# Fraction Word Problems

> Sum, difference, product and quotient in words, "fraction of the remainder" problems and working backwards to the start.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/fraction-word-problems/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Jeremy's pocket money
Every week Jeremy spends $\frac{1}{4}$ of his pocket money on chocolate. He saves $\frac{2}{3}$ of what is left. This week he saved 7.50 dollars. How much pocket money does he get?

His older sister draws a bar and cuts it into 4 equal parts. One part is chocolate, so 3 parts are left. She saves 2 of those 3 parts. So 2 parts are worth 7.50 dollars, 1 part is worth 3.75 dollars, and the whole bar of 4 parts is 15 dollars.

A drawing turned a confusing problem into easy steps. Fraction word problems are all about reading carefully: a fraction **of what**?

**In short:** In a fraction word problem, always ask "a fraction of what?": the whole amount, or what is left. Bar models help you see it, and if you know what a fraction of the total is, divide to find one part, then multiply.

## Before you start

- Adding, subtracting, multiplying and dividing fractions (the earlier chapters in this unit).
- Finding a fraction of an amount: divide by the bottom, multiply by the top.
- If $\frac{1}{4}$ of a number is 5, the number is $5 \times 4 = 20$.

## Operation words

| Words | Means |
|---|---|
| sum of | add |
| difference between | subtract the smaller from the bigger |
| product of | multiply |
| quotient of $a$ and $b$ | $a \div b$ |
| increase by | add |
| decrease by | subtract |
| square of $\frac{1}{4}$ | $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ |

### Worked example: Step by step
"Decrease the product of $\frac{3}{4}$ and $\frac{5}{8}$ by the square of $\frac{1}{4}$."

Product: $\frac{3}{4} \times \frac{5}{8} = \frac{15}{32}$. Square of $\frac{1}{4}$: $\frac{1}{16} = \frac{2}{32}$.

Decrease: $\frac{15}{32} - \frac{2}{32} = \frac{13}{32}$.

## "Of the whole" or "of the remainder"?

### Key idea: Read the fraction's "of"
- $\frac{1}{6}$ **of her money** means a fraction of the **starting** amount.
- $\frac{2}{3}$ **of what is left** means a fraction of the **remainder** after the first spending.

### Worked example: Both fractions of the whole
Nicole had 90 dollars. She spent $\frac{2}{5}$ of it on a dress and $\frac{1}{6}$ of it on a book.

Dress: $90 \div 5 \times 2 = 36$. Book: $90 \div 6 = 15$. Left: $90 - 36 - 15 = 39$ dollars.

## Bar models for the remainder

*Diagram: Bar model: a bar split into 4 equal parts, 1 part for chocolate; the other 3 parts are what is left, and 2 of those are savings worth 7.50 dollars*
*Jeremy's pocket money as 4 equal parts. Two parts are saved.*

### Worked example: Fraction of the remainder
Steve pays $\frac{1}{3}$ of his wages in tax. He saves $\frac{1}{8}$ of what remains. He saves 40 dollars. What does he earn?

After tax, $\frac{2}{3}$ remains. He saves $\frac{1}{8}$ of $\frac{2}{3}$, which is $\frac{2}{24} = \frac{1}{12}$ of his wages.

$\frac{1}{12}$ of his wages is 40, so his wages are $40 \times 12 = 480$ dollars.

### Worked example: Marbles in a box
$\frac{2}{5}$ of the marbles are red, $\frac{1}{4}$ are green, and $\frac{3}{4}$ of the **remaining** marbles are blue. There are 21 blue marbles. How many marbles are there?

Use twentieths: red $\frac{8}{20}$, green $\frac{5}{20}$, so $\frac{13}{20}$ are red or green and $\frac{7}{20}$ remain.

$\frac{3}{4}$ of the remainder is 21, so the remainder is 28.

$\frac{7}{20}$ of the box is 28, so $\frac{1}{20}$ is 4 and the box holds 80 marbles.

## Working backwards

When you know the **end** and want the **start**, undo each step in reverse.

### Worked example: Tom's stamps
Tom had 108 stamps. He gave 27 to Mark and some to Jane. Then he had $\frac{4}{9}$ of his stamps left. How many did Jane get?

$\frac{4}{9}$ of 108 is $12 \times 4 = 48$ left. He gave away $108 - 48 = 60$. Jane got $60 - 27 = 33$.

### Exam tip: Draw it
Whenever a problem says "of the remainder" or "the rest", draw a bar. Split it for the first fraction, then split the leftover piece for the second fraction.

### Challenge: Selective level: the remainder is a fraction of the start
Jane spent 200 dollars on a dress and $\frac{2}{7}$ of her **remaining** money on a bag. She then had $\frac{5}{12}$ of her **original** money left. How much did she start with?

After the dress she had $R$ dollars. After the bag she had $\frac{5}{7}$ of $R$. That equals $\frac{5}{12}$ of the start, so $R$ is $\frac{7}{12}$ of the start.

So the dress was the other $\frac{5}{12}$ of the start: $\frac{5}{12}$ is 200 dollars, $\frac{1}{12}$ is 40 dollars, and she started with 480 dollars.

Check: $480 - 200 = 280$, the bag is $\frac{2}{7}$ of 280 $= 80$, leaving 200, and $\frac{5}{12}$ of 480 is 200.

**Mind map**

```text
Fraction word problems
  Operation words
    Sum, difference
    Product, quotient
    Increase, decrease
  Of what?
    Of the whole
    Of the remainder
  Bar models
    Split the whole
    Split the leftover
  Work backwards
    Start from the end
    Undo each step
  Check
    Put the answer back in
```

## Quick summary

- Learn the words: sum, difference, product, quotient, increase, decrease, square.
- Always ask "a fraction **of what**?": the whole, or the remainder.
- A fraction of the remainder is a fraction of a fraction: $\frac{1}{8}$ of $\frac{2}{3}$ is $\frac{1}{12}$.
- Bar models make these problems visual.
- If a fraction of the total is known, divide to find one part, then multiply.
- Check your answer by running the story forwards.

## Quiz

Try these questions from the chapter quiz.

1. What is the sum of $\frac{2}{3}$, $\frac{3}{4}$ and $\frac{1}{12}$?
   - A. $\frac{6}{19}$
   - B. $1\frac{1}{4}$
   - C. $1\frac{1}{2}$
   - D. $\frac{17}{12}$
   - Answer: C. $1\frac{1}{2}$. In twelfths: $\frac{8}{12} + \frac{9}{12} + \frac{1}{12} = \frac{18}{12} = 1\frac{1}{2}$.

2. Increase the difference between $\frac{5}{6}$ and $\frac{2}{3}$ by $\frac{1}{6}$.
   - A. $\frac{1}{3}$
   - B. $\frac{1}{6}$
   - C. $\frac{2}{3}$
   - D. $\frac{1}{2}$
   - Answer: A. $\frac{1}{3}$. Difference: $\frac{5}{6} - \frac{4}{6} = \frac{1}{6}$. Add $\frac{1}{6}$: $\frac{2}{6} = \frac{1}{3}$.

3. Nicole had 90 dollars. She spent $\frac{2}{5}$ of it on a dress and $\frac{1}{6}$ of it on a book. How much was left?
   - A. 51 dollars
   - B. 39 dollars
   - C. 45 dollars
   - D. 30 dollars
   - Answer: B. 39 dollars. Dress 36, book 15. $90 - 51 = 39$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/fraction-word-problems/#quiz