Express one number as a percentage of another using (part ÷ whole) × 100
Convert units before calculating when needed
Apply to test scores, survey results, and comparisons
Compare two groups' performances using percentages
Real-World Applications
Education: Test scores are expressed as a percentage of the total marks available — scoring 36 out of 45 is converted to a percentage to compare performance across different papers or subjects.
Business: A company's market share is expressed as a percentage of total industry sales, allowing businesses and analysts to compare performance and track growth over time.
Healthcare: Survey results and clinical trial data are expressed as percentages — for example, the proportion of patients who responded to a treatment — to make findings clear and comparable.
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Example 2 — A student scores 48 out of 60. What percentage did they score?
Step 1
$(48 \div 60) \times 100$
Step 2
$0.8 \times 100$
Step 3
$= 80\%$
Example 3 — In a survey, 35 out of 50 people preferred Brand A. What % preferred Brand A?
Step 1
$(35 \div 50) \times 100$
Step 2
$0.70 \times 100$
Step 3
$= 70\%$
The "whole" is always the total group size or the maximum possible value.
✓ Quick Check 1 — The Core Method
Question 1 of 10
Same Units First — Converting Before Calculating
If the part and whole have different units, convert them to the same unit before calculating.
Always check units! You cannot express cm as a percentage of metres without converting first.
Example 4 — Express 25 cm as a percentage of 1.5 m
Step 1
Convert to same units: 1.5 m = 150 cm
Step 2
$(25 \div 150) \times 100$
Step 3
$= 16.7\%$ (to 1 d.p.)
Example 5 — Express 300 g as a percentage of 2 kg
Step 1
2 kg = 2000 g
Step 2
$(300 \div 2000) \times 100$
Step 3
$= 15\%$
Example 6 — Express 45 minutes as a percentage of 3 hours
Step 1
3 hours = 180 minutes
Step 2
$(45 \div 180) \times 100$
Step 3
$= 25\%$
✓ Quick Check 2 — Same Units First
Question 1 of 10
Real-Life Applications and Comparisons
Expressing numbers as percentages lets us compare results fairly — even when totals differ.
Example 7 — Student A: 42/60. Student B: 35/50. Who performed better?
Step 1
A: $(42 \div 60) \times 100 = 70\%$
Step 2
B: $(35 \div 50) \times 100 = 70\%$
Step 3
Equal performance — both scored 70%
Example 7b — Student A: 54/90. Student B: 40/80. Who performed better?
Step 1
A: $(54 \div 90) \times 100 = 60\%$
Step 2
B: $(40 \div 80) \times 100 = 50\%$
Step 3
Student A performed better (60% vs 50%) — even though B's raw score (40) looks close to A's (54), converting to percentages of their own totals reveals the real gap.
Example 8 — A survey of 200 people: 150 said they preferred online shopping. What %?
Step 1
$(150 \div 200) \times 100$
Step 2
$= 75\%$
Example 9 — A shop has 48 items. It sells 36. What % did it sell?
Step 1
$(36 \div 48) \times 100$
Step 2
$= 75\%$
Percentages allow fair comparison even when totals are different. Always convert to % before comparing.
✓ Quick Check 3 — Real-Life Applications and Comparisons
Question 1 of 10
Practice: Expressing as a Percentage
Calculate the percentage for each. Enter your answer (round to 1 d.p. where needed).
#
Question
Answer
Result
1
30 out of 40
2
48 out of 60
3
25 cm of 150 cm
4
300 g of 2000 g
5
45 min of 180 min
6
35 out of 50
7
150 out of 200
8
36 out of 48
9
84 out of 210
10
63 out of 90
⚠ Things to Watch Out For
Wrong order: A as a percentage of B = (A÷B)×100. Students often reverse it to (B÷A)×100. The quantity you're expressing goes on top.
Forgetting to multiply by 100: A÷B gives a decimal. Multiply by 100 to convert to a percentage. Missing this step gives the decimal (proportion), not the percentage.
Mismatched units: "30 minutes as a percentage of 2 hours" — convert first: 2 hours = 120 minutes. Then (30÷120)×100 = 25%.
Rounding too early: Keep full precision in the division step. Only round the final percentage to a sensible number of decimal places.
Percentages over 100: If A > B, the percentage exceeds 100%. This is mathematically correct — e.g. a profit that exceeds the original investment.