali takes $1\frac{1}{4}$ hours, Ravi takes 0.75 hours and Siti takes $\frac{2}{3}$ of an hour. Find the ratio of
(a) the time Ali takes to the time Ravi takes,
(b) the time Ravi takes to the time Siti takes.

ShowHide solution

All three times are in hours.

(a) Write both terms as decimals: $1\frac{1}{4} = 1.25$ , so the ratio is $1.25 : 0.75$ . Multiply every term by 100: $125 : 75$

The HCF of 125 and 75 is 25: $125 \div 25 : 75 \div 25 = 5 : 3$

(b) $\frac{2}{3}$ is a recurring decimal, so write both terms as fractions: $0.75 = \frac{3}{4}$ , and the ratio is $\frac{3}{4} : \frac{2}{3}$ .

The LCM of 4 and 3 is 12. Multiply every term by 12: $\frac{3}{4} \times 12 : \frac{2}{3} \times 12 = 9 : 8$

Check in minutes: the times are 75 minutes, 45 minutes and 40 minutes. $75 : 45 = 5 : 3$ and $45 : 40 = 9 : 8$ .

(a) The ratio of the time Ali takes to the time Ravi takes is $5 : 3$ .
(b) The ratio of the time Ravi takes to the time Siti takes is $9 : 8$ .

Question 12

Each of these has a mistake. Say what the mistake is and give the correct ratio.
(a) Ravi writes $\frac{1}{3} : \frac{1}{4} = 3 : 4$ .
(b) Hui Min writes $0.5 : 0.25 = 5 : 25 = 1 : 5$ .

ShowHide solution

(a) Ravi has used the denominators as the terms. The LCM of 3 and 4 is 12, and every term must be multiplied by it: $\frac{1}{3} \times 12 : \frac{1}{4} \times 12 = 4 : 3$

A check shows that $3 : 4$ cannot be right: $\frac{1}{3}$ is greater than $\frac{1}{4}$ , so the first term must be the greater one.

(b) Hui Min has multiplied 0.5 by 10 and 0.25 by 100. Every term must be multiplied by the same number. 0.25 has 2 decimal places, so multiply both by 100: $0.5 \times 100 : 0.25 \times 100 = 50 : 25$

The HCF of 50 and 25 is 25: $50 \div 25 : 25 \div 25 = 2 : 1$ . Check: 0.5 is twice 0.25.

(a) $\frac{1}{3} : \frac{1}{4} = 4 : 3$
(b) $0.5 : 0.25 = 2 : 1$

Key Points

Multiplying every term of a ratio by the same non-zero number gives an equivalent ratio. Use this to make every term a whole number, then divide every term by their HCF.
Decimals: multiply every term by 10, 100 or 1000, enough to clear the term with the most decimal places.
Fractions: multiply every term by the LCM of the denominators. When the denominators are the same, the ratio is the ratio of the numerators.
Dividing the first term by the second gives the same ratio: write the answer as a fraction in its simplest form. Its numerator is the first term and its denominator is the second.
A whole number in the ratio is multiplied by the same number as the other term.
Mixed numbers: change them to improper fractions first.
A fraction and a decimal: write both terms as decimals or both as fractions first. Use fractions when the decimal would recur.
The quantities must be in the same unit of measurement, the terms follow the order of the words, and the final ratio has whole-number terms with no common factor other than 1.