Percentage Basics and Conversions

What a Percentage Means

"Per cent" means "out of every hundred". So 35% is 35 parts out of 100:

$$35\% = \frac{35}{100} = 0.35$$

A percentage is just a fraction whose denominator is fixed at 100. Fixing the bottom makes comparing easy. 17 out of 25 and 33 out of 50 are hard to compare as they stand, but written as 68% and 66% the answer is obvious.

Percentage to fraction. Put the number over 100 and simplify.

• 45% = $\frac{45}{100} = \frac{9}{20}$
• 2.5% = $\frac{2.5}{100} = \frac{25}{1000} = \frac{1}{40}$
• 0.4% = $\frac{0.4}{100} = \frac{4}{1000} = \frac{1}{250}$
• $12\frac{1}{2}\% = \frac{25}{2} \times \frac{1}{100} = \frac{1}{8}$
• 150% = $\frac{150}{100} = \frac{3}{2}$

Fraction to percentage. Multiply by 100: $\frac{7}{20} \times 100 = 35$, so $\frac{7}{20}$ = 35%. In the same way $\frac{3}{8}$ = 37.5%.

Decimal and percentage. Move the decimal point two places. From a decimal to a percentage, move it right: 0.06 = 6%, 1.35 = 135%, 0.005 = 0.5%. To go back, move it left: 7.5% = 0.075.

Percentages above 100 and below 1. Both are allowed. 250% of 60 is 2.5 × 60 = 150, which is more than 60 itself. And 0.5% of 3000 is only 15. A percentage above 100 simply means "more than the whole".

The whole is 100%. If 35% of the students in a class are girls, the boys make up 100% − 35% = 65%. Many questions give you one part and expect you to see the other part this way.

Per cent versus percentage points

When a percentage itself goes up or down, there are two ways to describe the change. Suppose a school's pass rate goes from 40% to 50%.

• It rose by 10 percentage points, because 50 − 40 = 10.
• It rose by 25 per cent, because the rise of 10 is a quarter of the old 40.

Tax and interest rates work the same way. A GST rate moving from 12% to 18% is a rise of 6 percentage points, but the tax itself is 50% higher, since 6 is half of 12. When a question asks "by what per cent did the rate increase", it wants the second kind of answer. Percentage change is worked out fully in the section on percentage change and successive change.

Key Point: x% means $\frac{x}{100}$ of something. Always ask "per cent of what?", because the same 10% is a very different amount of different wholes.

Fractions as Percentages: 1/2 to 1/20

Knowing this table by heart saves more time than any other single thing in this chapter. Every row comes from one rule:

$$\frac{1}{n} = \frac{100}{n}\%$$

A few more are worth knowing too: $\frac{1}{25}$ = 4%, $\frac{1}{40}$ = 2.5%, $\frac{1}{50}$ = 2% and $\frac{1}{100}$ = 1%.

How to remember them. Most rows sit in families where each value is half of the one before:

• 50%, 25%, 12.5%, 6.25%, 3.125% for $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{16}$, $\frac{1}{32}$
• $33\frac{1}{3}\%$, $16\frac{2}{3}\%$, $8\frac{1}{3}\%$, $4\frac{1}{6}\%$ for $\frac{1}{3}$, $\frac{1}{6}$, $\frac{1}{12}$, $\frac{1}{24}$
• 20%, 10%, 5%, 2.5% for $\frac{1}{5}$, $\frac{1}{10}$, $\frac{1}{20}$, $\frac{1}{40}$
• $14\frac{2}{7}\%$ and $7\frac{1}{7}\%$ for $\frac{1}{7}$ and $\frac{1}{14}$

Reading the table backwards. When a question says $16\frac{2}{3}\%$ of 288, it really means $\frac{1}{6}$ of 288, which is 48. And $9\frac{1}{11}\%$ of 352 is just 352 ÷ 11 = 32. Questions with these odd-looking percentages are written so that the division comes out exact, which is a strong hint that the fraction route is the one the setter had in mind.

$\frac{1}{13}$, $\frac{1}{17}$ and $\frac{1}{19}$ rarely appear as exact values. If an option shows 7.69% or 5.26%, the setter is using these rows, and the rounded value is enough to pick the right option.

Multiples: Eighths, Sixths, Sevenths, Ninths and Elevenths

Once you know $\frac{1}{n}$, any $\frac{k}{n}$ is just k times it. For $\frac{3}{8}$: 3 × 12.5% = 37.5%. For $\frac{5}{7}$: $5 \times 14\frac{2}{7}\% = 71\frac{3}{7}\%$.

Two patterns make the awkward rows easy:

• Ninths: $\frac{k}{9}$ as a percentage repeats the digit k, as in $\frac{4}{9} = 44\frac{4}{9}\%$ and $\frac{7}{9} = 77\frac{7}{9}\%$.
• Elevenths: $\frac{k}{11}$ is 9k plus $\frac{k}{11}$. So $\frac{7}{11} = 63\frac{7}{11}\%$, because 9 × 7 = 63.

Using the families in calculation.

• 62.5% of 488 = $\frac{5}{8}$ × 488 = 61 × 5 = 305
• 87.5% of 3200 = 3200 − 400 = 2800, since it is one-eighth less than the whole
• $83\frac{1}{3}\%$ of 726 = $\frac{5}{6}$ × 726 = 121 × 5 = 605
• $71\frac{3}{7}\%$ of 1316 = 188 × 5 = 940, since 1316 ÷ 7 = 188
• $27\frac{3}{11}\%$ of 1199 = 109 × 3 = 327, since 1199 ÷ 11 = 109

Going the other way. Match an odd-looking percentage to its family. 56.25% ends in .25, which points to sixteenths: it is $\frac{9}{16}$. $58\frac{1}{3}\%$ has a third in it, which points to twelfths: it is $\frac{7}{12}$.

Comparing fractions. Turning fractions into percentages is a quick way to put them in order. For $\frac{3}{7}$, $\frac{5}{12}$ and $\frac{4}{9}$ the percentages are about 42.9%, 41.7% and 44.4%, so $\frac{4}{9}$ is the largest and $\frac{5}{12}$ is the smallest.

Finding x% of y

The basic formula is one line:

$$x\% \text{ of } y = \frac{x \times y}{100}$$

In practice you rarely need it. Build the percentage from easy pieces:

• 10% means moving the decimal point one place left: 10% of 840 = 84
• 1% means moving it two places: 1% of 840 = 8.4
• 5% is half of 10%, 50% is half the number, and 25% is half of 50%

So 35% of 840 = 30% + 5% = 252 + 42 = 294, and 17% of 600 = 10% + 7% = 60 + 42 = 102.

The swap rule. Both x% of y and y% of x equal $\frac{xy}{100}$, so they are always the same. Pick whichever is easier.

• 16% of 25 = 25% of 16 = 4
• 48% of 75 = 75% of 48 = 36
• 7% of 300 = 300% of 7 = 21

It follows that x% of y + y% of x is simply twice either one: 12% of 50 + 50% of 12 = 6 + 6 = 12.

A percentage of a percentage. "a% of b% of c" means take b% of c first, then a% of that:

$$a\% \text{ of } b\% \text{ of } c = \frac{a \times b \times c}{100 \times 100}$$

For 40% of 25% of 1600: 25% of 1600 = 400, and 40% of 400 = 160. As a single step, 40% of 25% is $\frac{40 \times 25}{100}$ = 10%, and 10% of 1600 = 160.

The trap is adding the percentages. 40% of 25% is 10%, not 65%. Percentages taken "of" each other multiply.

Mixing fractions and percentages. For $\frac{3}{5}$ of 45% of 2000: 45% of 2000 is 900, and $\frac{3}{5}$ of 900 is 540. Do whichever step cancels most easily first; the order does not change the answer.

Percentages above 100 and below 1. 125% of 96 = $\frac{5}{4}$ × 96 = 120. And 0.25% of 4800 = 4800 ÷ 400 = 12, because 0.25% = $\frac{1}{400}$.

Long expressions. In one like 37.5% of 640 + 14% of 50 − $16\frac{2}{3}\%$ of 414, treat each term on its own. $\frac{3}{8}$ of 640 = 240, 14% of 50 is the same as 50% of 14 = 7, and $\frac{1}{6}$ of 414 = 69. The total is 240 + 7 − 69 = 178.

Finding the Whole When a Part Is Given

Here the percentage and the part are known, and the whole is missing. If x% of N = P, then

$$N = \frac{P \times 100}{x}$$

The unitary way. If 35% of a number is 84, then 1% is 84 ÷ 35 = 2.4, and 100% is 240. The formula does the same thing in one step: 84 × 100 ÷ 35 = 240.

Jumping straight to another percentage. If 15% of a number is 63 and you want 40% of it, you don't need the number at all:

$$40\% \text{ of } N = 63 \times \frac{40}{15} = 168$$

The number itself is 420, and 40% of 420 is indeed 168. In the same way, if 20% of a number is 120, then 120% of it is 6 times as much, which is 720.

Use the fraction where you can. If $16\frac{2}{3}\%$ of a number is 45, then $\frac{1}{6}$ of the number is 45, so the number is 270. If 2.5% of a number is 18, then $\frac{1}{40}$ of it is 18 and the number is 720.

When the part is "the rest". In a class, 45% of the students are girls and there are 66 boys. The boys are the other 55% of the class, so 1% is 66 ÷ 55 = 1.2 and the class has 120 students. That leaves 54 girls. A common slip is to pair the 45% with the 66, which gives about 146.7 students, and a fraction of a student is a sure sign that something went wrong.

One percentage equal to another. If 24% of A equals 36% of 450, work out the known side first: 36% of 450 = 162. Then 24% of A = 162, so A = 162 × 100 ÷ 24 = 675.

Checking. Plug the answer back in. 24% of 675: 20% is 135 and 4% is 27, and 135 + 27 = 162. It takes ten seconds and catches most slips. Also, if your answer is smaller than the part (with a percentage below 100), you have divided the wrong way.

What Per Cent Is One Number of Another

To write A as a percentage of B:

$$\frac{A}{B} \times 100\%$$

The number after "of" goes in the denominator. "18 is what per cent of 72?" gives $\frac{18}{72}$ × 100 = 25%. Turn it round and "72 is what per cent of 18?" gives $\frac{72}{18}$ × 100 = 400%. Same two numbers, very different answers, and SSC puts both in the options.

The flip rule. If A is p% of B, then B is $\frac{100}{p} \times 100\%$ of A. A is 25% of B means B is 400% of A. A is 80% of B means B is 125% of A.

A part of a whole. If 27 of the 60 students in a class failed, then $\frac{27}{60}$ × 100 = 45% failed and the other 55% passed.

Decimals and fractions. "What per cent of 2.5 is 0.15?" Multiply both by 100 to clear the decimals: $\frac{15}{250}$ × 100 = 6%. "What per cent of $\frac{3}{4}$ is 0.6?" 0.6 ÷ 0.75 = 0.8, so the answer is 80%.

Ratios as percentages

If A : B = 3 : 5, then:

• A is $\frac{3}{5}$ = 60% of B
• B is $\frac{5}{3} = 166\frac{2}{3}\%$ of A
• A is $\frac{3}{8}$ = 37.5% of the total, and B is $\frac{5}{8}$ = 62.5% of the total

And back again: "A is 40% of B" means A : B = 40 : 100 = 2 : 5.

For a three-way split in the ratio 4 : 5 : 11, the total is 20 parts, so the shares are $\frac{4}{20}$ = 20%, $\frac{5}{20}$ = 25% and $\frac{11}{20}$ = 55%.

Two numbers measured against a third

If A is 20% of C and B is 50% of C, then A : B = 20 : 50. So A is 40% of B, and B is 250% of A. C cancels out, so you never need its value. The "more than" version of this question (A is 20% more than C, and so on) is in the percentage change section.

Percentages with Units

A percentage compares two amounts of the same kind, so both must be in the same unit before you divide. These are the conversions you need:

Some quick cases.

• 250 g as a percentage of 2 kg: $\frac{250}{2000}$ × 100 = 12.5%
• 45 paise as a percentage of ₹3: $\frac{45}{300}$ × 100 = 15%
• 27 minutes as a percentage of 3 hours: $\frac{27}{180}$ × 100 = 15%
• 7 hours 12 minutes as a percentage of a day: $\frac{432}{1440}$ × 100 = 30%
• 75 m as a percentage of 1.5 km: $\frac{75}{1500}$ × 100 = 5%
• 300 ml as a percentage of 2.4 litres: $\frac{300}{2400}$ × 100 = 12.5%

Convert the bigger unit into the smaller one. It keeps the numbers whole, as in every case above.

The other direction. "Find 12% of ₹5 in paise" means 12% of 500 paise, which is 60 paise. "How many minutes is 15% of a day?" 15% of 1440 = 216 minutes, which is 3 hours 36 minutes.

The trap. "40 paise is what per cent of ₹2?" Dividing 40 by 2 gives 2000%, which is absurd: 40 paise is less than ₹2, so the answer must be below 100%. In paise, ₹2 is 200, and $\frac{40}{200}$ × 100 = 20%. A quick "should this be more or less than 100%?" check catches almost every unit slip.

Mixed units in one quantity. Turn the whole thing into the smaller unit first. 1 hour 48 minutes is 108 minutes, not 1.48 hours. As a part of a day it is $\frac{108}{1440}$ = 7.5%. Using 1.48 hours by mistake gives about 6.17%, which is exactly the kind of wrong option you will see.

[SSC Tip] Time units come up often in these questions. Keep 1 day = 1440 minutes ready, and remember that 1% of a day is 14.4 minutes.

Percentage Equations: "Exceeds By", "Added To" and "Less Than"

Many questions describe a number only through its percentages. Write every phrase as a percentage of the same number N, and the question turns into a one-step equation.

"a% of a number exceeds its b% by k". The gap between the two percentages is (a − b)% of N, and that gap equals k:

$$N = \frac{k \times 100}{a - b}$$

If 65% of a number exceeds 40% of it by 120, then 25% of N = 120 and N = 480.

When a fraction is mixed in, change it to a percentage first. "$\frac{3}{5}$ of a number is 36 more than 45% of it": $\frac{3}{5}$ = 60%, so 15% of N = 36 and N = 240.

"Added to" and "subtracted from". "When 75% of a number is added to 75, the result is the number itself." So 75% of N + 75 = N. The 75 must fill the missing 25%, so 25% of N = 75 and N = 300.

"When 30% of a number is subtracted from 91, the result is the number itself." So 91 − 30% of N = N, which gives 130% of N = 91 and N = 70. Check: 30% of 70 is 21, and 91 − 21 = 70.

"Less than the number". "The sum of 40% and 35% of a number is 90 less than the number." So 75% of N = N − 90, which means 25% of N = 90 and N = 360.

Two unknowns. "The sum of two numbers is 280, and 40% of the first equals 30% of the second." From 40% of A = 30% of B, A : B = 30 : 40 = 3 : 4 (Percentage with Ratio, just below, explains why the numbers swap). Splitting 280 in the ratio 3 : 4 gives 120 and 160. Check: 40% of 120 = 48 and 30% of 160 = 48.

Several percentages of one number. "20% of a number + 35% of the same number = 154" is simply 55% of N = 154, so N = 280. Add or subtract the percentages first, and only then bring in the number.

[SSC Tip] In "exceeds by" questions the options usually include k × 100 ÷ a and k × 100 ÷ b as traps. In the first case above those are about 184.6 and 300. Only the difference (a − b) belongs in the denominator.

Percentage with Ratio

The cross rule. If a% of A = b% of B, then

$$\frac{A}{B} = \frac{b}{a}, \quad \text{so } A : B = b : a$$

The percentages swap sides. If 40% of A = 60% of B, then A : B = 60 : 40 = 3 : 2. This makes sense: A needs a smaller share to match B, so A must be the bigger number.

With fractions. "$\frac{2}{3}$ of A = 40% of B" means $\frac{2}{3}$A = $\frac{2}{5}$B. So $\frac{A}{B} = \frac{2}{5} \div \frac{2}{3} = \frac{3}{5}$, and A : B = 3 : 5. In other words, A is 60% of B.

Three quantities. "20% of A = 30% of B = $\frac{1}{6}$ of C." Call the common value k. Then

$$A = 5k, \quad B = \frac{10k}{3}, \quad C = 6k$$

Taking k = 3 clears the fraction straight away: A = 15, B = 10 and C = 18. So A : B : C = 15 : 10 : 18.

Using a total. "A and B together have ₹2,250, and 30% of A's money equals 45% of B's money. How much does B have?" By the cross rule A : B = 45 : 30 = 3 : 2. So B has $\frac{2}{5}$ of ₹2,250, which is ₹900, and A has ₹1,350.

Brackets on both sides. "If 50% of (x − y) = 30% of (x + y), what per cent of x is y?" Open the brackets:

$$0.5x - 0.5y = 0.3x + 0.3y$$

Collecting terms gives $0.2x = 0.8y$, so $y = \frac{x}{4}$. That means y is 25% of x, and x : y = 4 : 1.

Answering "what per cent". Once you have the ratio, any "what per cent" question is a single division. If A : B = 3 : 5, A is 60% of B and B is $166\frac{2}{3}\%$ of A. If $\frac{1}{4}$ of A = 35% of B, then A = 1.4B, so A is 140% of B.

[SSC Tip] The wrong option to watch for in this type is the unswapped ratio, a : b instead of b : a. Run a quick check: the number with the smaller percentage is the larger number.

Solved Examples

Question 1

2.5% written as a fraction is: (a) $\frac{1}{4}$ (b) $\frac{1}{400}$ (c) $\frac{1}{40}$ (d) $\frac{2}{5}$

Answer. Per cent means out of 100, so 2.5% = $\frac{2.5}{100}$. Multiplying the top and bottom by 10 gives $\frac{25}{1000} = \frac{1}{40}$. The answer is (c) $\frac{1}{40}$.

Question 2

$\frac{5}{16}$ written as a percentage is: (a) 32.5% (b) 31.25% (c) 3.125% (d) 31.5%

Answer. I know $\frac{1}{16}$ = 6.25%, so $\frac{5}{16}$ is 5 × 6.25% = 31.25%. The answer is (b) 31.25%.

Question 3

Which of these is the largest? (a) $\frac{3}{7}$ (b) 42% (c) 0.43 (d) $\frac{5}{12}$

Answer. I turn everything into a percentage. $\frac{3}{7} = 42\frac{6}{7}\%$, which is about 42.86%. 42% stays as it is. 0.43 is 43%. $\frac{5}{12} = 41\frac{2}{3}\%$. The biggest is 43%, so the answer is (c) 0.43. $\frac{3}{7}$ looks close, but it is just under 43%.

Question 4

$14\frac{2}{7}\%$ of 1638 is: (a) 273 (b) 224 (c) 234 (d) 243

Answer. $14\frac{2}{7}\%$ is the same as $\frac{1}{7}$, so I just divide by 7: 1638 ÷ 7 = 234. The answer is (c) 234. 273 is what you get by mixing it up with $16\frac{2}{3}\% = \frac{1}{6}$.

Question 5

$83\frac{1}{3}\%$ of 1452 + 37.5% of 1600 − $9\frac{1}{11}\%$ of 1001 = ? (a) 1729 (b) 1791 (c) 1701 (d) 1719

Answer. Each term uses a fraction from the table. $83\frac{1}{3}\% = \frac{5}{6}$, so the first term is 1452 ÷ 6 × 5 = 242 × 5 = 1210. 37.5% = $\frac{3}{8}$, so the second is 1600 ÷ 8 × 3 = 600. $9\frac{1}{11}\% = \frac{1}{11}$, so the third is 1001 ÷ 11 = 91. Total: 1210 + 600 − 91 = 1719. The answer is (d) 1719.

Question 6

The GST rate on a product was raised from 12% to 18%. Which statement is correct? (a) The rate rose by 6% (b) The rate rose by 50 percentage points (c) The rate rose by 6 percentage points, which is a 50% increase (d) The rate rose by 6 percentage points, which is a $33\frac{1}{3}\%$ increase

Answer. The rate went from 12 to 18, so it rose by 6 percentage points. As a per cent of the old rate, that rise is $\frac{6}{12}$ × 100 = 50%. So the tax is now 50% higher. The last option divides by the new rate 18, but the base is always the old value. The answer is (c) The rate rose by 6 percentage points, which is a 50% increase.

Question 7

48% of 75 + 75% of 48 = ? (a) 72 (b) 36 (c) 54 (d) 64

Answer. I swap the first term: 48% of 75 = 75% of 48. 75% is $\frac{3}{4}$, and $\frac{3}{4}$ of 48 = 36. So the sum is 36 + 36 = 72. The answer is (a) 72.

Question 8

30% of 40% of 25% of 8000 = ? (a) 7600 (b) 2400 (c) 24 (d) 240

Answer. I work from the inside out. 25% of 8000 = 2000. 40% of 2000 = 800. 30% of 800 = 240. The answer is (d) 240. Don't add 30 + 40 + 25 = 95% and take 95% of 8000: that gives 7600, but percentages taken "of" each other multiply.

Question 9

$\frac{2}{3}$ of 36% of 2500 + 12.5% of 640 = ? (a) 680 (b) 980 (c) 600 (d) 720

Answer. 36% of 2500 = 900, because 1% is 25 and 36 × 25 = 900. Then $\frac{2}{3}$ of 900 = 600. 12.5% is $\frac{1}{8}$, and 640 ÷ 8 = 80. Total: 600 + 80 = 680. The answer is (a) 680.

Question 10

If 35% of a number is 168, then 85% of the number is: (a) 480 (b) 398 (c) 408 (d) 418

Answer. If 35% is 168, then 1% is 168 ÷ 35 = 4.8. So 85% is 4.8 × 85 = 408. The answer is (c) 408.

Question 11

In a school, 38% of the students are girls and there are 434 boys. How many girls are there? (a) 266 (b) 700 (c) 434 (d) 280

Answer. If 38% are girls, the boys are the other 62%. So 62% of the school = 434, which makes 1% = 7 and the whole school 700. Girls = 700 − 434 = 266. The answer is (a) 266.

Question 12

If $12\frac{1}{2}\%$ of x is equal to 25% of 84, then x = ? (a) 42 (b) 168 (c) 186 (d) 84

Answer. 25% of 84 = 21. $12\frac{1}{2}\%$ is $\frac{1}{8}$, so $\frac{1}{8}$ of x = 21, which gives x = 168. The answer is (b) 168.

Question 13

0.36 is what per cent of 4.8? (a) 75% (b) 0.75% (c) 7.5% (d) $13\frac{1}{3}\%$

Answer. 4.8 comes after "of", so it is the whole and 0.36 is the part. $\frac{0.36}{4.8} = \frac{36}{480}$ = 0.075, which is 7.5%. The answer is (c) 7.5%.

Question 14

Two numbers are 30% and 75% of a third number. What per cent of the second number is the first? (a) 40% (b) 45% (c) 250% (d) 60%

Answer. I take the third number as 100. Then the first is 30 and the second is 75. The first as a per cent of the second is $\frac{30}{75}$ × 100 = 40%. The answer is (a) 40%.

Question 15

A and B share some money in the ratio 7 : 13. A's share is what per cent of the total? (a) $53\frac{11}{13}\%$ (b) 65% (c) 35% (d) $185\frac{5}{7}\%$

Answer. The total is 7 + 13 = 20 parts and A has 7 of them, so A's share is $\frac{7}{20}$ = 35%. The answer is (c) 35%.

Question 16

What per cent of 3.2 kg is 120 g? (a) 37.5% (b) 3.75% (c) 0.375% (d) 4.25%

Answer. First the same unit: 3.2 kg = 3200 g. Then $\frac{120}{3200}$ × 100 = 3.75%. The answer is (b) 3.75%.

Question 17

2 hours 24 minutes is what per cent of a day? (a) $9\frac{1}{3}\%$ (b) 10% (c) 12% (d) 2.4%

Answer. 2 hours 24 minutes = 144 minutes, and a day has 1440 minutes. $\frac{144}{1440} = \frac{1}{10}$, which is 10%. The answer is (b) 10%. If you write 2 hours 24 minutes as 2.24 hours, you get $9\frac{1}{3}\%$.

Question 18

75 paise is what per cent of ₹4.50? (a) $16\frac{2}{3}\%$ (b) 1.67% (c) 6% (d) 15%

Answer. ₹4.50 = 450 paise. $\frac{75}{450} = \frac{1}{6}$, which is $16\frac{2}{3}\%$. The answer is (a) $16\frac{2}{3}\%$.

Question 19

72% of a number exceeds $\frac{2}{5}$ of it by 112. The number is: (a) 280 (b) 350 (c) 400 (d) 320

Answer. $\frac{2}{5}$ is 40%. So 72% − 40% = 32% of the number is 112. Then 1% is 3.5, and the number is 350. The answer is (b) 350. Check: 72% of 350 = 252, $\frac{2}{5}$ of 350 = 140, and 252 − 140 = 112.

Question 20

When 60% of a number is added to 96, the result is the number itself. The number is: (a) 160 (b) 240 (c) 300 (d) 156

Answer. 60% of N + 96 = N, so 96 must be the other 40% of N. Then N = 96 × 100 ÷ 40 = 240. The answer is (b) 240.

Question 21

If 40% of (a − b) = 20% of (a + b), then b is what per cent of a? (a) 300% (b) 50% (c) $33\frac{1}{3}\%$ (d) $66\frac{2}{3}\%$

Answer. I open the brackets: 0.4a − 0.4b = 0.2a + 0.2b. Moving the terms across gives 0.2a = 0.6b, so a = 3b, which means b = $\frac{a}{3}$. That is $33\frac{1}{3}\%$ of a. The answer is (c) $33\frac{1}{3}\%$.

Question 22

If 25% of A = 40% of B = $\frac{2}{3}$ of C, then A : B : C = ? (a) 3 : 5 : 8 (b) 8 : 5 : 3 (c) 8 : 3 : 5 (d) 15 : 24 : 40

Answer. I call the common value k. 25% of A = k gives A = 4k. 40% of B = k gives B = $\frac{5k}{2}$. $\frac{2}{3}$ of C = k gives C = $\frac{3k}{2}$. Taking k = 2 gives A = 8, B = 5 and C = 3. So A : B : C = 8 : 5 : 3. The answer is (b) 8 : 5 : 3. Using 25 : 40 : $66\frac{2}{3}$ itself as the ratio, without swapping, leads to the last option.