Percentages: meaning and conversion

$30\%$ of the students passed. The shop offers a $25\%$ discount. The interest rate is $7\%$. The percent sign — $\%$ — means "per hundred." $30\%$ means $30$ out of every $100$, or the fraction $\dfrac{30}{100} = \dfrac{3}{10}$.

Percentages are everywhere in WAEC: in commerce (discounts, profit/loss), in statistics (percentage of students), in interest calculations, in growth and decay problems. The topic appears directly in nearly every paper.

This lesson introduces percentages, converts between percentages, fractions, and decimals, and lays the foundation for the operations covered in the next lesson.

Learning objectives

By the end of this lesson you should be able to:

• explain what a percentage means
• convert between percentages, fractions and decimals
• recall common percentage equivalents
• express one quantity as a percentage of another

Why percentages?

Percentages express proportions in a standard, comparable way. Whether you're talking about $30$ out of $100$ students, $30$ out of $50$ marks, or $300$ out of $1000$ NLe, the proportion is the same: $30\%$.

This makes percentages ideal for:

• Comparison: $80\%$ in Math vs $75\%$ in English shows which is better, regardless of total marks.
• Standardisation: different quizzes have different totals, but percentages put them on the same scale.
• Finance: interest rates of $5\%$, $7\%$, $10\%$ are directly comparable.
• Daily life: discounts of $15\%, 25\%, 50\%$ make pricing transparent.

Converting between forms

Percentage → Fraction

Divide by $100$ and simplify.

• $25\% = \dfrac{25}{100} = \dfrac{1}{4}$.
• $40\% = \dfrac{40}{100} = \dfrac{2}{5}$.
• $60\% = \dfrac{60}{100} = \dfrac{3}{5}$.
• $75\% = \dfrac{75}{100} = \dfrac{3}{4}$.
• $100\% = 1$.
• $150\% = \dfrac{150}{100} = \dfrac{3}{2} = 1\dfrac{1}{2}$.
• $250\% = \dfrac{250}{100} = \dfrac{5}{2} = 2\dfrac{1}{2}$.
• $5\% = \dfrac{5}{100} = \dfrac{1}{20}$.
• $12\dfrac{1}{2}\% = \dfrac{12.5}{100} = \dfrac{25}{200} = \dfrac{1}{8}$.

Percentage → Decimal

Move the decimal point two places to the left.

• $50\% = 0.5$.
• $75\% = 0.75$.
• $100\% = 1.00 = 1$.
• $25\% = 0.25$.
• $5\% = 0.05$.
• $150\% = 1.5$.
• $0.5\% = 0.005$.
• $7\dfrac{1}{2}\% = 7.5\% = 0.075$.

Fraction → Percentage

Multiply by $100$.

• $\dfrac{1}{2} = 0.5 = 50\%$.
• $\dfrac{1}{4} = 0.25 = 25\%$.
• $\dfrac{1}{5} = 0.2 = 20\%$.
• $\dfrac{1}{8} = 0.125 = 12.5\%$.
• $\dfrac{2}{3} = 0.\overline{6} = 66\dfrac{2}{3}\%$.
• $\dfrac{3}{8} = 0.375 = 37.5\%$.
• $\dfrac{7}{20} = 0.35 = 35\%$.

Decimal → Percentage

Multiply by $100$.

• $0.6 = 60\%$.
• $0.3 = 30\%$.
• $0.04 = 4\%$.
• $1.5 = 150\%$.
• $0.0025 = 0.25\%$.

Common percentage equivalents to memorise

These show up so often that knowing them by heart saves time on WAEC.

Expressing one quantity as a percentage of another

To express $A$ as a percentage of $B$: $\dfrac{A}{B} \times 100\%$.

Example 1

Express $15$ as a percentage of $60$.

$\dfrac{15}{60} \times 100 = 25\%$.

Example 2

Aminata scored $42$ out of $60$ in a Math test. What percentage did she score?

$\dfrac{42}{60} \times 100 = 70\%$.

Example 3

A class of $40$ students had $32$ pass the test. What was the pass rate?

$\dfrac{32}{40} \times 100 = 80\%$.

Example 4

$25$ cm out of $1$ m. Express as a percentage.

Convert to same units: $25$ cm and $100$ cm. $\dfrac{25}{100} \times 100 = 25\%$.

Example 5

$200$ grams as a percentage of $1.5$ kg.

Convert to same units: $200$ g and $1500$ g. $\dfrac{200}{1500} \times 100 = \dfrac{200}{15} \approx 13.33\%$.

Example 6

Mohamed earned $4500$ NLe last month. His expenses were $3000$ NLe. What percentage of his income did he spend? What percentage did he save?

Spending: $\dfrac{3000}{4500} \times 100 = \dfrac{300}{4.5} = \dfrac{2}{3} \times 100 = 66\dfrac{2}{3}\%$. Saving: $100\% - 66\dfrac{2}{3}\% = 33\dfrac{1}{3}\%$.

Or directly: $1500/4500 \times 100 = 33\dfrac{1}{3}\%$. ✓

Example 7

In an examination, Aminata got $54$ out of $80$ in Mathematics and $48$ out of $70$ in English. In which subject did she perform better?

Math: $\dfrac{54}{80} \times 100 = 67.5\%$. English: $\dfrac{48}{70} \times 100 \approx 68.57\%$.

She performed slightly better in English.

Comparing percentages

Often you'll need to determine which of two percentages represents more.

Example 8

Which is larger: $30\%$ of $200$ or $25\%$ of $240$?

First: $30\% \times 200 = 60$. Second: $25\% \times 240 = 60$. Both are $60$. They're equal.

Example 9

Aminata scored $80\%$ in a test out of $50$. Mohamed scored $75\%$ in a test out of $80$. Who got more correct answers?

Aminata: $80\% \times 50 = 40$ correct. Mohamed: $75\% \times 80 = 60$ correct.

Mohamed got more correct, even though his percentage was lower (he had more total to start with).

This illustrates a key idea: percentages aren't directly comparable across different totals unless you check the absolute numbers.

Percentages over $100\%$ and under $1\%$

Percentages aren't bounded between $0$ and $100$. They can exceed $100\%$ (when the part is larger than the whole) and be less than $1\%$ (when the part is much smaller than the whole).

Examples

• A factory's output this year is $130\%$ of last year's. (More than the whole — output grew by $30\%$.)
• Only $0.5\%$ of the population was sampled. (Half a percent — a small fraction.)
• The interest is $7.5\%$ per annum. (Between $0$ and $100$.)

Example 10

A company's sales last year were $400000$ NLe. This year's sales were $520000$ NLe. Express this year's sales as a percentage of last year's.

$\dfrac{520000}{400000} \times 100 = 130\%$.

The sales are $130\%$ of last year's, i.e., a $30\%$ increase.

Example 11

Out of $500$ candidates, only $3$ scored above $90\%$. What percentage of candidates is this?

$\dfrac{3}{500} \times 100 = 0.6\%$.

Very small.

Sierra Leonean context examples

Example 12

A WAEC class has $80$ students. $35$ passed Mathematics. What percentage passed?

$\dfrac{35}{80} \times 100 = 43.75\%$.

Example 13

Aminata bought a piece of fabric at $\dfrac{2}{3}$ of its original price of $300$ NLe. What was the discount, expressed as a percentage?

Discount = $1 - \dfrac{2}{3} = \dfrac{1}{3}$ of original. As percentage: $\dfrac{1}{3} \times 100 = 33\dfrac{1}{3}\%$.

Verify: she paid $\dfrac{2}{3} \times 300 = 200$ NLe. She saved $100$ NLe out of $300$, which is $\dfrac{100}{300} = 33\dfrac{1}{3}\%$. ✓

Example 14

A Sierra Leonean village has $500$ people. Of these, $300$ are children, $150$ are adults, and $50$ are elderly. Find the percentage of each group.

Children: $\dfrac{300}{500} \times 100 = 60\%$. Adults: $\dfrac{150}{500} \times 100 = 30\%$. Elderly: $\dfrac{50}{500} \times 100 = 10\%$. Check: $60\% + 30\% + 10\% = 100\%$. ✓

WASSCE examination focus

Practice problems

1. Convert to a fraction in simplest form. (a) $40\%$ (b) $75\%$ (c) $35\%$ (d) $87\dfrac{1}{2}\%$ (e) $150\%$

2. Convert to a decimal. (a) $25\%$ (b) $87.5\%$ (c) $5\%$ (d) $0.2\%$ (e) $200\%$

3. Convert to a percentage. (a) $\dfrac{3}{8}$ (b) $0.85$ (c) $\dfrac{5}{6}$ (d) $1.4$ (e) $\dfrac{17}{20}$

4. Express the first as a percentage of the second. (a) $15$ out of $25$. (b) $30$ out of $40$. (c) $250$ g out of $4$ kg. (d) $45$ minutes out of $3$ hours. (e) $150$ NLe out of $400$ NLe.

5. Aminata scored $48$ out of $75$ on a test. What percentage did she score?

6. Mohamed earned $6000$ NLe and spent $4500$ NLe. What percentage of his earnings did he spend? What percentage did he save?

7. Out of $200$ candidates, $35$ scored distinctions. What percentage scored distinctions?

8. A class has $50$ students. $30$ are girls. What percentage are girls? boys?

9. A factory produced $480$ items, of which $24$ were defective. Percentage defective? Percentage good?

10. Foday's earnings rose from $4000$ NLe to $5000$ NLe. Express the new earnings as a percentage of the old.

11. Which is larger: $\dfrac{3}{5}$, $0.55$, or $58\%$?

12. Of $150$ products, $30$ were sold. What percentage were sold? What percentage are unsold?

13. A water tank is $60\%$ full. If the tank holds $250$ litres, how much water is in it?

14. In a survey, $36\%$ of $250$ people preferred tea. How many preferred tea?

Worked solutions

1. (a) $\dfrac{2}{5}$. (b) $\dfrac{3}{4}$. (c) $\dfrac{7}{20}$. (d) $\dfrac{7}{8}$. (e) $\dfrac{3}{2} = 1.5$.

2. (a) $0.25$. (b) $0.875$. (c) $0.05$. (d) $0.002$. (e) $2$.

3. (a) $37.5\%$. (b) $85\%$. (c) $83\dfrac{1}{3}\%$. (d) $140\%$. (e) $85\%$.

4. (a) $60\%$. (b) $75\%$. (c) $\dfrac{250}{4000} \times 100 = 6.25\%$. (d) $\dfrac{45}{180} \times 100 = 25\%$. (e) $\dfrac{150}{400} \times 100 = 37.5\%$.

5. $\dfrac{48}{75} \times 100 = 64\%$.

6. Spent: $\dfrac{4500}{6000} \times 100 = 75\%$. Saved: $25\%$.

7. $\dfrac{35}{200} \times 100 = 17.5\%$.

8. Girls: $60\%$. Boys: $40\%$.

9. Defective: $\dfrac{24}{480} \times 100 = 5\%$. Good: $95\%$.

10. $\dfrac{5000}{4000} \times 100 = 125\%$.

11. Convert all to decimals: $\dfrac{3}{5} = 0.6, 0.55, 0.58$. Largest: $0.6 = \dfrac{3}{5}$.

12. Sold: $\dfrac{30}{150} \times 100 = 20\%$. Unsold: $80\%$.

13. $60\% \times 250 = 150$ litres.

14. $36\% \times 250 = 90$ people.

What's next

In the next lesson we cover finding percentages of quantities — the core operation $A\% \times B$ used in nearly every percentage problem: discounts, taxes, interest, statistics, and more.