The Remainder Model

The word remainder changes the parent of a fraction. If $\frac{2}{5}$ of the books on a trolley are storybooks, then $\frac{1}{3}$ of the remaining books are comics, that $\frac{1}{3}$ is not cut from all the books. It is cut from the leftover $\frac{3}{5}$. Name each whole before you calculate and the model stays honest.

Parents: ask your child to point to the parent whole for each fraction and predict the next cut before revealing the worked model.

By the end you will be able to identify each fraction's parent whole, enlarge the remainder without changing its amount, and solve a two-stage remainder problem.

Name the parent whole

A parent whole is the amount the fraction is talking about. In “$\frac{2}{5}$ of all the books”, the parent is the entire trolley. In “$\frac{1}{3}$ of the remaining books”, the parent is only the books left after storybooks are set aside.

The remainder is still part of the original whole, but it can become a new parent for the next fraction. That is the key move.

Cut the original bar, then re-cut the leftover

Work through this trolley problem:

$\frac{2}{5}$ of the books on a trolley are storybooks. $\frac{1}{3}$ of the remaining books are comics. The rest are information books. There are 48 information books. How many books are on the trolley?

First cut all books into 5 equal shares. Storybooks take 2 shares, leaving a remainder of 3 shares. Now look only at those 3 remainder shares. $\frac{1}{3}$ of them is 1 share of comics, leaving 2 shares of information books.

The diagram below enlarges the three-share remainder so that its inner cut is easy to see. The enlargement is a focus view: it does not create more books.

The information books occupy 2 original shares. Use the known information count to value a share, then read the five-share original bar.

Find the whole trolley

$$2\text{ shares} = 48\text{ books}$$

$$1\text{ share} = 48 \div 2 = 24\text{ books}$$

$$\text{Whole trolley} = 5 \times 24 = 120\text{ books}$$

Re-cut into one common small unit

The two cuts can be combined into one exact grid. Five original shares, each split into three small pieces, make 15 equal small units. In that grid:

Make equal small units

$$\text{Storybooks} = 2 \times 3 = 6\text{ small units}$$

$$\text{Comics} = 1 \times 3 = 3\text{ small units}$$

$$\text{Information} = 2 \times 3 = 6\text{ small units}$$

$$\text{Whole trolley} = 5 \times 3 = 15\text{ small units}$$

The 48 information books fill 6 small units. The re-cut makes each piece comparable without changing the number of books.

Value the small unit

$$1\text{ small unit} = 48 \div 6 = 8\text{ books}$$

$$\text{Whole trolley} = 15 \times 8 = 120\text{ books}$$

Use the adjustable experiment below. Predict which part is the parent, then step the value of one small unit until the information section matches 48.

🔬

Trolley books: re-cut the remainder

🤔 Predict first: After $\frac{2}{5}$ are storybooks, the $\frac{1}{3}$ applies to which parent?

All 5 partsThe 3-part remainder

Check the whole before you add fractions

The nested model gives a clean check:

Check each parent whole

$$\frac{2}{5} \times 120 = 48\text{ storybooks}$$

$$\text{Remainder} = 120 - 48 = 72\text{ books}$$

$$\frac{1}{3} \times 72 = 24\text{ comics}$$

$$\text{Information} = 72 - 24 = 48\text{ books}$$

The categories add to 48 + 24 + 48 = 120. Notice that $\frac{1}{3}$ of the original 120 would be 40, but comics are 24 because the fraction was taken from the smaller remainder of 72.

Watch out — easily mixed up

Quick recap

Graduated practice

These move from naming the parent to handling two fractions that share the original whole. Open each solution only after you have named its parent whole.

1. Art books and biographies

$\frac{1}{4}$ of a shelf is art books. $\frac{2}{3}$ of the remaining books are biographies. The final 18 books are atlases. How many books are on the shelf?

Show solution

Use 12 original shares. Art uses 3, leaving 9. Biographies use 6 of those 9, leaving 3 shares for atlases.

Practice 1

$$3\text{ shares} = 18\text{ atlases}$$

$$1\text{ share} = 18 \div 3 = 6\text{ books}$$

$$\text{Shelf} = 12 \times 6 = 72\text{ books}$$

Check: 18 art, 36 biographies, 18 atlases.

2. Beads left after yellow

$\frac{2}{7}$ of 84 beads are yellow. $\frac{1}{3}$ of the remainder are blue. How many beads are neither yellow nor blue?

Show solution

Yellow beads are $\frac{2}{7}$ of 84. Then calculate the second fraction from the remainder.

Practice 2

$$\frac{2}{7} \times 84 = 24\text{ yellow beads}$$

$$\text{Remainder} = 84 - 24 = 60\text{ beads}$$

$$\frac{1}{3} \times 60 = 20\text{ blue beads}$$

$$\text{Neither} = 60 - 20 = 40\text{ beads}$$

3. Two parts of the same total

$\frac{3}{10}$ of 90 cards are red and $\frac{2}{5}$ of the same 90 cards are blue. The rest are green. How many are green?

Show solution

Both fractions name the original 90 cards, so neither fraction is taken from a remainder.

Practice 3

$$\frac{3}{10} \times 90 = 27\text{ red cards}$$

$$\frac{2}{5} \times 90 = 36\text{ blue cards}$$

$$\text{Green} = 90 - 27 - 36 = 27\text{ cards}$$

The words “the same 90 cards” keep both fractions on the original parent whole.

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

1. Ben read $\frac{1}{3}$ of a book, then $\frac{1}{2}$ of the remainder the next day. The $\frac{1}{2}$ applies to…

   The whole bookThe part left after the first day$\frac{1}{2}$ the whole book

2. After spending $\frac{2}{5}$ of some money, what fraction is left as the remainder?

   $\frac{3}{5}$$\frac{2}{5}$$\frac{5}{2}$

3. A bar is split into 5 shares and each share is re-cut into 3 equal small units. How many units make the whole bar?

   81215

4. A boy spends $\frac{1}{2}$ of his money, then $\frac{1}{4}$ of the remainder. What fraction of the original money is the second amount?

   $\frac{1}{4}$$\frac{1}{8}$$\frac{1}{2}$

5. In a remainder model, 6 equal small units carry 48 information books. What is one unit?

   6 books48 books8 books

6. Why is $\frac{2}{5}$ + $\frac{1}{3}$ wrong for the trolley problem?

   The fractions have different parent wholesFractions cannot be addedThe answer would be a whole number

7. A shelf has 12 original shares. The final atlas part is 3 shares and equals 18 books. What is the total shelf size?

   54 books72 books216 books

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