Practice Maths

L28 — Solving Problems with Fractions

Key Terms

fraction of a quantity
Multiply the fraction by the amount. E.g., 34 of 20 = 34 × 20 = 15.
finding the whole
If you know a fractional part and its value, divide the value by the fraction to find the original whole. E.g., 34 of ? = 15 → ? = 15 ÷ 34 = 20.
inverse operations
Opposite operations used to “undo” a calculation. Multiplication and division are inverse operations.

Key Methods

Finding a fraction of a quantity: multiply the fraction by the number.

34 of 20 = 34 × 20 = 604 = 15

Finding the whole when given a fraction: divide by the fraction.

If 34 of a number is 15, then the whole = 15 ÷ 34 = 15 × 43 = 20

Worked Example

A school library has 240 books. 38 of the books are fiction. How many fiction books are there?

Step 1 — Identify: 38 of 240

Step 2 — Multiply: 38 × 240 = 7208

Step 3 — Simplify: 720 ÷ 8 = 90 fiction books

“Of” Means Multiply

The most important idea in this lesson: whenever you see the word “of” between a fraction and a quantity, it means multiply. “34 of 20” = 34 × 20. “13 of 90” = 13 × 90.

Why does this work? “34 of 20” means “find 34 of the way from 0 to 20.” One quarter of 20 is 5, so three quarters is 15. 34 × 20 = 604 = 15. Both methods agree!

A Practical Method: Divide Then Multiply

For fractions like ab of something, here is a reliable two-step method:

  1. Divide by the denominator to find “one part.”
  2. Multiply by the numerator to find “that many parts.”

Example: 38 of 40. Divide 40 by 8 = 5 (one eighth). Multiply 5 by 3 = 15 (three eighths). So 38 of 40 = 15.

“Of” = multiply. You can also write it as one operation: 38 × 40 = 1208 = 15. Both methods give the same answer.

Finding the Whole When You’re Given the Part

This is the reverse problem. “34 of a number is 24. What is the number?” You have the fraction AND the result, but not the original.

Method: divide the given value by the fraction.

  • 24 ÷ 34 = 24 × 43 = 963 = 32. The original number was 32.
  • Check: 34 × 32 = 24 ✓
Common Mistake: In “finding the whole” problems, students sometimes multiply instead of dividing. If 35 of a number is 21, multiplying 21 × 35 gives 12.6 — clearly wrong (the answer should be bigger than 21, not smaller). If the answer seems too small, you’ve probably multiplied instead of divided.

Multi-Step Problems: Take It One Step at a Time

Some problems chain multiple fraction operations. For example: “Emma saves 25 of her weekly earnings of $75 for 4 weeks.”

  • Step 1: How much does she save each week? 25 × $75 = $30.
  • Step 2: Over 4 weeks: $30 × 4 = $120.

Label each step clearly and you will rarely make errors.

  1. Calculate

    1. 12 of 60
    2. 34 of 80
    3. 25 of 45
    4. 38 of 40
    5. 49 of 63

  2. Calculate

    1. 23 of 99
    2. 58 of 64
    3. 710 of 120
    4. 56 of 72
    5. 37 of 84

  3. Girls in a Class

    A class has 30 students. 25 are girls. How many girls are there?

  4. Red Lollies

    A bag contains 56 lollies. 37 are red. How many are red?

  5. Finding the Whole

    1. 34 of a number is 24. What is the number?
    2. 25 of a number is 16. What is the number?
    3. 58 of a number is 35. What is the number?

  6. Finding the Full Amount

    25 of an amount is $30. What is the full amount?

  7. Pocket Money

    Sam spends 38 of his weekly pocket money of $48. How much does he spend?

  8. Sports Day

    56 of the students in a school of 420 attend the sports day. How many attend?

  9. Emma’s Savings

    Emma saves 25 of her earnings. She earns $75 per week. After 4 weeks, how much has she saved?

  10. The Farm

    A farmer uses 35 of his land for crops and 14 for grazing. What fraction is used for other purposes? If the farm is 240 hectares, how many hectares is this?

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