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MATHEMATICSNumber & Arithmetic
CURRENT TOPICPercentages
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● Master Topic Guide Verified 11+ Curriculum

Percentages Master Guide 📊

Master percentage definitions, FDP conversions, finding percentage of amounts, reverse percentages, and bar model word problems for 11+ exams!

Practice Levels:Warm-UpCore ModePro Mode
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PERCENTAGES OVERVIEW

A percentage means "out of 100". We write it using the % symbol. ($25\% = \frac25100 = 0.25$)

1. WHAT IS A PERCENTAGE?

A percentage shows a part of a whole "out of 100". The whole is 100%.

25%
═ 25 out of 100
═ 25/100 ═ 0.25
2. PERCENTAGE, FRACTION & DECIMAL

Percentages, fractions and decimals can show the same value.

PercentageFractionDecimal
1%1/1000.01
5%5/100 or 1/200.05
10%10/100 or 1/100.10
25%25/100 or 1/40.25
50%50/100 or 1/20.50
75%75/100 or 3/40.75
100%100/100 or 11.00
3. BAR MODEL & PIE CHART

Percentages can be shown in different visual ways.

25%
25%
25%
25%
■ 75% shaded
□ 25% not shaded
4. EQUIVALENT PERCENTAGES

Different percentages and fractions can be equivalent.

50% ═ 1/2 ═ 0.5 ═ 2/4 ═ 4/8
1/2
5. FINDING % OF AN AMOUNT
Formula: (% / 100) × Amount
10% of 80 ═ (10/100) × 80 ═ 8
25% of 60 ═ (25/100) × 60 ═ 15
50% of 90 ═ (50/100) × 90 ═ 45
75% of 40 ═ (75/100) × 40 ═ 30
5% of 200 ═ (5/100) × 200 ═ 10
6. FINDING THE WHOLE AMOUNT
Formula: Whole ═ (Part / %) × 100
20 is 25% of ? ➔ (20/25)×100 ═ 80
15 is 30% of ? ➔ (15/30)×100 ═ 50
36 is 60% of ? ➔ (36/60)×100 ═ 60
7. PERCENTAGES GREATER THAN 100%

Percentages can be more than 100%. (150% means 150 out of 100 = 1.5)

120% ═ 1.2175% ═ 1.75250% ═ 2.5
Number Line: 0 ──────── 100% ──── 150% ──── 200%
8. REAL LIFE EXAMPLES
🏷️ Sale: £50 shirt reduced 20% ➔ 20% is £10 ➔ Sale price £40
🔋 Battery: 75% charged ➔ 75 out of 100 parts charged
📝 Test score: 64 out of 80 ➔ (64/80)×100 ═ 80%
🔄

EQUIVALENT FRACTIONS, DECIMALS & PERCENTAGES

Different ways to show the same value. Fractions, decimals and percentages are equivalent.

KEY IDEAS
  • ★ Equivalent means equal in value.
  • ★ Size is the same, just written differently.
  • ★ You can convert between all three!
EQUIVALENT FRACTIONS, DECIMALS & PERCENTAGES – QUICK REFERENCE TABLE
FRACTIONDECIMALPERCENTAGE
1/4
0.25
25%
25%
25%
25%
25%
1/2
0.50
50%
25%
25%
25%
25%
3/4
0.75
75%
25%
25%
25%
25%
1/1
1.00
100%
25%
25%
25%
25%
1/5
0.20
20%
25%
25%
25%
25%
2/5
0.40
40%
25%
25%
25%
25%
3/5
0.60
60%
25%
25%
25%
25%
4/5
0.80
80%
25%
25%
25%
25%
1/8
0.125
12.5%
25%
25%
25%
25%
3/8
0.375
37.5%
25%
25%
25%
25%
5/8
0.625
62.5%
25%
25%
25%
25%
7/8
0.875
87.5%
25%
25%
25%
25%
1/10
0.10
10%
25%
25%
25%
25%
FRACTION TO DECIMAL

1. Divide numerator by denominator.

3/4 ═ 3 ÷ 4 ═ 0.75
DECIMAL TO PERCENTAGE

1. Multiply decimal by 100.

0.75 × 100 ═ 75%
PERCENTAGE TO DECIMAL

1. Divide percentage by 100.

75% ÷ 100 ═ 0.75
PERCENTAGE TO FRACTION

1. Write percentage over 100.
2. Simplify the fraction.

75% ═ 75/100 ═ 3/4
COMMON EQUIVALENT FRACTIONS (USEFUL TO KNOW)
1/2 ═ 2/4 ═ 4/8
1/3 ═ 2/6 ═ 3/9
1/4 ═ 2/8 ═ 3/12
3/4 ═ 6/8 ═ 9/12
REMEMBER

★ All three show the exact same amount!

★ It's just a different way of writing it!

INTERACTIVE FDP GRID VISUALIZER

Hover over any grid to see its percentage representation!

50% (1/2)50% out of 100
25% (1/4)25% out of 100
10% (1/10)10% out of 100
75% (3/4)75% out of 100
20% (1/5)20% out of 100
1% (1/100)1% out of 100
📊

PERCENTAGE WORD PROBLEMS USING BAR MODELS

Bar models help us understand percentage problems by showing the whole as 100% and parts as equal sections.

1. FINDING A PERCENTAGE OF AN AMOUNT

Q: 25% of a number is 30. What is the number?

25% (30)
25% (30)
25% (30)
25% (30)

25% is 1/4 of the whole. If 25% = 30, then 100% = 30 × 4 = 120.

Answer: The number is 120.
2. FINDING THE WHOLE AMOUNT

Q: 60% of a number is 72. What is the number?

10%
10%
10%
10%
10%
10%
10%
10%
10%
10%

60% is 6 lots of 10%. 72 ÷ 6 = 12 (one lot of 10%). 100% = 12 × 10 = 120.

Answer: The number is 120.
3. FINDING A PART (PERCENTAGE)

Q: A bag has 80 sweets. 30% of the sweets are red. How many are red?

8
8
8
8
8
8
8
8
8
8

10% of 80 = 8. 30% = 3 lots of 10% = 8 × 3 = 24 sweets.

Answer: 24 sweets are red.
4. FINDING A PERCENTAGE

Q: In a class of 40 students, 16 students like pizza. What percentage likes pizza?

Each 10% block = 40 ÷ 10 = 4 students. 16 students = 4 lots of 10% = 40%.

Answer: 40% of the class likes pizza.
5. MULTI-STEP WORKING BACKWARDS

Q: Sam gives £30 to his sister, spends half of the rest on presents, and has £20 left. How much did he start with?

Work backwards: £20 left is half of remaining ➔ remaining before presents = £40. Before £30 sister gift = £40 + £30 = £70.

Answer: Sam started with £70.
6. SALE DISCOUNT COMPARISON

Q: A jacket is £80. It is in a sale with 25% off. What is the sale price?

Whole 100% = £80. Discount 25% = £20. Sale price 75% = £80 − £20 = £60.

Answer: The sale price is £60.
INTERACTIVE PERCENTAGES CHAMPION QUIZ11+ Exam Practice
1. Find 15% of 80:
2. If 20% of a number is 16, what is the number?
3. Reduce £50 by 20%:
4. 16 out of 40 students like pizza. What % is this?

🚀 Ready to Test Your Knowledge?

Take our graded 11+ Percentages assessment now!