Fraction, Decimal, Percentage Problems

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📚 Part 1: Multiple Choice (Quick checks)

1. What is 1/4 expressed as a decimal?

0.25

0.4

0.75

2. Which fraction is equivalent to 40%?

2/5

4/10

1/4

3. Select all that are equivalent to 0.5

1/2

50%

5/10

2/5

4. Estimate: Which is closest to 25% of 360?

100

5. Which decimal is equivalent to 7/8?

0.875

0.78

0.87

6. What percentage is equivalent to 0.35?

3.5%

35%

350%

✏️ Part 2: Short Answers — Differentiated

Level 1: Foundations (with worked examples and scaffolding)

📝 Worked Example: Finding a fraction of a number

Example: Find 1/3 of 12

Method 1 - Division: Divide the number by the denominator (bottom number)
12 ÷ 3 = 4
So 1/3 of 12 = 4

Method 2 - Multiplication: Multiply the number by the fraction
1/3 × 12 = 12/3 = 4

Visual: Imagine 12 objects split into 3 equal groups. Each group has 4 objects.

7. Find 1/2 of 48. (Hint: What do you divide 48 by to find half?)

📝 Worked Example: Converting decimals to percentages

Example: Convert 0.25 to a percentage

Rule: To convert a decimal to a percentage, multiply by 100 and add the % symbol

Step 1: 0.25 × 100 = 25
Step 2: Add % symbol = 25%

Quick tip: Move the decimal point 2 places to the right!

0.25 → 2.5 → 25 → 25%

8. Write 0.75 as a percentage. (Remember: multiply by 100 or move decimal 2 places right)

📝 Worked Example: Converting fractions to decimals

Example: Convert 1/4 to a decimal

Method: Divide the top number by the bottom number

Calculation: 1 ÷ 4 = 0.25

Visual thinking: 1/4 means "1 out of 4 equal parts" which is the same as 0.25

Memory aid: Think of money - 1/4 of a dollar is 25 cents = $0.25

9. Convert 3/5 to a decimal. (Divide the top by the bottom: 3 ÷ 5 = ?)

10. What is 1/4 of 80? (Use the same method as question 7)

📝 Worked Example: Converting percentages to fractions

Example: Convert 25% to a fraction in simplest form

Step 1: Write the percentage as a fraction over 100
25% = 25/100

Step 2: Simplify by finding common factors
25/100 = 25÷25/100÷25 = 1/4

Check: Does 1/4 = 25%? Yes! 1÷4 = 0.25 = 25%

11. Write 60% as a fraction in simplest form. (Start with 60/100, then simplify)

12. What is 1/5 of 35? (Hint: Divide 35 by 5)

13. Convert 0.4 to a percentage. (Use the same method as question 8)

14. Write 2/10 as a decimal. (Remember: 2 ÷ 10 = ?)

🎯 Quick Reference Guide

Common Conversions to Remember:

1/2 = 0.5 = 50%
1/4 = 0.25 = 25%
3/4 = 0.75 = 75%
1/5 = 0.2 = 20%
1/10 = 0.1 = 10%

Conversion Rules:

Fraction → Decimal: Divide top ÷ bottom
Decimal → Percentage: × 100 and add %
Percentage → Fraction: Put over 100 and simplify

Level 2 (Beginning Year 6): Apply and calculate

15. What is 15% of $240? Show your working.

16. A recipe needs 3/4 cup of sugar. You only have a 1/2 cup measure. How many half-cups are needed?

17. Calculate 2/3 of 72. Show your method.

18. A shop increases prices by 20%. If a toy originally cost $15, what is the new price?

19. Convert 5/8 to both a decimal and percentage. Show your working.

20. Find 3/4 of 120. Show your calculation method.

21. What is 30% of 150? Show your working.

22. A class has 24 students. If 5/8 of them wear glasses, how many students wear glasses?

Level 3 (Year 6 — Extended problems)

23. A backpack costs $120. It is reduced by 25%, then an extra 10% is taken off the reduced price. Calculate the final price. Show all steps.

24. Estimation challenge: The school expects 1,250 visitors. About 18% will buy a sausage at $3 each and 12% will buy a drink at $2 each. Estimate the total revenue from sausages and drinks. Write your estimate and show key calculations.

25. Multi-step problem: Sarah saves 3/8 of her weekly allowance of $24. She spends 40% of her savings on a book and 1/3 of the remaining money on a gift. How much money does she have left? Show all calculations.

26. Challenge: A pizza is cut into 12 equal slices. Tom eats 25% of the pizza, Lisa eats 1/3 of the pizza, and Jake eats 0.2 of the pizza. How many slices are left? Express your answer as a fraction of the whole pizza and as a number of slices.

27. Real-world application: A store has a "buy 2, get 1 free" sale on books that normally cost $18 each. If you buy 6 books, what percentage discount do you receive compared to the normal price? Show your reasoning.

28. Complex conversion: Order these values from smallest to largest: 7/12, 0.58, 59%, 3/5. Show your working by converting all to the same form.

29. Investment problem: Maya invests $800. After one year, her investment increases by 12.5%. She then withdraws 3/8 of the new total. How much money does she have left in her investment? Show all steps.

30. Sports statistics: In a basketball season, James made 7/10 of his free throws, Sarah made 68% of hers, and Alex made 0.72 of his. Who had the best free throw percentage? Convert all to percentages and order from best to worst performance.

31. Recipe scaling: A cake recipe serves 8 people and uses 2.5 cups of flour. If you want to make enough cake for 14 people, how much flour do you need? Express your answer as both a decimal and a mixed number. Show your reasoning.

📋 Answer Key

Part 1: Multiple Choice

1. What is 1/4 expressed as a decimal?

Answer: 0.25

2. Which fraction is equivalent to 40%?

Answer: 2/5 (or 4/10 - both are correct)

3. Select all that are equivalent to 0.5

Answer: 1/2, 50%, 5/10 (all correct except 2/5)

4. Estimate: Which is closest to 25% of 360?

Answer: 90 (25% of 360 = 90 exactly)

5. Which decimal is equivalent to 7/8?

Answer: 0.875

6. What percentage is equivalent to 0.35?

Answer: 35%

Part 2: Short Answers

Level 1 Solutions

7. Find 1/2 of 48.

Answer: 24

Working: 48 ÷ 2 = 24

8. Write 0.75 as a percentage.

Answer: 75%

Working: 0.75 × 100 = 75%

9. Convert 3/5 to a decimal.

Answer: 0.6

Working: 3 ÷ 5 = 0.6

10. What is 1/4 of 80?

Answer: 20

Working: 80 ÷ 4 = 20

11. Write 60% as a fraction in simplest form.

Answer: 3/5

Working: 60% = 60/100 = 6/10 = 3/5

12. What is 1/5 of 35?

Answer: 7

Working: 35 ÷ 5 = 7

13. Convert 0.4 to a percentage.

Answer: 40%

Working: 0.4 × 100 = 40%

14. Write 2/10 as a decimal.

Answer: 0.2

Working: 2 ÷ 10 = 0.2

Level 2 Solutions

15. What is 15% of $240? Show your working.

Answer: $36

Working: 15% = 15/100 = 0.15
0.15 × $240 = $36

16. A recipe needs 3/4 cup of sugar. You only have a 1/2 cup measure. How many half-cups are needed?

Answer: 1.5 half-cups (or 1 and a half)

Working: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 1.5

17. Calculate 2/3 of 72. Show your method.

Answer: 48

Working: 2/3 × 72 = (2 × 72) ÷ 3 = 144 ÷ 3 = 48

18. A shop increases prices by 20%. If a toy originally cost $15, what is the new price?

Answer: $18

Working: 20% of $15 = 0.20 × $15 = $3
New price = $15 + $3 = $18

19. Convert 5/8 to both a decimal and percentage. Show your working.

Answer: 0.625 and 62.5%

Working: 5 ÷ 8 = 0.625
0.625 × 100% = 62.5%

20. Find 3/4 of 120. Show your calculation method.

Answer: 90

Working: 3/4 × 120 = (3 × 120) ÷ 4 = 360 ÷ 4 = 90

21. What is 30% of 150? Show your working.

Answer: 45

Working: 30% = 30/100 = 0.3
0.3 × 150 = 45

22. A class has 24 students. If 5/8 of them wear glasses, how many students wear glasses?

Answer: 15 students

Working: 5/8 × 24 = (5 × 24) ÷ 8 = 120 ÷ 8 = 15

Level 3 Solutions

23. A backpack costs $120. It is reduced by 25%, then an extra 10% is taken off the reduced price. Calculate the final price. Show all steps.

Answer: $81

Working:
Step 1: 25% of $120 = 0.25 × $120 = $30
Step 2: Price after first reduction = $120 - $30 = $90
Step 3: 10% of $90 = 0.10 × $90 = $9
Step 4: Final price = $90 - $9 = $81

24. Estimation challenge: The school expects 1,250 visitors. About 18% will buy a sausage at $3 each and 12% will buy a drink at $2 each. Estimate the total revenue from sausages and drinks.

Answer: Approximately $975

Working:
Sausage buyers: 18% of 1,250 ≈ 225 people
Sausage revenue: 225 × $3 = $675
Drink buyers: 12% of 1,250 ≈ 150 people
Drink revenue: 150 × $2 = $300
Total revenue: $675 + $300 = $975

Answer: $3.60

Working:
Step 1: Sarah's savings = 3/8 × $24 = $9
Step 2: Spent on book = 40% × $9 = 0.4 × $9 = $3.60
Step 3: Money left after book = $9 - $3.60 = $5.40
Step 4: Spent on gift = 1/3 × $5.40 = $1.80
Step 5: Money left = $5.40 - $1.80 = $3.60

Answer: 3 slices left, which is 1/4 of the pizza

Working:
Tom eats: 25% = 1/4 = 3 slices
Lisa eats: 1/3 = 4 slices
Jake eats: 0.2 = 1/5 = 2.4 slices ≈ 2 slices
Total eaten: 3 + 4 + 2 = 9 slices
Slices left: 12 - 9 = 3 slices = 3/12 = 1/4 of pizza

Answer: 33.33% (or 33⅓%) discount

Working:
Normal price for 6 books: 6 × $18 = $108
With sale: Pay for 4 books, get 2 free = 4 × $18 = $72
Savings: $108 - $72 = $36
Percentage discount: ($36 ÷ $108) × 100% = 33.33%

28. Complex conversion: Order these values from smallest to largest: 7/12, 0.58, 59%, 3/5. Show your working by converting all to the same form.

Answer: 0.58, 7/12, 59%, 3/5

Working (converting to decimals):
7/12 = 0.583̄
0.58 = 0.58
59% = 0.59
3/5 = 0.6
Order: 0.58 < 0.583̄ < 0.59 < 0.6

Answer: $562.50

Working:
Step 1: 12.5% increase = 12.5% of $800 = 0.125 × $800 = $100
Step 2: New total = $800 + $100 = $900
Step 3: Amount withdrawn = 3/8 × $900 = $337.50
Step 4: Money left = $900 - $337.50 = $562.50

Answer: Alex (72%), James (70%), Sarah (68%)

Working:
James: 7/10 = 0.7 = 70%
Sarah: 68% = 68%
Alex: 0.72 = 72%
Order from best to worst: Alex (72%), James (70%), Sarah (68%)

Answer: 4.375 cups or 4⅜ cups

Working:
Step 1: Find the scaling factor = 14 ÷ 8 = 1.75
Step 2: New flour amount = 2.5 × 1.75 = 4.375 cups
Step 3: Convert to mixed number: 4.375 = 4 + 0.375 = 4 + 3/8 = 4⅜ cups