Drawing a ratio as bars turns a word problem into something you can see
Draw each share as a bar
A bar model draws each share in a ratio as a bar made of equal parts. Every part is the same size, and the bars start at the same line, one under the other.
For the ratio 3 : 5, draw one bar of 3 parts and one bar of 5 parts. Because all the parts are the same size, the 5-part bar is longer, and you can see how much longer it is.
Every part is the same size, so the bar of 5 parts reaches 2 parts further than the bar of 3 parts.
The difference is counted in parts
Lined up at the start, the two bars differ by 5 − 3 = 2 parts. That is true whatever one part turns out to be worth. If one part is worth 4, the shares are 3 × 4 = 12 and 5 × 4 = 20, and the difference is 2 × 4 = 8. If one part is worth 10, the difference is 2 × 10 = 20.
The gap between the bars is 2 parts long: the 2 parts of the longer bar that the shorter bar does not reach.
Where the known amount goes
A ratio problem usually tells you one real amount, and the bar model shows which parts it fills. A total fills all the parts, 3 + 5 = 8 of them. One share fills only its own parts. A difference fills only the gap, 5 − 3 = 2 parts. Mark the known amount on the bars, divide it by the number of parts it fills, and you know what one part is worth.
Two friends share marbles in the ratio 3 : 5, and one friend has 16 more marbles than the other. The 16 fills the gap of 2 parts, so one part is 16 ÷ 2 = 8 marbles. The shares are 3 × 8 = 24 and 5 × 8 = 40. Check: 40 − 24 = 16.
The difference of 16 fills the 2 parts of the gap, so each part is 8. The shares are 24 and 40.
The usual mistakes
Dividing a difference by the total number of parts. The 16 fills only the 2 parts of the gap, not all 8 parts, so one part is 16 ÷ 2 = 8, not 16 ÷ 8 = 2.
Giving a share when the question asks how many more. If one part is worth 8, the larger share is 40, but it has only 2 × 8 = 16 more than the smaller share.
Adding the shares when the question asks how many more. 24 + 40 = 64 is how many marbles there are altogether, not how far apart the two shares are.
Ratio · Sharing a Total in a Given Ratio
QuestionAhmad and Bala share 104 marbles in the ratio 5 : 3. (a) How many marbles does Ahmad get? (b) How many more marbles does Ahmad get than Bala?
1.Draw Ahmad's share as 5 equal units and Bala's share as 3 equal units.
Ahmad has 5 equal units and Bala has 3. 2.Count all the units: 5 + 3 = 8 units, and 8 units = 104 marbles.
The 5 + 3 = 8 units are all 104 marbles. 3.1 unit = 104 ÷ 8 = 13 marbles.
1 unit = 104 ÷ 8 = 13 marbles. 4.(a) Ahmad gets 5 × 13 = 65 marbles.
(a) Ahmad gets 5 × 13 = 65 marbles. 5.(b) Ahmad has 5 − 3 = 2 units more than Bala, which is 2 × 13 = 26 marbles. Check: 65 + 39 = 104.
(b) Ahmad has 2 units more than Bala: 2 × 13 = 26 marbles.
Answer: (a) 65 marbles; (b) 26 marbles
Watch out for
- Dividing 104 by 5 to find Ahmad's share. The 104 marbles are all 8 units, so divide by 8 to find one unit first.
- Giving 39 as the answer to part (b). 39 is Bala's share. The question asks how many more Ahmad gets, which is the difference between the two shares.
Ratio and proportion · An Art Club of Boys and Girls: The Members, Then the Paintbrushes Given Out
QuestionThe ratio of the number of boys to the number of girls in an art club was 4 : 7. There were 36 more girls than boys. (a) How many members were there in the art club altogether? (b) Each girl was given 3 paintbrushes and each boy was given 2 paintbrushes. How many paintbrushes were given out altogether?
1.Draw the boys as 4 equal units and the girls as 7 units of the same size.
Boys : girls = 4 : 7. The boys are 4 units and the girls are 7 units of the same size. 2.The girls' bar is 7 − 4 = 3 units longer than the boys' bar, and those 3 units are the 36 more girls. So 1 unit is 36 ÷ 3 = 12 children.
The girls' bar is 3 units longer, and those 3 units are the 36 more girls: 1 unit is 36 ÷ 3 = 12. 3.(a) There are 4 + 7 = 11 units altogether, so the club has 11 × 12 = 132 members.
(a) There are 4 + 7 = 11 units, so the club has 11 × 12 = 132 members. 4.The boys are 4 × 12 = 48 children and the girls are 7 × 12 = 84 children.
There are 4 × 12 = 48 boys and 7 × 12 = 84 girls. 5.(b) The boys get 48 × 2 = 96 paintbrushes and the girls get 84 × 3 = 252, so 96 + 252 = 348 paintbrushes were given out. Check: 84 − 48 = 36 more girls than boys.
(b) A boy gets 2 paintbrushes and a girl gets 3: 48 × 2 + 84 × 3 = 348 paintbrushes.
Answer: (a) 132 members; (b) 348 paintbrushes
Watch out for
- Dividing the 36 by all 11 units. The 36 is how many more girls there are than boys, which is only the 3 units the girls' bar has beyond the boys' bar, so 1 unit is 36 ÷ 3 = 12.
- Multiplying the 132 members by 2 + 3 = 5 paintbrushes, or by one of the two numbers. A boy gets 2 paintbrushes and a girl gets 3, so count the boys and the girls on their own: 48 × 2 + 84 × 3 = 348.
Your turn
1.2 : 5 with one part worth 5. How much bigger is the larger share?