Ratios of Quantities in Different Units

Finding the ratio of two quantities that are measured in different units, such as 40 cm and 2 m: change them to the same unit first, then simplify.

Before you start

A ratio compares two quantities of the same kind. Its terms are written in the order of the words, and a ratio has no unit of measurement: in a class of 18 boys and 24 girls, the ratio of boys to girls is $18 : 24$ .

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 highest common factor (HCF) of the terms.

The HCF of 18 and 24 is 6, so $18 : 24 = 18 \div 6 : 24 \div 6 = 3 : 4$ .

The Same Unit First

Ali has a ribbon 40 cm long. Siti has a ribbon 2 m long. What is the ratio of the length of Ali's ribbon to the length of Siti's ribbon?

The two lengths are given in different units of measurement: centimetres and metres. Suppose we write the two numbers down as they stand and simplify.

Wrong

$40 : 2 = 20 : 1$

This says that Ali's ribbon is 20 times as long as Siti's. But 2 m is much longer than 40 cm, so the answer cannot be right. The 40 counts centimetres and the 2 counts metres. The two numbers count different things, so they cannot be compared.

Right

Change both lengths to the same unit of measurement first. 1 m = 100 cm, so 2 m = $2 \times 100 = 200$ cm. Now both numbers count centimetres.

40 cm and 200 cm give the ratio $40 : 200 = 1 : 5$

The last step divides both terms by their HCF, 40. The ratio $1 : 5$ says that Siti's ribbon is 5 times as long as Ali's, which agrees with the two lengths.

The bar models below show what each answer says. In a bar model, a "unit" is one of the equal parts of a bar. It is not a unit of measurement.

What $20 : 1$ says: a bar of 20 units for Ali's ribbon and a bar of 1 unit for Siti's. The bar for 40 cm comes out 20 times as long as the bar for 2 m, which is not how the ribbons are.

What $1 : 5$ says: a bar of 1 unit for Ali's ribbon and a bar of 5 units for Siti's. Each unit of the bar model stands for 40 cm, so Siti's bar stands for $5 \times 40 = 200$ cm, which is 2 m.

Once the units of measurement are the same, they are left out of the ratio: 40 centimetres compared with 200 centimetres is the same comparison as 40 compared with 200. This is why a ratio has no unit of measurement. In your working, write the two quantities with their units first (40 cm and 200 cm), and then the ratio without them ( $40 : 200$ ).

Two quantities can be changed to the same unit of measurement only when they are of the same kind: two lengths, two masses, two volumes, two lengths of time or two amounts of money. A volume of 300 ml and a mass of 200 g cannot be changed to the same unit, so they cannot be compared by a ratio.

Rule: the ratio of two quantities in different units

Change the quantities to the same unit of measurement.
Write the two numbers as a ratio, in the order of the words, leaving out the unit of measurement.
Write the ratio in its simplest form: divide every term by the HCF of the terms.

Do not write the numbers as a ratio, and do not simplify, until the units of measurement are the same.

Changing Units of Measurement

These are the units of measurement used in this lesson, and how each larger unit is changed to the smaller one. You met them in primary school.

Kind of quantity	Conversion	To change the larger unit to the smaller	For example
Length	1 cm = 10 mm	multiply by 10	3 cm = $3 \times 10 = 30$ mm
Length	1 m = 100 cm	multiply by 100	2 m = $2 \times 100 = 200$ cm
Length	1 km = 1000 m	multiply by 1000	4 km = $4 \times 1000 = 4000$ m
Mass	1 kg = 1000 g	multiply by 1000	1.5 kg = $1.5 \times 1000 = 1500$ g
Volume	1 litre = 1000 ml	multiply by 1000	2 litres = $2 \times 1000 = 2000$ ml
Time	1 h = 60 min	multiply by 60	2 h = $2 \times 60 = 120$ min
Time	1 min = 60 s	multiply by 60	3 min = $3 \times 60 = 180$ s
Money	$1 = 100 cents	multiply by 100	$4 = $4 \times 100 = 400$ cents

To change a quantity to a smaller unit of measurement, multiply: it takes more of the smaller unit to make the same amount. To change it to a larger unit, divide. Take care with time. Hours, minutes and seconds go in 60s, while the other units go in 10s, 100s or 1000s.

Example 1: Length, mass and volume

Write each ratio in its simplest form.
(a) 45 cm to 3 m
(b) 2 kg to 800 g
(c) 350 ml to 2 litres

Solution

(a) The units of measurement are centimetres and metres. Change the metres to centimetres: 3 m = $3 \times 100 = 300$ cm.

45 cm and 300 cm give the ratio $45 : 300$ . The HCF of 45 and 300 is 15, so $45 : 300 = 45 \div 15 : 300 \div 15 = 3 : 20$ .

(b) Change the kilograms to grams: 2 kg = $2 \times 1000 = 2000$ g.

The 2 kg comes first in the question, so 2000 g comes first: 2000 g and 800 g give the ratio $2000 : 800$ . The HCF of 2000 and 800 is 400, so $2000 : 800 = 5 : 2$ .

You may also divide by common factors in steps: $2000 : 800 = 20 : 8$ (dividing both terms by 100) $= 5 : 2$ (dividing both terms by 4).

350 ml and 2000 ml give the ratio $350 : 2000$ . The HCF of 350 and 2000 is 50, so $350 : 2000 = 7 : 40$ .

(a) The ratio of 45 cm to 3 m is $3 : 20$ .
(b) The ratio of 2 kg to 800 g is $5 : 2$ .
(c) The ratio of 350 ml to 2 litres is $7 : 40$ .

Example 2: Time and money

Write each ratio in its simplest form.
(a) 36 s to 2 min
(b) 1.5 h to 50 min
(c) 60 cents to $3

Solution

(a) 1 min = 60 s, so 2 min = $2 \times 60 = 120$ s.

36 s and 120 s give the ratio $36 : 120$ . The HCF of 36 and 120 is 12, so $36 : 120 = 3 : 10$ .

(b) 1 h = 60 min, so 1.5 h = $1.5 \times 60 = 90$ min. It is not 150 min, because an hour has 60 minutes, not 100.

90 min and 50 min give the ratio $90 : 50$ . The HCF of 90 and 50 is 10, so $90 : 50 = 9 : 5$ .

60 cents and 300 cents give the ratio $60 : 300$ . The HCF of 60 and 300 is 60, so $60 : 300 = 1 : 5$ .

(a) The ratio of 36 s to 2 min is $3 : 10$ .
(b) The ratio of 1.5 h to 50 min is $9 : 5$ .
(c) The ratio of 60 cents to $3 is $1 : 5$ .

Which Unit to Choose

In every example so far, the quantity in the larger unit of measurement was changed to the smaller unit. The next example shows why.

Example 3: Choosing the unit

Two bags of sugar, A and B, have masses of 750 g and 1.5 kg. Find the ratio of the mass of bag B to the mass of bag A in its simplest form.

Solution

The units of measurement are grams and kilograms. The gram is the smaller one, so change the kilograms to grams: 1.5 kg = $1.5 \times 1000 = 1500$ g.

The question asks for B to A, so the mass of bag B comes first: 1500 g and 750 g give the ratio $1500 : 750$ .

$1500 = 2 \times 750$ , so 750 is a factor of both terms and the HCF is 750.

$1500 : 750 = 1500 \div 750 : 750 \div 750 = 2 : 1$

The ratio of the mass of bag B to the mass of bag A is $2 : 1$ .

We could have changed the grams to kilograms instead: 750 g = $750 \div 1000 = 0.75$ kg. The table shows both choices.

Unit chosen	Bag B	Bag A	Ratio of B to A
grams (the smaller unit)	1500 g	750 g	$1500 : 750 = 2 : 1$
kilograms (the larger unit)	1.5 kg	0.75 kg	$1.5 : 0.75$

Both ratios are correct, and they are equivalent. But $1.5 : 0.75$ is not in its simplest form, because its terms are not whole numbers. More work is needed to turn it into $2 : 1$ , and that work is taught in the lesson on ratios with fractions and decimals. The grams gave whole numbers at once.

With time, the larger unit often gives a fraction instead of a decimal: 45 min is $\frac{45}{60} = \frac{3}{4}$ h.

Rule: choosing the unit

Change every quantity to the smallest unit of measurement in the question. You then multiply and never divide, so a whole number stays a whole number.

In this lesson a quantity given as a decimal or a fraction, such as 1.5 kg or $\frac{3}{4}$ h, also becomes a whole number in the smaller unit (1500 g and 45 min). So when you choose the smallest unit, every ratio here has whole-number terms before you simplify it.

Quantities Written with Two Units

A quantity is often written with two units of measurement together, such as 1 h 20 min or 2 m 5 cm. Change the part in the larger unit to the smaller unit, then add the part that is already in the smaller unit.

1 h 20 min = 60 min + 20 min = 80 min

2 m 5 cm = 200 cm + 5 cm = 205 cm

3 kg 50 g = 3000 g + 50 g = 3050 g

An amount of money such as $2.40 is of the same sort: it is 2 dollars and 40 cents, so $2.40 = 200 cents + 40 cents = 240 cents. The quick way is to multiply the number of dollars by 100: $2.40 \times 100 = 240$ .

Example 4: One quantity with two units

Write each ratio in its simplest form.
(a) 1 h 20 min to 45 min
(b) 35 cm to 2 m 5 cm
(c) $2.40 to 90 cents

Solution

(a) 1 h 20 min = 60 min + 20 min = 80 min.

80 min and 45 min give the ratio $80 : 45$ . The HCF of 80 and 45 is 5, so $80 : 45 = 16 : 9$ .

(b) 2 m 5 cm = 200 cm + 5 cm = 205 cm. It is not 250 cm, and it is not 25 cm.

35 cm and 205 cm give the ratio $35 : 205$ . The HCF of 35 and 205 is 5, so $35 : 205 = 7 : 41$ . Both 7 and 41 are prime numbers, so the ratio cannot be simplified further.

240 cents and 90 cents give the ratio $240 : 90$ . The HCF of 240 and 90 is 30, so $240 : 90 = 8 : 3$ .

(a) The ratio of 1 h 20 min to 45 min is $16 : 9$ .
(b) The ratio of 35 cm to 2 m 5 cm is $7 : 41$ .
(c) The ratio of $2.40 to 90 cents is $8 : 3$ .

Example 5: Both quantities with two units

A watermelon has a mass of 4 kg 200 g. A papaya has a mass of 1 kg 350 g. Find the ratio of the mass of the watermelon to the mass of the papaya in its simplest form.

Solution

Change both masses to grams. Watermelon: 4 kg 200 g = 4000 g + 200 g = 4200 g. Papaya: 1 kg 350 g = 1000 g + 350 g = 1350 g.

Watermelon to papaya: 4200 g and 1350 g give the ratio $4200 : 1350$ .

Find the HCF by prime factorisation: $4200 = 2^{3} \times 3 \times 5^{2} \times 7$ and $1350 = 2 \times 3^{3} \times 5^{2}$ . The HCF is $2 \times 3 \times 5^{2} = 150$ .

$4200 : 1350 = 4200 \div 150 : 1350 \div 150 = 28 : 9$

Or divide by common factors in steps: $4200 : 1350 = 420 : 135$ (dividing both terms by 10) $= 84 : 27$ (dividing both terms by 5) $= 28 : 9$ (dividing both terms by 3).

The ratio of the mass of the watermelon to the mass of the papaya is $28 : 9$ .

Word Problems

In a word problem, first pick out the two quantities that are to be compared and the order in which the question names them. Sometimes one of them has to be worked out first. Then follow the three steps of the rule.

Example 6: Mixing a drink

Mrs Goh makes a jug of lime drink by mixing 250 ml of lime cordial with 2 litres of water. Find, in its simplest form, the ratio of
(a) the volume of cordial to the volume of water,
(b) the volume of cordial to the volume of the whole drink.

Solution

(a) Change the litres to millilitres: 2 litres = $2 \times 1000 = 2000$ ml.

Cordial to water: 250 ml and 2000 ml give the ratio $250 : 2000$ . Since $2000 = 8 \times 250$ , the HCF of 250 and 2000 is 250, so $250 : 2000 = 1 : 8$ .

The bar model shows this ratio. Each unit of the bar model stands for 250 ml. (The millilitre is the unit of measurement; the "1 unit" of the bar model is one equal part of a bar.)

(b) The whole drink is the cordial and the water together: 250 ml + 2000 ml = 2250 ml.

Cordial to the whole drink: 250 ml and 2250 ml give the ratio $250 : 2250 = 1 : 9$ , dividing both terms by 250. In the bar model, the whole drink is $1 + 8 = 9$ units and the cordial is 1 of them.

(a) The ratio of the volume of cordial to the volume of water is $1 : 8$ .
(b) The ratio of the volume of cordial to the volume of the whole drink is $1 : 9$ .

Example 7: Part of a concert

A school concert lasts 1 h 30 min. The choir sings for 24 min of that time. Find the ratio of the time for which the choir sings to the time taken by the rest of the concert, in its simplest form.

Solution

Change the length of the concert to minutes: 1 h 30 min = 60 min + 30 min = 90 min.

The rest of the concert takes $90 - 24 = 66$ min.

Choir to the rest: 24 min and 66 min give the ratio $24 : 66$ . The HCF of 24 and 66 is 6, so $24 : 66 = 4 : 11$ .

Check the two times: 24 min + 66 min = 90 min, which is 1 h 30 min.

The ratio of the time for which the choir sings to the time taken by the rest of the concert is $4 : 11$ .

Common Mistakes

Writing the ratio before the units are the same. The ratio of 40 cm to 2 m is not $40 : 2 = 20 : 1$ . Change 2 m to 200 cm first: $40 : 200 = 1 : 5$ .
Reading a quantity with two units as one number. 2 m 5 cm is 205 cm, not 250 cm or 25 cm. 1 h 20 min is 80 min, not 120 min.
Treating time as if it went in 100s. An hour has 60 minutes, so 1.5 h is 90 min, not 150 min, and 1 h 20 min is not 1.2 h.
Multiplying by the wrong number. 1 m is 100 cm, but 1 km is 1000 m and 1 kg is 1000 g.
Losing the order. The terms follow the order of the words, not the size of the quantities or of their units: the ratio of 2 kg to 800 g is $5 : 2$ , not $2 : 5$ .
Leaving a unit of measurement in the answer. Write $1 : 5$ , not "1 cm : 5 cm". A ratio has no unit of measurement.

Practice

Work each question on paper before you open its solution.

Question 1

Change each quantity to the unit of measurement shown.
(a) 3.2 m to centimetres
(b) 2 kg 40 g to grams
(c) 1 h 35 min to minutes
(d) $3.05 to cents

ShowHide solution

(a) 1 m = 100 cm, so 3.2 m = $3.2 \times 100 = 320$ cm.

(b) 2 kg = 2000 g, so 2 kg 40 g = 2000 g + 40 g = 2040 g.

(d) $1 = 100 cents, so $3.05 = $3.05 \times 100 = 305$ cents.

(a) 3.2 m = 320 cm
(b) 2 kg 40 g = 2040 g
(c) 1 h 35 min = 95 min
(d) $3.05 = 305 cents

Question 2

Write each ratio of lengths in its simplest form.
(a) 70 cm to 2 m
(b) 4 km to 500 m
(c) 8 mm to 6 cm

ShowHide solution

(a) 2 m = $2 \times 100 = 200$ cm. 70 cm and 200 cm give the ratio $70 : 200$ . The HCF of 70 and 200 is 10, so $70 : 200 = 7 : 20$ .

(b) 4 km = $4 \times 1000 = 4000$ m. The 4 km comes first, so 4000 m and 500 m give the ratio $4000 : 500$ . The HCF of 4000 and 500 is 500, so $4000 : 500 = 8 : 1$ .

(c) 6 cm = $6 \times 10 = 60$ mm. 8 mm and 60 mm give the ratio $8 : 60$ . The HCF of 8 and 60 is 4, so $8 : 60 = 2 : 15$ .

(a) $7 : 20$
(b) $8 : 1$
(c) $2 : 15$

Question 3

Write each ratio of masses or volumes in its simplest form.
(a) 150 g to 2 kg
(b) 1.2 kg to 900 g
(c) 400 ml to 1.5 litres
(d) 3 litres to 750 ml

ShowHide solution

(a) 2 kg = $2 \times 1000 = 2000$ g. 150 g and 2000 g give the ratio $150 : 2000$ . The HCF of 150 and 2000 is 50, so $150 : 2000 = 3 : 40$ .

(b) 1.2 kg = $1.2 \times 1000 = 1200$ g. 1200 g and 900 g give the ratio $1200 : 900$ . The HCF of 1200 and 900 is 300, so $1200 : 900 = 4 : 3$ .

(c) 1.5 litres = $1.5 \times 1000 = 1500$ ml. 400 ml and 1500 ml give the ratio $400 : 1500$ . The HCF of 400 and 1500 is 100, so $400 : 1500 = 4 : 15$ .

(d) 3 litres = $3 \times 1000 = 3000$ ml. 3000 ml and 750 ml give the ratio $3000 : 750$ . Since $3000 = 4 \times 750$ , the HCF is 750, so $3000 : 750 = 4 : 1$ .

(a) $3 : 40$
(b) $4 : 3$
(c) $4 : 15$
(d) $4 : 1$

Question 4

Write each ratio of times or amounts of money in its simplest form.
(a) 40 min to 2 h
(b) $\frac{3}{4}$ h to 20 min
(c) 45 s to 3 min
(d) 40 cents to $5
(e) $12 to 75 cents

ShowHide solution

(a) 2 h = $2 \times 60 = 120$ min. 40 min and 120 min give the ratio $40 : 120$ . The HCF of 40 and 120 is 40, so $40 : 120 = 1 : 3$ .

(b) $\frac{3}{4}$ h = $\frac{3}{4} \times 60 = 45$ min. 45 min and 20 min give the ratio $45 : 20$ . The HCF of 45 and 20 is 5, so $45 : 20 = 9 : 4$ .

(c) 3 min = $3 \times 60 = 180$ s. 45 s and 180 s give the ratio $45 : 180$ . Since $180 = 4 \times 45$ , the HCF is 45, so $45 : 180 = 1 : 4$ .

(d) $5 = $5 \times 100 = 500$ cents. 40 cents and 500 cents give the ratio $40 : 500$ . The HCF of 40 and 500 is 20, so $40 : 500 = 2 : 25$ .

(e) $12 = $12 \times 100 = 1200$ cents. 1200 cents and 75 cents give the ratio $1200 : 75$ . Since $1200 = 16 \times 75$ , the HCF is 75, so $1200 : 75 = 16 : 1$ .

(a) $1 : 3$
(b) $9 : 4$
(c) $1 : 4$
(d) $2 : 25$
(e) $16 : 1$

Question 5

Ravi says, "The ratio of 500 m to 2 km is $500 : 2$ , which is $250 : 1$ in its simplest form." Explain his mistake and find the correct ratio.

ShowHide solution

Ravi wrote the numbers as a ratio while the units of measurement were still different. The 500 counts metres and the 2 counts kilometres, so the two numbers cannot be compared.

His answer says that 500 m is 250 times as long as 2 km. But 2 km is the longer distance.

Change the kilometres to metres first: 2 km = $2 \times 1000 = 2000$ m.

500 m and 2000 m give the ratio $500 : 2000$ . Since $2000 = 4 \times 500$ , the HCF is 500, so $500 : 2000 = 1 : 4$ .

Ravi did not change the two distances to the same unit of measurement first. The ratio of 500 m to 2 km is $1 : 4$ .

Question 6

Write each ratio in its simplest form.
(a) 1 h 45 min to 50 min
(b) 60 cm to 1 m 8 cm
(c) $3.60 to 84 cents
(d) 2 kg 250 g to 1 kg 500 g
(e) 2 min 40 s to 1 min 12 s

ShowHide solution

(a) 1 h 45 min = 60 min + 45 min = 105 min. 105 min and 50 min give the ratio $105 : 50$ . The HCF of 105 and 50 is 5, so $105 : 50 = 21 : 10$ .

(b) 1 m 8 cm = 100 cm + 8 cm = 108 cm. 60 cm and 108 cm give the ratio $60 : 108$ . The HCF of 60 and 108 is 12, so $60 : 108 = 5 : 9$ .

(c) $3.60 = $3.60 \times 100 = 360$ cents. 360 cents and 84 cents give the ratio $360 : 84$ . The HCF of 360 and 84 is 12, so $360 : 84 = 30 : 7$ .

(d) 2 kg 250 g = 2000 g + 250 g = 2250 g and 1 kg 500 g = 1000 g + 500 g = 1500 g. 2250 g and 1500 g give the ratio $2250 : 1500$ . The HCF of 2250 and 1500 is 750, so $2250 : 1500 = 3 : 2$ .

(e) 2 min 40 s = 120 s + 40 s = 160 s and 1 min 12 s = 60 s + 12 s = 72 s. 160 s and 72 s give the ratio $160 : 72$ . The HCF of 160 and 72 is 8, so $160 : 72 = 20 : 9$ .

(a) $21 : 10$
(b) $5 : 9$
(c) $30 : 7$
(d) $3 : 2$
(e) $20 : 9$

Question 7

Hui Min and Ali each find the ratio of 800 m to 2 km. Hui Min changes both distances to metres. Ali changes both distances to kilometres.
(a) Write the ratio that each of them gets before simplifying.
(b) Whose ratio has whole-number terms? Write it in its simplest form.

ShowHide solution

(a) Hui Min: 2 km = $2 \times 1000 = 2000$ m, so 800 m and 2000 m give her the ratio $800 : 2000$ .

Ali: 800 m = $800 \div 1000 = 0.8$ km, so 0.8 km and 2 km give him the ratio $0.8 : 2$ .

(b) Hui Min's ratio has whole-number terms. The HCF of 800 and 2000 is 400, so $800 : 2000 = 2 : 5$ .

Ali's ratio is correct too, but it is not in its simplest form, because 0.8 is not a whole number. Changing to the smaller unit of measurement, the metre, avoided the decimal.

(a) Hui Min gets $800 : 2000$ and Ali gets $0.8 : 2$ .
(b) Hui Min's ratio has whole-number terms. In its simplest form, the ratio of 800 m to 2 km is $2 : 5$ .

Question 8

At a school canteen, a plate of chicken rice costs $2.80 and a cup of barley drink costs 80 cents. Find the ratio of the price of the chicken rice to the price of the barley drink in its simplest form.

ShowHide solution

Change the dollars to cents: $2.80 = $2.80 \times 100 = 280$ cents.

Chicken rice to barley drink: 280 cents and 80 cents give the ratio $280 : 80$ .

The HCF of 280 and 80 is 40, so $280 : 80 = 280 \div 40 : 80 \div 40 = 7 : 2$ .

The ratio of the price of the chicken rice to the price of the barley drink is $7 : 2$ .

Question 9

Siti spent 1 h 20 min on her maths homework and 50 min on her science homework. Find, in its simplest form, the ratio of
(a) the time she spent on maths to the time she spent on science,
(b) the time she spent on science to the total time she spent on both.

ShowHide solution

(a) 1 h 20 min = 60 min + 20 min = 80 min.

Maths to science: 80 min and 50 min give the ratio $80 : 50$ . The HCF of 80 and 50 is 10, so $80 : 50 = 8 : 5$ .

(b) Total time = 80 min + 50 min = 130 min.

Science to total: 50 min and 130 min give the ratio $50 : 130$ . The HCF of 50 and 130 is 10, so $50 : 130 = 5 : 13$ .

(a) The ratio of the time spent on maths to the time spent on science is $8 : 5$ .
(b) The ratio of the time spent on science to the total time is $5 : 13$ .

Question 10

A blue whale is 29.9 m long. A great white shark is 490 cm long. Find the ratio of the length of the blue whale to the length of the great white shark in its simplest form.

ShowHide solution

Change the metres to centimetres: 29.9 m = $29.9 \times 100 = 2990$ cm.

Blue whale to great white shark: 2990 cm and 490 cm give the ratio $2990 : 490$ .

Find the HCF by prime factorisation: $2990 = 2 \times 5 \times 13 \times 23$ and $490 = 2 \times 5 \times 7^{2}$ . The HCF is $2 \times 5 = 10$ .

$2990 : 490 = 2990 \div 10 : 490 \div 10 = 299 : 49$

Since $299 = 13 \times 23$ and $49 = 7 \times 7$ , the two terms have no common factor other than 1, so the ratio is in its simplest form.

The ratio of the length of the blue whale to the length of the great white shark is $299 : 49$ .

Question 11

A rectangular vegetable plot in a school garden has a length of 4 m 20 cm and a breadth of 90 cm. Find, in its simplest form, the ratio of
(a) the length of the plot to its breadth,
(b) the breadth of the plot to its perimeter.

ShowHide solution

(a) 4 m 20 cm = 400 cm + 20 cm = 420 cm.

Length to breadth: 420 cm and 90 cm give the ratio $420 : 90$ . The HCF of 420 and 90 is 30, so $420 : 90 = 14 : 3$ .

(b) Perimeter = $2 \times (420 + 90) = 2 \times 510 = 1020$ cm.

Breadth to perimeter: 90 cm and 1020 cm give the ratio $90 : 1020$ . The HCF of 90 and 1020 is 30, so $90 : 1020 = 3 : 34$ .

(a) The ratio of the length of the plot to its breadth is $14 : 3$ .
(b) The ratio of the breadth of the plot to its perimeter is $3 : 34$ .

Question 12

A jug holds 1.25 litres of barley drink. Mr Tan pours 350 ml of it into a glass. Find the ratio of the volume of drink in the glass to the volume of drink left in the jug, in its simplest form.

ShowHide solution

Change the litres to millilitres: 1.25 litres = $1.25 \times 1000 = 1250$ ml.

Volume left in the jug = $1250 - 350 = 900$ ml.

Glass to jug: 350 ml and 900 ml give the ratio $350 : 900$ . The HCF of 350 and 900 is 50, so $350 : 900 = 7 : 18$ .

Check the two volumes: 350 ml + 900 ml = 1250 ml, which is 1.25 litres.

The ratio of the volume of drink in the glass to the volume of drink left in the jug is $7 : 18$ .

Key Points

The terms of a ratio must count the same unit of measurement. Change the quantities to the same unit before you write the ratio.
Then write the numbers as a ratio in the order of the words, without the unit of measurement, and divide every term by the HCF of the terms to write the ratio in its simplest form.
Change to the smallest unit of measurement in the question, so that you multiply and the terms are whole numbers. Changing to the larger unit often brings in a decimal or a fraction.
A quantity with two units is changed part by part: 1 h 20 min = 60 min + 20 min = 80 min, 2 m 5 cm = 205 cm and $2.40 = 240 cents.
Conversions: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m, 1 kg = 1000 g, 1 litre = 1000 ml, 1 h = 60 min, 1 min = 60 s, $1 = 100 cents.
The answer has no unit of measurement: the ratio of 40 cm to 2 m is $1 : 5$ .