Dividing a Quantity in a Given Ratio

Sharing money, lengths, mixtures and angles in a given ratio: count the units, find the value of 1 unit, find each share, and check that the shares add up to the whole.

Before you start

A ratio such as $3 : 5$ can be drawn as a bar model: a bar of 3 equal units and a bar of 5 equal units. In this lesson a unit means one of these equal parts. Centimetres, grams and minutes are called units of measurement.

A ratio is in its simplest form when its terms are whole numbers with no common factor other than 1. Divide every term by the HCF: $12 : 18 = 2 : 3$ , dividing both terms by 6.

A ratio with fractions or decimals is changed to whole numbers by multiplying every term by the same number: $\frac{1}{2} : \frac{1}{3} = 3 : 2$ (multiply by 6, the LCM of the denominators) and $0.6 : 1.5 = 6 : 15 = 2 : 5$ (multiply by 10, then divide by 3).

Quantities must be in the same unit of measurement before their ratio is written: 40 cm to 1 m is $40 : 100 = 2 : 5$ .

Dividing a Quantity into Two Shares

Mei Ling and Ravi share $72 in the ratio $3 : 5$ . This means that the $72 is split into two shares with nothing left over, and that for every $3 Mei Ling receives, Ravi receives $5. It does not mean that they receive $3 and $5.

The order of the terms follows the order of the names. Mei Ling is named first, so the first term, 3, is hers. In this kind of problem the total is known and the shares are not.

Example 1: Sharing money between two people

Mei Ling and Ravi share $72 in the ratio $3 : 5$ . How much does each of them receive?

Solution

Draw a bar model. Mei Ling's share is a bar of 3 equal units and Ravi's share is a bar of 5 of the same units. The two bars together are the $72, so $72 is written on the bracket at the right. Each share is marked "?" because it is not known yet.

Count the units: $3 + 5 = 8$ . The total number of units is 8.

The 8 units together are the whole amount: 8 units = $72.

Find the value of 1 unit: $72 \div 8 = 9$ , so 1 unit = $9.

Mei Ling has 3 units: $3 \times 9 = 27$ , so her share is $27.

Ravi has 5 units: $5 \times 9 = 45$ , so his share is $45.

Check: $27 + 45 = 72$ , so the two shares add up to the $72. Also, dividing both terms of $27 : 45$ by 9 gives $3 : 5$ , so the shares are in the given ratio.

Mei Ling receives $27 and Ravi receives $45.

Rule: dividing a quantity in a given ratio

Add the terms of the ratio. The sum is the total number of units.
Divide the quantity by the total number of units. The answer is the value of 1 unit.
Multiply the value of 1 unit by each term of the ratio. The answers are the shares.
Check that the shares add up to the quantity.

On the bar model, the quantity that is divided goes on the bracket and each share is a bar marked "?".

Three Shares

The same four steps divide a quantity into three shares, or more. The bar model has one bar for each share.

Example 2: Three shares

Altogether 176 students from School P, School Q and School R take part in a parade. The numbers of students from the three schools are in the ratio $5 : 6 : 11$ . How many students from each school take part?

Solution

Draw one bar for each school: 5 units, 6 units and 11 units. The three bars together are the 176 students.

Total number of units: $5 + 6 + 11 = 22$

22 units = 176 students

1 unit = 8 students, since $176 \div 22 = 8$

School P has 5 units: $5 \times 8 = 40$ students

School Q has 6 units: $6 \times 8 = 48$ students

School R has 11 units: $11 \times 8 = 88$ students

Check: $40 + 48 + 88 = 176$

40 students from School P, 48 students from School Q and 88 students from School R take part.

Things that are counted and things that can be cut

Students, marbles and books are counted in whole numbers, so, once the ratio is in its simplest form, the value of 1 unit must come out as a whole number. In Example 2 the ratio $5 : 6 : 11$ is in its simplest form, and 1 unit is a whole number: $176 \div 22 = 8$ . A group of 165 students could not be divided in the ratio $5 : 6 : 11$ , because $165 \div 22 = 7.5$ and there is no such thing as half a student.

Money, lengths, masses and volumes can be cut into smaller amounts, so for them 1 unit may be a decimal or a fraction. If $10 is shared in the ratio $1 : 3$ , then 4 units = $10 and 1 unit = $2.50. The shares are $2.50 and $7.50, which add up to $10.

Shares as Fractions of the Whole

Put the two bars of Example 1 end to end. They make one bar, which is the whole $72, cut into 8 equal units. Mei Ling's share is 3 of the 8 units, so it is $\frac{3}{8}$ of the $72. Ravi's share is 5 of the 8 units, so it is $\frac{5}{8}$ of the $72.

Mei Ling's share: $\frac{3}{8} \times 72 = 27$ , so $27.

Ravi's share: $\frac{5}{8} \times 72 = 45$ , so $45.

These are the answers of Example 1.

Why the two ways agree

To find $\frac{3}{8}$ of 72, divide 72 by 8 and multiply the answer by 3: $\frac{3}{8} \times 72 = 3 \times (72 \div 8) = 3 \times 9 = 27$ . Dividing by 8 is finding the value of 1 unit, and multiplying by 3 is taking 3 units. So the fraction way is the units way written on one line. It is the same division and the same multiplication, and the two ways always give the same answer.

Rule: each share as a fraction of the whole

When a quantity is divided in a ratio of whole numbers, each share is a fraction of the whole quantity. The numerator is the share's own term of the ratio, and the denominator is the sum of all the terms.

In the ratio $3 : 5$ the sum of the terms is 8, so the shares are $\frac{3}{8}$ and $\frac{5}{8}$ of the quantity.

In the ratio $1 : 2 : 4$ the sum of the terms is 7, so the shares are $\frac{1}{7}$ , $\frac{2}{7}$ and $\frac{4}{7}$ of the quantity.

The fractions add up to 1, because the shares together are the whole quantity: $\frac{3}{8} + \frac{5}{8} = 1$ .

The fraction way is quick when only one share is wanted. The units way shows every share at once. Use the one you find clearer, and check in the same way: the shares add up to the whole.

Example 3: Using fractions of the whole

A school bookshop has 425 markers. Each marker is red, orange or green, and the numbers of red, orange and green markers are in the ratio $21 : 24 : 40$ . Find the number of markers of each colour.

Solution

Draw one bar for each colour. There are too many units to draw one by one, so each bar is drawn as a single block with its number of units written inside.

Total number of units: $21 + 24 + 40 = 85$

The red markers are 21 of the 85 units, so they are $\frac{21}{85}$ of the 425 markers: $\frac{21}{85} \times 425 = 21 \times (425 \div 85) = 21 \times 5 = 105$

Orange markers: $\frac{24}{85} \times 425 = 24 \times 5 = 120$

Green markers: $\frac{40}{85} \times 425 = 40 \times 5 = 200$

In each line, $425 \div 85 = 5$ is worked out first. That 5 is the value of 1 unit: 85 units = 425 markers, so 1 unit = 5 markers.

Check: $105 + 120 + 200 = 425$

There are 105 red markers, 120 orange markers and 200 green markers.

When the Ratio Must Be Changed First

A bar model is made of whole units, so the ratio must have whole-number terms before the units are counted. Sometimes the ratio is not given ready to use: the quantities are in different units of measurement, or the terms are fractions or decimals, or the terms have a common factor. Write the ratio in its simplest form first. Then divide the quantity as before.

Example 4: Different units of measurement

Ali and Ravi help at their uncle's fruit stall. Ali works for 1 hour 30 minutes and Ravi works for 2 hours 15 minutes. Their uncle gives them $60 to share in the ratio of the times they worked. How much does each boy receive?

Solution

Put both times in the same unit of measurement, minutes: 1 h 30 min = 90 min and 2 h 15 min = 135 min.

Ali's time : Ravi's time $= 90 : 135$ . The HCF of 90 and 135 is 45, so divide both terms by 45: $90 : 135 = 2 : 3$ .

Draw the bar model: 2 units for Ali and 3 units for Ravi, $60 in all.

Total number of units: $2 + 3 = 5$

5 units = $60, so 1 unit = $12, since $60 \div 5 = 12$

Ali has 2 units: $2 \times 12 = 24$ , so $24. Ravi has 3 units: $3 \times 12 = 36$ , so $36.

Check: $24 + 36 = 60$

The ratio $90 : 135$ would give the same shares, but with 225 units and an awkward value for 1 unit. In its simplest form, $2 : 3$ , there are only 5 units.

Ali receives $24 and Ravi receives $36.

Example 5: A ratio of fractions

Mrs Goh shares $150 between Siti and Hui Min in the ratio $\frac{1}{2} : \frac{1}{3}$ . How much does each girl receive?

Solution

Clear the fractions. The LCM of the denominators 2 and 3 is 6, so multiply both terms by 6: $\frac{1}{2} \times 6 = 3$ and $\frac{1}{3} \times 6 = 2$ . So $\frac{1}{2} : \frac{1}{3} = 3 : 2$ .

Draw the bar model: 3 units for Siti and 2 units for Hui Min, $150 in all.

Total number of units: $3 + 2 = 5$

5 units = $150, so 1 unit = $30, since $150 \div 5 = 30$

Siti has 3 units: $3 \times 30 = 90$ , so $90. Hui Min has 2 units: $2 \times 30 = 60$ , so $60.

Check: $90 + 60 = 150$

Do not take $\frac{1}{2}$ of $150 and $\frac{1}{3}$ of $150. That gives $75 and $50, which add up to only $125, so $25 would be left over. The terms $\frac{1}{2}$ and $\frac{1}{3}$ compare the two shares with each other. They are not fractions of the $150. Siti's share is $\frac{3}{5}$ of the $150 and Hui Min's share is $\frac{2}{5}$ of it.

Siti receives $90 and Hui Min receives $60.

Example 6: A ratio of decimals in a drink

Hui Min makes 2 litres of fruit punch by mixing orange juice, pineapple juice and soda water in the ratio $1.5 : 1 : 2.5$ . How many millilitres of each does she use?

Solution

Clear the decimals. Multiply every term by 10, then divide every term by the HCF, 5: $1.5 : 1 : 2.5 = 15 : 10 : 25 = 3 : 2 : 5$

The answer is wanted in millilitres, so write the whole in millilitres: 2 litres = 2000 ml.

Draw one bar for each drink: 3 units, 2 units and 5 units, 2000 ml in all.

Total number of units: $3 + 2 + 5 = 10$

10 units = 2000 ml, so 1 unit = 200 ml, since $2000 \div 10 = 200$

Orange juice: $3 \times 200 = 600$ , so 600 ml. Pineapple juice: $2 \times 200 = 400$ , so 400 ml. Soda water: $5 \times 200 = 1000$ , so 1000 ml.

Check: $600 + 400 + 1000 = 2000$

She uses 600 ml of orange juice, 400 ml of pineapple juice and 1000 ml of soda water.

Lengths, Mixtures and Triangles

Anything that has a total can be divided in a ratio. In each problem, ask first: what is the whole? It may be the length before it is cut, the mass of the whole mixture, the sum of the angles or the perimeter. The whole goes on the bracket of the bar model.

Example 7: Cutting a length

Mr Tan cuts a rope 4.5 m long into two pieces in the ratio $2 : 7$ . Find the length of each piece.

Solution

The whole is the 4.5 m of rope. Draw 2 units for the shorter piece and 7 units for the longer piece.

Total number of units: $2 + 7 = 9$

9 units = 4.5 m, so 1 unit = 0.5 m, since $4.5 \div 9 = 0.5$ . A rope can be cut anywhere, so 1 unit need not be a whole number of metres.

Shorter piece: $2 \times 0.5 = 1$ , so 1 m. Longer piece: $7 \times 0.5 = 3.5$ , so 3.5 m.

Check: $1 + 3.5 = 4.5$

The pieces are 1 m and 3.5 m long.

Example 8: A mixture

To lay a path in the school garden, Mr Tan mixes cement, sand and stones in the ratio $1 : 2 : 4$ by mass. He needs 84 kg of the mixture. What mass of each does he use?

Solution

"By mass" means that the ratio compares the masses of the three materials. The whole is the 84 kg of mixture. Draw 1 unit for the cement, 2 units for the sand and 4 units for the stones.

Total number of units: $1 + 2 + 4 = 7$

7 units = 84 kg, so 1 unit = 12 kg, since $84 \div 7 = 12$

Cement: $1 \times 12 = 12$ , so 12 kg. Sand: $2 \times 12 = 24$ , so 24 kg. Stones: $4 \times 12 = 48$ , so 48 kg.

Check: $12 + 24 + 48 = 84$

He uses 12 kg of cement, 24 kg of sand and 48 kg of stones.

Example 9: The angles and the sides of a triangle

(a) In triangle $ABC$ , angle $A$ , angle $B$ and angle $C$ are in the ratio $2 : 3 : 4$ . Find the three angles.
(b) In triangle $PQR$ , the sides $PQ$ , $QR$ and $RP$ are in the ratio $4 : 5 : 6$ . The perimeter of the triangle is 45 cm. Find the length of each side.

Solution

(a) The question gives no total, but the total is known from primary school: the angles of a triangle add up to $180^\circ$ . So the whole is $180^\circ$ .

Total number of units: $2 + 3 + 4 = 9$

9 units $= 180^\circ$ , so 1 unit $= 20^\circ$ , since $180 \div 9 = 20$

Angle $A = 2 \times 20^\circ = 40^\circ$ , angle $B = 3 \times 20^\circ = 60^\circ$ and angle $C = 4 \times 20^\circ = 80^\circ$

Check: $40 + 60 + 80 = 180$

(b) The perimeter is the sum of the three sides, so the whole is 45 cm.

Total number of units: $4 + 5 + 6 = 15$

15 units = 45 cm, so 1 unit = 3 cm, since $45 \div 15 = 3$

$PQ = 4 \times 3 = 12$ cm, $QR = 5 \times 3 = 15$ cm and $RP = 6 \times 3 = 18$ cm

Check: $12 + 15 + 18 = 45$

(a) Angle $A = 40^\circ$ , angle $B = 60^\circ$ and angle $C = 80^\circ$ .
(b) $PQ = 12$ cm, $QR = 15$ cm and $RP = 18$ cm.

Common Mistakes

Dividing by a term of the ratio instead of by the total number of units. To share $72 in the ratio $3 : 5$ , do not work out $72 \div 3$ or $72 \div 5$ . The $72 is all 8 units, so divide by 8.
Giving the shares to the wrong people. The terms follow the order of the names. When Mei Ling and Ravi share in the ratio $3 : 5$ , Mei Ling has the 3 units and Ravi has the 5 units.
Mixing up two fractions. In the ratio $3 : 5$ , Mei Ling's share is $\frac{3}{8}$ of the whole amount. It is $\frac{3}{5}$ of Ravi's share, not $\frac{3}{5}$ of the whole amount.
Using the fractions in a ratio as fractions of the whole. To share in the ratio $\frac{1}{2} : \frac{1}{3}$ , change the ratio to $3 : 2$ first. Half of the amount and a third of the amount do not add up to the amount.
Counting units before the units of measurement agree. The ratio of 1 hour 30 minutes to 45 minutes is not $130 : 45$ . In minutes it is $90 : 45 = 2 : 1$ .
Leaving out the check. If the shares do not add up to the whole, something has gone wrong.

Practice

Work each question on paper, with a bar model, before you open its solution.

Question 1

Ali and Mei Ling share 108 stickers in the ratio $5 : 4$ . How many stickers does each of them get?

ShowHide solution

Ali has 5 units and Mei Ling has 4 units. The two bars together are the 108 stickers.

Total number of units: $5 + 4 = 9$

9 units = 108 stickers, so 1 unit = 12 stickers, since $108 \div 9 = 12$

Ali has 5 units: $5 \times 12 = 60$ . Mei Ling has 4 units: $4 \times 12 = 48$ .

Check: $60 + 48 = 108$

Ali gets 60 stickers and Mei Ling gets 48 stickers.

Question 2

Jenny and her mother sell a flat for $ $504\,000$ and share the money in the ratio $2 : 3$ .
(a) What fraction of the money is Jenny's share? Use it to find how much Jenny receives.
(b) Find her mother's share and check your answers.

ShowHide solution

(a) Total number of units: $2 + 3 = 5$ . Jenny is named first, so the 2 units are hers. Her share is 2 of the 5 units, which is $\frac{2}{5}$ of the money.

$\frac{2}{5} \times 504\,000 = 2 \times 100\,800 = 201\,600$ , since $504\,000 \div 5 = 100\,800$

(b) Her mother has 3 of the 5 units, which is $\frac{3}{5}$ of the money: $\frac{3}{5} \times 504\,000 = 3 \times 100\,800 = 302\,400$

Check: $201\,600 + 302\,400 = 504\,000$

(a) Jenny's share is $\frac{2}{5}$ of the money. She receives $ $201\,600$ .
(b) Her mother receives $ $302\,400$ .

Question 3

Mr Tan gives out 180 exercise books to Class 1A, Class 1B and Class 1C in the ratio $3 : 4 : 5$ . How many exercise books does each class get?

ShowHide solution

Draw one bar for each class: 3 units, 4 units and 5 units. The three bars together are the 180 exercise books.

Total number of units: $3 + 4 + 5 = 12$

12 units = 180 exercise books, so 1 unit = 15 exercise books, since $180 \div 12 = 15$

Class 1A: $3 \times 15 = 45$ . Class 1B: $4 \times 15 = 60$ . Class 1C: $5 \times 15 = 75$ .

Check: $45 + 60 + 75 = 180$

Class 1A gets 45 exercise books, Class 1B gets 60 and Class 1C gets 75.

Question 4

A strip of wood 3.6 m long is cut into three pieces in the ratio $1 : 3 : 5$ . Find the length of each piece in centimetres.

ShowHide solution

The answer is wanted in centimetres, so write the whole in centimetres: 3.6 m = 360 cm.

Total number of units: $1 + 3 + 5 = 9$

9 units = 360 cm, so 1 unit = 40 cm, since $360 \div 9 = 40$

The pieces are $1 \times 40 = 40$ cm, $3 \times 40 = 120$ cm and $5 \times 40 = 200$ cm long.

Check: $40 + 120 + 200 = 360$

The pieces are 40 cm, 120 cm and 200 cm long.

Question 5

The angles of a triangle are in the ratio $2 : 3 : 5$ .
(a) Find the three angles.
(b) What kind of triangle is it?

ShowHide solution

(a) The angles of a triangle add up to $180^\circ$ , so the whole is $180^\circ$ .

Total number of units: $2 + 3 + 5 = 10$

10 units $= 180^\circ$ , so 1 unit $= 18^\circ$ , since $180 \div 10 = 18$

The angles are $2 \times 18^\circ = 36^\circ$ , $3 \times 18^\circ = 54^\circ$ and $5 \times 18^\circ = 90^\circ$ .

Check: $36 + 54 + 90 = 180$

(b) One of the angles is $90^\circ$ , which is a right angle.

(a) The angles are $36^\circ$ , $54^\circ$ and $90^\circ$ .
(b) It is a right-angled triangle.

Question 6

The sides of a triangle are in the ratio $3 : 5 : 7$ . Its perimeter is 1.2 m. Find the length of each side in centimetres.

ShowHide solution

The perimeter is the sum of the three sides, so it is the whole. In centimetres, 1.2 m = 120 cm.

Total number of units: $3 + 5 + 7 = 15$

15 units = 120 cm, so 1 unit = 8 cm, since $120 \div 15 = 8$

The sides are $3 \times 8 = 24$ cm, $5 \times 8 = 40$ cm and $7 \times 8 = 56$ cm long.

Check: $24 + 40 + 56 = 120$

The sides are 24 cm, 40 cm and 56 cm long.

Question 7

Ali and Ravi buy a pack of 35 game cards together. Ali pays 80 cents and Ravi pays $2. They share the cards in the ratio of the amounts they paid. How many cards does each boy get?

ShowHide solution

Put both amounts in the same unit of measurement, cents: $2 = 200 cents.

Ali's amount : Ravi's amount $= 80 : 200$ . The HCF of 80 and 200 is 40, so divide both terms by 40: $80 : 200 = 2 : 5$ .

Draw the bar model: 2 units for Ali and 5 units for Ravi, 35 cards in all.

Total number of units: $2 + 5 = 7$

7 units = 35 cards, so 1 unit = 5 cards, since $35 \div 7 = 5$

Ali has 2 units: $2 \times 5 = 10$ . Ravi has 5 units: $5 \times 5 = 25$ .

Check: $10 + 25 = 35$

Ali gets 10 cards and Ravi gets 25 cards.

Question 8

Mr Tan and Mrs Goh open a drinks stall together. Mr Tan puts in $\frac{3}{4}$ of the money needed to start it and Mrs Goh puts in the rest. In its first year the stall makes a profit of $ $16\,000$ , which they share in the ratio of the amounts they put in. How much does each of them receive?

ShowHide solution

Mrs Goh puts in the rest of the money: $1 - \frac{3}{4} = \frac{1}{4}$ of it.

Mr Tan's amount : Mrs Goh's amount $= \frac{3}{4} : \frac{1}{4}$ . Multiply both terms by 4: the ratio is $3 : 1$ .

Total number of units: $3 + 1 = 4$

4 units = $ $16\,000$ , so 1 unit = $4000, since $16\,000 \div 4 = 4000$

Mr Tan has 3 units: $3 \times 4000 = 12\,000$ , so $ $12\,000$ . Mrs Goh has 1 unit, so $4000.

Check: $12\,000 + 4000 = 16\,000$

Mr Tan receives $ $12\,000$ and Mrs Goh receives $4000.

Question 9

Mrs Goh mixes peanuts and raisins in the ratio $2.5 : 1$ by mass to make 700 g of snack mix. What mass of each does she use?

ShowHide solution

Clear the decimal. Multiply both terms by 10, then divide both terms by the HCF, 5: $2.5 : 1 = 25 : 10 = 5 : 2$

Total number of units: $5 + 2 = 7$

7 units = 700 g, so 1 unit = 100 g, since $700 \div 7 = 100$

Peanuts: $5 \times 100 = 500$ , so 500 g. Raisins: $2 \times 100 = 200$ , so 200 g.

Check: $500 + 200 = 700$

She uses 500 g of peanuts and 200 g of raisins.

Question 10

Mr Tan shares $260 among Ali, Siti and Ravi in the ratio $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$ .
(a) How much does each of them receive?
(b) Explain why Ali does not receive half of the $260.

ShowHide solution

(a) Clear the fractions. The LCM of the denominators 2, 3 and 4 is 12, so multiply every term by 12: $\frac{1}{2} \times 12 = 6$ , $\frac{1}{3} \times 12 = 4$ and $\frac{1}{4} \times 12 = 3$ . The ratio is $6 : 4 : 3$ , which is in its simplest form.

Total number of units: $6 + 4 + 3 = 13$

13 units = $260, so 1 unit = $20, since $260 \div 13 = 20$

Ali has 6 units: $6 \times 20 = 120$ . Siti has 4 units: $4 \times 20 = 80$ . Ravi has 3 units: $3 \times 20 = 60$ .

Check: $120 + 80 + 60 = 260$

(b) The terms of the ratio compare the three shares with one another. They are not fractions of the $260. Ali has 6 of the 13 units, so his share is $\frac{6}{13}$ of the $260, which is a little less than half of it.

Half, a third and a quarter of the money could not all be given out: $\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12}$ , which is more than the whole.

(a) Ali receives $120, Siti receives $80 and Ravi receives $60.
(b) The fractions in the ratio compare the shares with one another. Ali's share is $\frac{6}{13}$ of the $260, not $\frac{1}{2}$ of it.

Question 11

The length and the breadth of a rectangular garden plot are in the ratio $5 : 3$ . The perimeter of the plot is 64 m. Find
(a) the length and the breadth of the plot,
(b) the area of the plot.

ShowHide solution

(a) The perimeter is made of 2 lengths and 2 breadths. So 1 length and 1 breadth together are half of the perimeter: $64 \div 2 = 32$ , which is 32 m. This 32 m is the whole that is divided in the ratio $5 : 3$ .

Total number of units: $5 + 3 = 8$

8 units = 32 m, so 1 unit = 4 m, since $32 \div 8 = 4$

Length: $5 \times 4 = 20$ , so 20 m. Breadth: $3 \times 4 = 12$ , so 12 m.

Check: $20 + 12 = 32$ , and the perimeter is $2 \times 32 = 64$ , as given.

(b) Area $= 20 \times 12 = 240$

(a) The length is 20 m and the breadth is 12 m.
(b) The area is 240 $\text{m}^{2}$ .

Question 12

(a) Mrs Goh shares $45 between her son and her daughter in the ratio $3 : 7$ . How much does each of them receive?
(b) Can a class of 45 students be divided into two groups in the ratio $3 : 7$ ? Explain.
(c) Which of these numbers of students can be divided into two groups in the ratio $3 : 7$ ?
30, 36, 40, 45, 50

ShowHide solution

(a) Total number of units: $3 + 7 = 10$

10 units = $45, so 1 unit = $4.50, since $45 \div 10 = 4.5$ . Money can be cut into dollars and cents, so 1 unit need not be a whole number of dollars.

Her son has 3 units: $3 \times 4.5 = 13.5$ , so $13.50. Her daughter has 7 units: $7 \times 4.5 = 31.5$ , so $31.50.

Check: $13.5 + 31.5 = 45$

(b) There are 10 units again. 10 units = 45 students would make 1 unit = 4.5 students, and the groups would have 13.5 and 31.5 students. Students are counted in whole numbers, so it cannot be done.

(c) The ratio $3 : 7$ is in its simplest form, so the value of 1 unit must be a whole number, and the number of students must divide exactly by 10, the total number of units. Of the five numbers, 30, 40 and 50 do. They give groups of 9 and 21, 12 and 28, and 15 and 35.

(a) Her son receives $13.50 and her daughter receives $31.50.
(b) No. The ratio $3 : 7$ is in its simplest form, and 1 unit would be 4.5 students, which is not a whole number.
(c) 30, 40 and 50

Key Points

To divide a quantity in a given ratio: add the terms to find the total number of units, divide the quantity by that total to find the value of 1 unit, then multiply the value of 1 unit by each term to find each share.
On the bar model, the whole goes on the bracket and each share is a bar of its own number of units, marked "?".
Always check that the shares add up to the whole.
Each share is also a fraction of the whole: its own term over the sum of the terms. In the ratio $3 : 5$ the shares are $\frac{3}{8}$ and $\frac{5}{8}$ of the whole. This is the units way written on one line.
Before counting units, put the quantities in the same unit of measurement, clear fractions and decimals, and write the ratio in its simplest form.
Things that are counted need a whole number for 1 unit, once the ratio is in its simplest form. Money and measurements can be cut, so for them 1 unit may be a decimal.
Sometimes the whole is not stated but known: the angles of a triangle add up to $180^\circ$ , and the length and the breadth of a rectangle together are half of its perimeter.