Quick percentage tricks

Once you can compute $1\%$ and $10\%$ in your head, you can compute almost any reasonable percent of any reasonable number in seconds.

Concept

The $10\%$ trick. $10\%$ of any number is that number with its decimal point moved one place left.

• $10\%$ of $\mathbb{?}\,750 = \mathbb{?}\,75$.
• $10\%$ of $\mathbb{?}\,4200 = \mathbb{?}\,420$.
• $10\%$ of $\mathbb{?}\,86 = \mathbb{?}\,8.6$.

The $1\%$ trick. $1\%$ is $1/100$ of the number, decimal point shifted two places.

• $1\%$ of $\mathbb{?}\,750 = \mathbb{?}\,7.50$.

Build any percent from $10\%$ and $1\%$.

• $20\% = 2 \times 10\%$.
• $30\% = 3 \times 10\%$.
• $5\% = \tfrac{1}{2} \times 10\%$.
• $15\% = 10\% + 5\%$.
• $12\% = 10\% + 2 \times 1\%$.
• $17\% = 10\% + 5\% + 2 \times 1\%$.

Example: $17\%$ of $\mathbb{?}\,600$.

• $10\% = 60$. $5\% = 30$. $1\% = 6$, so $2\% = 12$.
• Total: $60 + 30 + 12 = \mathbb{?}\,102$.

The "switch" trick. "$x\%$ of $y = y\%$ of $x$." This is because both equal $\dfrac{xy}{100}$.

• $4\%$ of $25 = 25\%$ of $4 = 1$.
• $50\%$ of $36 = 36\%$ of $50 = 18$.

The trick is useful when one of the numbers is much friendlier than the other.

Sales tax trick. If you pay $\mathbb{?}\,P$ including a $t\%$ tax, the pre-tax amount is $\dfrac{P}{1 + t/100}$. For $t = 5\%$, divide by $1.05$. For $t = 18\%$, divide by $1.18$.

Reverse percentage. If $x\%$ of $N$ is $K$, then $N = K \cdot \dfrac{100}{x}$.

• $20\%$ of $N = 60$ $\to$ $N = 60 \times 5 = 300$.
• $25\%$ of $N = 90$ $\to$ $N = 90 \times 4 = 360$.

Quick increase/decrease. To increase $\mathbb{?}\,P$ by $r\%$, multiply by $1 + r/100$. To decrease, multiply by $1 - r/100$.

Worked examples

Example 1. Find $15\%$ of $\mathbb{?}\,480$ mentally.

• $10\% = 48$. $5\% = 24$. Sum: $48 + 24 = \mathbb{?}\,72$.

Example 2. Find $4\%$ of $250$ using the switch trick.

• $4\%$ of $250 = 250\%$ of $4 = 2.5 \times 4 = 10$.

Example 3. A bill includes $5\%$ tax and totals $\mathbb{?}\,1050$. Find the pre-tax amount.

• $1050 / 1.05 = 1000$.

Example 4. $40\%$ of a number is $96$. Find the number.

• $N = 96 \times \dfrac{100}{40} = 96 \times 2.5 = 240$.

Try it yourself

1. Find $20\%$ of $\mathbb{?}\,850$ mentally.
2. Find $12\%$ of $\mathbb{?}\,750$.
3. Compute $8\%$ of $25$ using the switch trick.
4. $30\%$ of a number is $144$. Find the number.
5. After an $18\%$ GST, a bill is $\mathbb{?}\,1180$. Find the pre-tax amount.
6. Increase $\mathbb{?}\,5000$ by $15\%$.
7. Decrease $\mathbb{?}\,4000$ by $12\%$.
8. Find $7\%$ of $\mathbb{?}\,820$ using $10\% \pm$ shifts.

Activity

Mental maths sprint. Time yourself doing $20$ percentage problems of the kind above. Then repeat tomorrow. Within a week, $15$ seconds per problem should be comfortable. Cashiers and shopkeepers train themselves exactly this way.