Parts of a wholeGrades 5–6
Ann and Ben share 35 sweets in the ratio 3 to 4. How many does each get? In these tasks a ratio says how many equal parts each one gets, not how many things. Counting the parts first and finding what one part is worth turns every ratio task into a division.
Ann □ □ □ Ben □ □ □ □ 7 parts = 35
3 to 4 means: for every 3 equal parts of one, 4 of the other. Together that is 7 equal parts, not 3 or 4.
Divide the total by the number of parts to find one part, then multiply by each one's number: 35 : 7 = 5, so 3 parts are 15 and 4 parts are 20.
A ratio stays the same when both of its numbers are multiplied by the same number: 3 to 4 is 6 to 8 and 30 to 40. That is how a recipe grows.
Whatever the ratio, add its numbers up first. That is how many equal parts the whole is cut into.
| Shared | Parts | One part and the shares |
|---|---|---|
| 35 sweets, 3 to 4 | 3 + 4 = 7 | 35 : 7 = 5, so 15 and 20 |
| 60 cards, 1 to 2 to 3 | 1 + 2 + 3 = 6 | 60 : 6 = 10, so 10, 20 and 30 |
| 45 beads, 2 to 7 | 2 + 7 = 9 | 45 : 9 = 5, so 10 and 35 |
Draw one box for every part of the ratio, a row of boxes for each one who shares. Count all the boxes, find what one box holds, then fill in the shares. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 5–6
Max and Lena pick 45 mushrooms together. For every 4 mushrooms Max picks, Lena picks 5. How many mushrooms does Lena pick?
Max □ □ □ □ Lena □ □ □ □ □
Answer 25 mushrooms.
Example 2 · Grades 5–6
A choir has 4 boys for every 7 girls, and there are 9 more girls than boys. How many children sing in the choir?
boys □ □ □ □
girls □ □ □ □ □ □ □
└ 9 ┘Answer 33 children.
Example 3 · Grades 5–6
Two boxes hold marbles in the ratio 3 to 1. After 10 marbles are moved from the first box to the second, the boxes hold as many marbles each. How many marbles are there in all?
before A □ □ □ B □ after A □ □ B □ □
Answer 40 marbles.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
The numbers of the ratio are taken for shares of the total: 4 to 5 is read as 4 out of 5, or the 45 mushrooms are divided by 5 instead of by the 4 + 5 = 9 parts.
What to say
“How many equal parts are there in all?”
The child adds where the ratio needs multiplying: a recipe for 6 more people gets 6 more eggs, when the eggs must grow in step with the people.
What to say
“If there are twice as many people, what happens to the eggs?”
A share is found and the wrong one is given: Max's mushrooms instead of Lena's, or one part, 3 children, instead of the whole choir.
What to say
“Whose share does the question ask about?”
Make a drink or a dough by a ratio together: one cup of juice to three of water. Then make twice as much, and see what doubles.
For any sharing at home, of sweets or of chores, draw a box for every part and fill them in. The boxes do the thinking.
Look at the scale of a map or a model car: 1 to 50 means every centimetre stands for 50. Measure something together and scale it up.