Finding percentages

Two ques­tions come up again and again with per­cent­ages. "What is 5% of 20?" and "5 is what per cent of 20?" They sound alike, but they are dif­fer­ent. In this les­son you will learn to answer both, and then use them in word prob­lems about num­bers, time, weights and elec­tions.

One num­ber as a per­cent­age of another#

Express 5 as a per­cent­age of 20.

Think of it this way: out of 20, we have 5. Out of 100, how many would we have? Here are three ways to work it out.

First way: the uni­tary method.

Out of 20 we have 5.

Out of 1 we have $\displaystyle \frac{5}{20}$.

Out of 100 we have $\displaystyle \frac{5}{20} \times 100 = 25$.

Sec­ond way: scale up. 100 is $\displaystyle \frac{100}{20} = 5$ times 20. So mul­ti­ply 5 by the same amount: $\displaystyle \frac{100}{20} \times 5 = 25$.

Third way: pro­por­tion. Let the answer be $x$. Then $20 : 5 = 100 : x$, so $20 \times x = 5 \times 100$ and $\displaystyle x = \frac{5}{20} \times 100 = 25$.

So 5 is 25% of 20.

Rule. To express $x$ as a per­cent­age of $y$:

$$\displaystyle \left(\frac{x}{y} \times 100\right)\%$$

A per­cent­age of a num­ber#

What is 5% of 20?

5% means 5 out of every 100. So take $\displaystyle \frac{5}{100}$ of 20:

$$\displaystyle \frac{5}{100} \times 20 = 1$$

What is 10% of 150?

$$\displaystyle \frac{10}{100} \times 150 = 15$$

Rule. To find $x\%$ of $y$:

$$\displaystyle \frac{x}{100} \times y$$

The word "of" means mul­ti­ply.

Exam­ple 1#

Find 36% of 30.

$$\displaystyle \frac{36}{100} \times 30 = \frac{1080}{100} = 10.8$$

Exam­ple 2#

0.01 is what per cent of 0.1?

$$\displaystyle \left(\frac{0.01}{0.1} \times 100\right)\% = \left(\frac{1}{10} \times 100\right)\% = 10\%$$

Here $\displaystyle \frac{0.01}{0.1} = \frac{1}{10}$, because 0.01 is one tenth of 0.1.

Exam­ple 3#

What per cent of 4 is 3?

$$\displaystyle \left(\frac{3}{4} \times 100\right)\% = 75\%$$

Find­ing the num­ber#

Some­times the num­ber itself is unknown. Call it $x$ and write the sen­tence as an equa­tion.

Exam­ple 4#

35% of a num­ber is 12 less than 50% of the same num­ber. Find the num­ber.

Let the num­ber be $x$.

$$\displaystyle \begin{aligned} \frac{35}{100}x &= \frac{50}{100}x - 12 \\ 12 &= \frac{50x}{100} - \frac{35x}{100} \\ 12 &= \frac{15x}{100} \\ x &= \frac{12 \times 100}{15} = 80 \end{aligned}$$

The num­ber is 80. Check: 35% of 80 is 28 and 50% of 80 is 40. And $40 - 12 = 28$.

Exam­ple 5#

A num­ber is 42 more than 16% of itself. Find the num­ber.

Let the num­ber be $x$.

$$\displaystyle \begin{aligned} x &= \frac{16}{100}x + 42 \\ x - \frac{16}{100}x &= 42 \\ \frac{84x}{100} &= 42 \\ x &= \frac{42 \times 100}{84} = 50 \end{aligned}$$

The num­ber is 50. Check: 16% of 50 is 8, and $8 + 42 = 50$.

Exam­ple 6#

15% of 40 is greater than 25% of a num­ber by 2. Find the num­ber.

Let the num­ber be $x$.

$$\displaystyle \begin{aligned} \frac{15}{100} \times 40 &= \frac{25}{100}x + 2 \\ 6 &= \frac{25x}{100} + 2 \\ 4 &= \frac{25x}{100} \\ x &= \frac{400}{25} = 16 \end{aligned}$$

The num­ber is 16.

Per­cent­ages with units#

Before you divide, put both amounts in the same unit, just as with ratios.

Exam­ple 7#

What per cent of a day is 3 hours?

A day has 24 hours.

$$\displaystyle \left(\frac{3}{24} \times 100\right)\% = \left(\frac{1}{8} \times 100\right)\% = 12.5\%$$

So 3 hours is $\displaystyle 12\frac{1}{2}\%$ of a day.

Exam­ple 8#

What per­cent­age of 4 kg is 500 g?

4 kg = 4000 g.

$$\displaystyle \left(\frac{500}{4000} \times 100\right)\% = \frac{25}{2}\% = 12.5\%$$

Exam­ple 9#

What per cent of 2 met­ric tonnes is 40 quin­tals?

1 quin­tal = 100 kg and 1 met­ric tonne = 1000 kg = 10 quin­tals. So 2 met­ric tonnes = 20 quin­tals.

$$\displaystyle \left(\frac{40}{20} \times 100\right)\% = 200\%$$

40 quin­tals is twice 2 tonnes, so the answer is more than 100%.

A word prob­lem: an elec­tion#

Exam­ple 10#

In an elec­tion between two can­di­dates, 20% of the votes were invalid. One can­di­date got 55% of the valid votes. There were 7500 votes in all. How many valid votes did the other can­di­date get?

If 20% were invalid, then $100 - 20 = 80$ per cent were valid.

$$\displaystyle \text{valid votes} = \frac{80}{100} \times 7500 = 6000$$

The first can­di­date got 55% of the valid votes. So the other can­di­date got $100 - 55 = 45$ per cent of them.

$$\displaystyle \frac{45}{100} \times 6000 = 2700$$

The other can­di­date got 2700 votes.

Be care­ful to take 45% of the valid votes, 6000, and not of all 7500 votes.

Prac­tice#

1. Find 20% of 350.
2. Find 12.5% of 640.
3. Express 18 as a per­cent­age of 72.
4. What per­cent­age of 5 m is 75 cm?
5. What per­cent­age of an hour is 45 sec­onds?
6. 40% of a num­ber is 18 more than 25% of the same num­ber. Find the num­ber.
7. A num­ber is 48 more than 20% of itself. Find the num­ber.
8. Asha scored 342 marks out of 450. What per­cent­age did she score?
9. In an elec­tion between two can­di­dates, there were 9000 votes and 10% were invalid. The win­ner got 60% of the valid votes. How many votes did the loser get?
10. 0.2 is what per cent of 0.05?

Answers#