Percentages

The word per cent comes from Latin per centum, meaning "out of one hundred". A percentage is just a ratio whose second part is fixed at $100$. Saying "$30\%$" is the same as "$30$ out of $100$" or "$\tfrac{30}{100} = 0.3$".

Concept

Conversions.

• Percent $\to$ fraction: $30\% = \dfrac{30}{100} = \dfrac{3}{10}$.
• Percent $\to$ decimal: $30\% = 0.30$ (just shift the decimal $2$ places left).
• Fraction $\to$ percent: multiply by $100$. $\dfrac{3}{4} = 75\%$.
• Decimal $\to$ percent: multiply by $100$. $0.4 = 40\%$.

Finding a percent of a quantity. "$p\%$ of $x$" means $\dfrac{p}{100} \cdot x$.

Example: $15\%$ of $80 = \dfrac{15}{100} \cdot 80 = 12$.

Expressing one quantity as a percent of another. Marks $54$ out of $60$: $\dfrac{54}{60} \times 100 = 90\%$.

Percentage change. When a quantity goes from $a$ to $b$:

$\text{percentage change} = \frac{b - a}{a} \times 100.$

Positive $\to$ increase. Negative $\to$ decrease.

Example: a $\mathbb{?}\,500$ shirt is now $\mathbb{?}\,400$. Change $= (400 - 500)/500 \times 100 = -20\%$. A $20\%$ decrease.

Useful mental percentages.

To compute $20\%$ of $\mathbb{?}\,800$, take one-fifth: $\mathbb{?}\,160$. To compute $25\%$ of $\mathbb{?}\,640$, take one-fourth: $\mathbb{?}\,160$.

Reverse problems. "$20\%$ of what is $80$?" Set up the equation: $\dfrac{20}{100} \cdot x = 80$. So $x = 80 \cdot 5 = 400$.

Worked examples

Example 1. Express $\dfrac{7}{20}$ as a percent.

• $\dfrac{7}{20} = 0.35 = 35\%$.

Example 2. Find $12\%$ of $250$.

• $\dfrac{12}{100} \cdot 250 = 30$.

Example 3. A score increased from $60$ to $75$. Find the percent increase.

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

Example 4. A book is sold for $\mathbb{?}\,540$ after a $10\%$ discount. Find the original price.

• Let original $= x$. After $10\%$ off: $x \cdot 0.9 = 540$.
• $x = 540 / 0.9 = 600$.

Try it yourself

1. Express $0.85$ and $\dfrac{3}{8}$ as percentages.
2. Find $18\%$ of $250$.
3. What is $40\%$ of $\mathbb{?}\,720$?
4. $24$ is what percent of $80$?
5. A salary rose from $\mathbb{?}\,40{,}000$ to $\mathbb{?}\,46{,}000$. Find the percent increase.
6. After a $15\%$ discount a bag costs $\mathbb{?}\,1700$. Find its marked price.
7. A class has $48$ students of whom $36$ pass. What is the pass percentage?
8. If $35\%$ of a number is $84$, find the number.

Activity

Sale hunt. Visit any nearby shop or website with a sale on. Pick $5$ items with discounts. For each, compute the selling price yourself before checking the displayed amount. Discrepancies (rare but real) make for an instructive maths-meets-shopping lesson.