Practice Maths

L44 — Expressing as Fractions

Key Terms

fraction of a quantity
Expressing part A as a fraction of total B: AB. Both quantities must be in the same unit.
percentage
A fraction with denominator 100, written using the % symbol. To convert a fraction to a percentage, multiply by 100. E.g. 34 = 75%.
unit conversion
Changing quantities to the same unit before comparing. E.g. to express 30 cm as a fraction of 1 m, first convert 1 m = 100 cm.
simplify
Reduce a fraction to its lowest terms by dividing numerator and denominator by the HCF.
whole
The total amount used as the denominator when expressing a part as a fraction. Always identify “out of what total?”

Expressing A as a Fraction of B

A as a fraction of B = AB — both quantities must be in the same units before you write the fraction.

To convert to a percentage: multiply the fraction by 100.

Unit conversion reminder: 1 hour = 60 minutes, 1 km = 1000 m, 1 kg = 1000 g, 1 L = 1000 mL, 1 m = 100 cm.

Hot Tip: Always check that both quantities are in the same unit before forming the fraction. 15 minutes as a fraction of 1 hour means you must first write 1 hour as 60 minutes!

Worked Example 1 — 15 minutes as a fraction of 1 hour

Step 1: Convert to same units: 1 hour = 60 minutes.

Step 2: Fraction = 1560 = ¼

Step 3: As a percentage: ¼ × 100 = 25%

Worked Example 2 — 250 g as a fraction of 1 kg

Step 1: Convert: 1 kg = 1000 g.

Step 2: Fraction = 2501000 = ¼

Step 3: As a percentage: 25%

The Power of Proportional Thinking

Expressing one quantity as a fraction or percentage of another is a proportional thinking skill — it lets you compare quantities that might be different sizes. “30 out of 40 on a test” and “42 out of 60 on another test” — which is better? Only by expressing both as percentages (75% vs 70%) can you compare fairly.

This skill is used constantly: sports batting averages, rainfall comparisons, election vote percentages, nutritional labels, quality control in factories. It’s the mathematical basis of comparison.

Same Units: The Non-Negotiable Step

Before writing ANY fraction that compares two quantities, check they’re in the same unit. This is the most important step and the most commonly skipped one:

  • Wrong: 400 m as a fraction of 2 km → 4002 = 200 (nonsense!).
  • Correct: Convert 2 km = 2000 m, then 4002000 = 15 = 20%.

The rule: both quantities in the fraction must measure the same thing in the same unit.

Remember: Always write both quantities in the same unit before forming the fraction. Check: metres/metres, grams/grams, minutes/minutes. If you spot different units, convert before proceeding. No exceptions!

Three-Step Process for Every Problem

  1. Identify A (the part) and B (the whole). Ask: “out of what total?” to find B.
  2. Convert to same units if needed.
  3. Write AB, simplify, then ×100 for percentage if required.

Finding the Whole from a Fraction

Sometimes you’re given the fraction and the result, and need to find the original. “40% of a class is 20 students — how many students in total?”

  • 40% of total = 20 → 0.4 × total = 20 → total = 20 ÷ 0.4 = 50 students.
Common Mistake: Using the wrong denominator. “8 blue marbles out of 20 total” → fraction of blue = 820 (not 812, not 1220). The denominator is always the TOTAL, not the count of the other category.

  1. Express as a fraction in simplest form

    1. 3 out of 9
    2. 4 out of 20
    3. 6 out of 15
    4. 8 out of 24
    5. 12 out of 16
    6. 15 out of 25
    7. 18 out of 30
    8. 7 out of 21

  2. Express each as a percentage

    1. 1 out of 4
    2. 3 out of 5
    3. 2 out of 8
    4. 3 out of 20
    5. 7 out of 10
    6. 4 out of 25
    7. 9 out of 30
    8. 11 out of 50

  3. Unit conversions required — express as a fraction in simplest form

    1. 30 cm as a fraction of 1 m
    2. 20 minutes as a fraction of 1 hour
    3. 400 mL as a fraction of 1 L
    4. 250 g as a fraction of 1 kg
    5. 45 minutes as a fraction of 1 hour
    6. 600 m as a fraction of 1 km
    7. 75 cm as a fraction of 1 m
    8. 500 mL as a fraction of 2 L

  4. Identify the “whole” and check same-unit rule

    1. Explain why you cannot directly write 5 minutes2 hours as a fraction without converting first.
    2. A jug holds 750 mL. A bottle holds 1.5 L. What fraction of the bottle's volume is the jug?
    3. Tom scored 36 points out of a possible 48. Write this as a fraction in simplest form, then as a percentage.
    4. True or false: 400 m as a fraction of 2 km equals 15. Show your working.

  5. Word problems

    1. Mia scored 18 out of 24 on a maths test. What fraction did she get correct? Express as a percentage.
    2. A soccer team won 9 games out of 15 played. What fraction of games did they win? What percentage is that?
    3. Jake saved $35 out of his weekly earnings of $140. What fraction of his earnings did he save? Express as a percentage.
    4. A bag of lollies has 8 red, 12 blue and 20 green. What fraction of the bag is green? What fraction is not green?
    5. In PE class, 15 out of 25 students could complete 10 push-ups. The next month, 20 out of 25 could. Express both results as percentages and find the improvement.

  6. Express as a fraction and percentage — mixed units

    Convert to the same units, then express as a simplified fraction and as a percentage.

    1. 4 km as a fraction of 16 km
    2. 35 minutes as a fraction of 2 hours
    3. 80 cm as a fraction of 2 m
    4. 200 g as a fraction of 1.5 kg
    5. 450 mL as a fraction of 3 L
    6. 15 minutes as a fraction of 2 hours
    7. $1.20 as a fraction of $4.80
    8. 50 seconds as a fraction of 5 minutes

  7. Identify what the “whole” is

    1. A student answered 18 questions correctly and got 6 wrong. What fraction did they get correct?
    2. Out of 5 boys and 15 girls in a class, what fraction of students are boys? What fraction are girls?
    3. A bag contains 4 red, 3 blue and 5 green marbles. What fraction are not red?
    4. A team scored 42 goals and conceded 18 during a season. What fraction of total goals were scored by the team?
    5. In a car park, there are 12 sedans, 8 SUVs and 4 vans. What fraction of vehicles are SUVs? Write as a fraction and percentage.

  8. Percentage change and comparison

    1. Last month, a shop sold 60 items. This month it sold 75 items. Express this month's sales as a fraction and percentage of last month's.
    2. A plant was 40 cm tall. After two weeks it grew 12 cm. What fraction and percentage of the original height did it grow?
    3. Ella's test score improved from 56 out of 80 to 69 out of 80. Express both scores as percentages and calculate the percentage improvement.
    4. A road is 120 km. Roadworks cover 45 km. What fraction is affected? If repairs are complete on 30 km, what fraction and percentage of roadworks are done?

  9. Fraction as rate and density

    1. A survey of 200 people found that 150 prefer tea to coffee. Write this as a fraction, a decimal and a percentage.
    2. In a school of 480 students, 120 are in Year 7. What fraction of the school are Year 7 students? What percentage?
    3. A factory produces 2400 widgets per day. Quality checks reject 72. What fraction passes quality checks? What percentage is rejected?
    4. A sports team has 18 players. During a match, 12 players were on the field. Express the fraction on the field. If 6 players scored, what fraction of the team scored?

  10. Multi-step problems connecting fractions, decimals and percentages

    1. A coat originally cost $180. It is on sale for $135. What fraction of the original price is the sale price? What is the percentage discount?
    2. A class of 30 students sat two tests. In Test 1, 35 passed. In Test 2, 70% passed. How many more students passed Test 2 than Test 1?
    3. A farmer harvested 840 kg of wheat. She sold 58 of it and kept the rest. How much wheat did she keep? Express the kept amount as a percentage of the total.
    4. Three friends share the cost of a $240 gift. The first pays 13, the second pays 35%, and the third pays the rest. Who pays the most? How much does each person pay?
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