Parts of a wholeGrades 5–6
Ben eats a third of a pizza, and Ann eats half of what is left. Who ate more? In these tasks a fraction is always a fraction of something, and most of the difficulty is knowing of what: of the whole, or of what was left. A strip of paper cut into equal parts shows it better than any rule.
●●● ●●● ●●● ○○○ 3/4 of 12 = 9
A fraction is always a fraction of something. “A quarter of the rest” and “a quarter of the whole” are different numbers, so first find what the fraction is taken of.
To find 3/4 of 20, divide by 4 and take 3 parts: 15. To find the whole when 3/4 of it is 15, divide by 3 for one part and take 4 parts: 20.
What is given away and what is left together make one whole. If 1/3 and 1/4 are given away, 1 − 1/3 − 1/4 = 5/12 is left.
Whatever the question, find what one part is worth first, then take as many parts as it needs.
| Given | One part | Asked for |
|---|---|---|
| 3/4 of 20 | 20 : 4 = 5 | 3 parts, 15 |
| 3/4 of a number is 15 | 15 : 3 = 5 | the number, 4 parts, 20 |
| 2/5 of a number is 14 | 14 : 2 = 7 | the number, 5 parts, 35 |
Draw the whole as a long strip and cut it into as many equal parts as the fraction's bottom number says. When a share is taken of what is left, draw what is left as a new strip. Then find what one part is worth. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 5–6
A baker bakes 60 rolls. By noon he sells half of them, and by evening two thirds of the rest. How many rolls are left?
Answer 10 rolls.
Example 2 · Grades 5–6
On Monday Tom reads 2/7 of a book, and on Tuesday 3/7 of it. Then 40 pages are left. How many pages are in the book?
■ ■ ■ ■ ■ □ □
40 pagesAnswer 140 pages.
Example 3 · Grades 5–6
Granny sells eggs at the market. The first buyer takes half of her eggs and half an egg more. The second takes half of the eggs left and half an egg more, and so does the third. Then all the eggs are sold, and nobody broke an egg. How many eggs did Granny have?
7 → 3 → 1 → 0
Answer 7 eggs.
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 share is taken of the wrong amount: two thirds of all 60 rolls instead of the 30 left, or the 40 pages left are taken for the part Tom read.
What to say
“Two thirds of what? Point to it on the strip.”
A number found on the way becomes the answer: the 20 rolls sold in the evening instead of those left, or the 100 pages Tom read instead of the whole book.
What to say
“What exactly does the question ask for?”
The work stops one step short: one seventh of the book is found, 20 pages, and never turned into the whole book; or the rolls sold at noon are taken away and the evening's are forgotten.
What to say
“Is that the answer, or a step on the way to it?”
Cut an apple, a pizza or a sheet of paper into equal parts and name them: halves, thirds, quarters. Then take some away and ask what part is left.
When a fraction comes up in a recipe or a game, ask “a third of what?”. The answer is often the whole task.
Draw the strip together for any sharing at home: a long rectangle cut into equal cells. It works for every task in this topic.