Percentages: Complete Guide, Formulas & Practice Questions

Now that you have gone through the fractions topic, you will already be somewhat familiar with percentages. Percentages are among the most frequently tested topics in the aptitude and admission tests of major universities, and they also appear in the SAT, mostly inside word problems on discounts, growth, interest, data and mixtures.

Work through each concept, study the four examples (the two marked IBA level are in the style of IBA past-paper questions), and then attempt the 12 practice questions at the end.

Contents
  1. Understanding percent
  2. Percent of a number
  3. Finding the percent and the whole (is / of)
  4. Percentage change
  5. Adding a percent: percentage increase
  6. Subtracting a percent: percentage decrease
  7. Reverse percentages: finding the original
  8. Percentages in mixtures and solutions
  9. Aptitude-test shortcuts
  10. Practice questions

1. Understanding Percent

A percent is a fraction, a special fraction that always has 100 in the denominator: x100, whatever the value of x. Instead of writing the 100 every time, we use the symbol %. Whenever you see % after a number, that number is a percent:

15% = 15100    25% = 25100
  • Zero percent: 0% = 0100 = 0. A zero in the numerator is fine (only a zero denominator is forbidden).
  • Percent above 100%: an “improper fraction percentage” is perfectly possible: 125% = 125100 = 1.25. It means more than the whole.

Converting between forms

  • Percent → decimal: divide by 100, which moves the decimal point two places left. 65% = 0.65.
  • Decimal → percent: multiply by 100, which moves the decimal point two places right. 0.65 = 65%.
  • Fraction → percent: divide to get the decimal, then multiply by 100. 38 = 0.375 = 37.5%.
PercentFractionDecimal
10%1100.1
12.5%180.125
20%150.2
25%140.25
3313%130.333…
50%120.5
6623%230.666…
75%340.75
100%11
200%22

Understanding the concept

“Percent” literally means “per hundred”. So 37% means 37 out of every 100, and the three forms (fraction, decimal, percent) are just three ways of writing one value. Memorising the table above lets you convert instantly, which matters in timed tests. In calculations, it is usually safest to turn a percent into a decimal or fraction first.

Example 1Write 15%, 125% and 0% as fractions and decimals.
15% = 15100 = 320 = 0.15
125% = 125100 = 54 = 1.25
0% = 0 ÷ 100 = 0

125% is greater than 1 because it is more than the whole.

Example 2Express 0.065 and 38 as percentages.
0.065 × 100 = 6.5%
38 = 0.375 → 0.375 × 100 = 37.5%

Moving the decimal point two places right is the same as multiplying by 100.

Example 3IBA levelThe fraction 0.12 + 0.180.75 is equivalent to what percent?
0.300.75 = 3075 = 25 = 0.4 = 40%

Always simplify the numerator first, then divide.

Example 4IBA levelArrange 38, 0.38, 37% and 13 in ascending order.

Convert everything to percent.

38 = 37.5%   0.38 = 38%   37%   13 = 33.3%

Ascending order: 13, 37%, 38, 0.38.

2. Percent of a Number

The word “of” usually means multiplication. To find p% of a number x, convert p% to a decimal or fraction and multiply:

p% of x = p100 × x
Be careful what the percent is “of”. In “35% of those who participate”, the whole is the participants, not the original total. Work step by step from the inside out.

Understanding the concept

Percent problems are often built in layers: “p% of q% of a number” or “p% of 23 of 15″. Evaluate each layer in order, always using the result of the previous step as the new whole. Because multiplication is commutative, p% of q% of x equals (p × q ÷ 10,000) × x, which is a quick way to combine layers.

Example 1What is 25% of 36?

25% = 14, so we need one quarter of 36.

14 × 36 = 9
Example 2A school has 640 students. 62.5% take part in at least one extracurricular activity, and 35% of those students take part in sports. How many play a sport?

Find 62.5% of 640 first.

0.625 × 640 = 400

Now 35% of those 400 (not of 640):

0.35 × 400 = 140

So 140 students take part in sports.

Example 3IBA levelWhat is 20% of 34 of 80?

Inner layer: 34 of 80 = 60. Outer layer: 20% of 60.

0.2 × 60 = 12
Example 4IBA level40% of 25% of a number is 30. Find the number.

Combine the layers: 40% × 25% = 0.4 × 0.25 = 0.1 (that is 10% of the number).

0.1 × N = 30 → N = 300

Check: 25% of 300 = 75, and 40% of 75 = 30 ✓.

3. Finding the Percent and the Whole (Is / Of)

In a percent problem the whole goes with the word “of” and the part goes with the word “is”. The relationship is:

isof = %100    i.e.   part = (percent ÷ 100) × whole

There are three types of question, depending on which of the three quantities is missing:

  • Find the part: What is 25% of 36? Multiply.
  • Find the percent: 18 is what percent of 3? Divide part by whole, then × 100.
  • Find the whole: 13 is 40% of what number? Divide the part by the percent (as a decimal).

Understanding the concept

Identify the whole first: it is the number that follows “of” (or “percent of”). Then write the single equation part = percent × whole and solve for whichever quantity is unknown. A percent can be greater than 100% when the part is bigger than the whole, as in “18 is 600% of 3”.

Example 118 is what percent of 3?

Let the percent be m (as a multiplier): m × 3 = 18, so m = 6.

6 × 100% = 600%

The part is larger than the whole, so the percent exceeds 100%.

Example 224 is 40% of what number?
0.40 × N = 24 → N = 24 ÷ 0.40 = 60

Check: 40% of 60 = 24 ✓.

Example 3IBA levelIn a hall, 150 of the 200 seats were occupied. What percent of the seats were NOT occupied?

Unoccupied seats = 200 − 150 = 50.

50200 × 100% = 25%

Check: occupied = 75%, and 75% + 25% = 100% ✓. Always read whether the question asks for the occupied or the unoccupied part.

Example 4IBA levelIf 35% of a number is 49, what is 60% of the same number?

Find the number first.

0.35 × N = 49 → N = 140
60% of 140 = 0.6 × 140 = 84

Shortcut: 60% ÷ 35% × 49 = 84 (use proportion without finding N).

4. Percentage Change

Percentage change measures how large an increase or decrease is relative to the original value.

Percent change = New value − Original valueOriginal value × 100%
  • Increase: percent increase = IncreaseOriginal value × 100%
  • Decrease: percent decrease = DecreaseOriginal value × 100%
The denominator is the original value, not the new one. This is one of the easiest mistakes to make under time pressure.

Understanding the concept

The “original” is the value you started from. If a price goes from 40 to 50, the change is 10 relative to 40, which is 25%. If it goes from 50 back to 40, the change is 10 relative to 50, which is only 20%. The same absolute change gives different percentages because the base differs. A positive result means increase; a negative result means decrease.

Example 1The number of subscribers to a website rose from 2,400 to 3,060. Find the percent increase.
Increase = 3060 − 2400 = 660
6602400 × 100% = 0.275 × 100% = 27.5%
Example 2A price falls from 80 to 62. Find the percent decrease.
Decrease = 80 − 62 = 18
1880 × 100% = 22.5%
Example 3IBA levelSara earns $16 per hour and is promised a raise of 60 cents per hour. What is her percent increase in hourly pay?

Convert 60 cents to $0.60. The original pay is $16.

0.6016 × 100% = 3.75%
Example 4IBA levelA value rises from 40 to 50 and then falls back to 40. Find the percent increase and the percent decrease.
Increase: 1040 × 100% = 25%
Decrease: 1050 × 100% = 20%

The same change of 10 gives different percentages because the original value (the denominator) changes from 40 to 50.

5. Adding a Percent: Percentage Increase

When a quantity is increased by p% you keep 100% of the original and add p% more, so the new amount is (100 + p)% of the original.

New value = Original × (1 + p100)

For an 18% increase, the multiplier is 1 + 0.18 = 1.18. For a 300% increase, it is 1 + 3 = 4; for a 400% increase, 1 + 4 = 5.

Successive increases compound. A 15% increase followed by a 20% increase is not a 35% increase, because the second applies to the already increased amount. Multiply the multipliers: 1.15 × 1.20 = 1.38 (a 38% increase).
Vocabulary matters: “What is 20% of 500?” means 500 × 0.20 = 100, but “500 is increased by 20%” means 500 × 1.20 = 600.

Understanding the concept

“Increased by 300%” does not mean “becomes 300% of the original”. It means the original plus 300% of it, which is 400% of the original. This single wording trap appears again and again in IBA-style questions, so always add 100% to the stated increase to get the multiplier.

Example 1A company’s revenue was $480,000. It increased by 15% in the first year and by 20% in the second year. Find the revenue after the second increase.

First year: multiplier 1.15.

480,000 × 1.15 = 552,000

Second year: the 20% applies to $552,000, not $480,000.

552,000 × 1.20 = 662,400

Revenue = $662,400. (Adding 15% + 20% = 35% would give $648,000, which is wrong.)

Example 2Increase 650 by 18%.
650 × 1.18 = 767

Check: 18% of 650 = 117, and 650 + 117 = 767 ✓.

Example 3IBA levelIf the result of increasing a by 300% is b, then a is what percent of b?

Increase by 300% means the multiplier is 1 + 3 = 4, so b = 4a.

ab = a4a = 14 = 25%
Example 4IBA levelA number is increased by 20% and then by 25%. What is the total percent increase?

Multiply the multipliers.

1.20 × 1.25 = 1.50

The total increase is 50%, not 45%.

6. Subtracting a Percent: Percentage Decrease

When a quantity decreases by p%, subtract p% from 100%. A 25% decrease leaves 75%, so the multiplier is 0.75.

New value = Original × (1 − p100)
Discount then tax: they are applied to different bases, so you cannot simply combine them. A 30% discount followed by a 7.5% tax is not a 22.5% net discount. Multiply the multipliers: 0.70 × 1.075.
Equivalent single change: for successive changes the combined multiplier is the product of the individual multipliers. An increase of a% followed by a decrease of a% always leaves you below the start.

Understanding the concept

A decrease takes the base down, so the next percentage works on a smaller number. That is why an increase followed by an equal percent decrease does not return to the start: 20% up then 20% down gives 1.2 × 0.8 = 0.96, a 4% loss. To find the total percent change from several steps, multiply the multipliers and subtract 1.

Example 1A jacket costs $320. It is reduced by 30% in a sale, and then a 7.5% sales tax is added. What is the final price?

Discount: multiplier 0.70.

320 × 0.70 = 224

Tax on the discounted price: multiplier 1.075.

224 × 1.075 = 240.80

Final price = $240.80.

Example 2A phone costs Rs. 25,000. Find its price after a 12% discount.
25,000 × (1 − 0.12) = 25,000 × 0.88 = 22,000

The price is Rs. 22,000.

Example 3IBA levelA price is increased by 25% and then decreased by 20%. What is the net effect?
1.25 × 0.80 = 1.00

The multiplier is exactly 1, so the final price equals the original: no net change.

Example 4IBA levelSuccessive discounts of 20% and 10% are given. What single discount is equivalent?

The multipliers are 0.80 and 0.90.

0.80 × 0.90 = 0.72

The customer pays 72%, so the equivalent discount is 100% − 72% = 28% (not 30%).

7. Reverse Percentages: Finding the Original

Often the final amount is given and you must find the original. Because the final amount equals the original times a multiplier, divide by the multiplier:

Original = Final valueMultiplier
  • After a p% increase: original = final ÷ (1 + p/100).
  • After a p% decrease: original = final ÷ (1 − p/100).
  • After several steps: divide by the product of the multipliers.
Common mistake: adding the percentage back to the final amount. After a 20% discount the final price is 80% of the original, so the original is final ÷ 0.8, not final × 1.2.

Understanding the concept

Write the relationship as an equation: Original × multiplier = Final. The unknown is the original, so divide both sides by the multiplier. This is the idea behind questions such as “the customer paid p dollars including tax on a discounted price; what was the original price in terms of p?” The answer is p divided by the product of all the multipliers.

Example 1After a 20% discount the price of a bag is Rs. 1,600. Find the original price.

The price is 80% of the original.

Original = 1,600 ÷ 0.80 = 2,000
Example 2A restaurant bill is Rs. 2,160 including 8% tax. Find the bill before tax.
Before tax = 2,160 ÷ 1.08 = 2,000
Example 3IBA levelAfter a 25% discount and then a 10% tax, Ali paid Rs. 4,950. Find the original price.

Multipliers: 0.75 and 1.10, so the combined multiplier is 0.75 × 1.10 = 0.825.

Original = 4,950 ÷ 0.825 = 6,000
Example 4IBA levelA salary was raised by 10% and then by 20% and is now Rs. 66,000. What was the original salary?
1.10 × 1.20 = 1.32
Original = 66,000 ÷ 1.32 = 50,000

Check: 50,000 × 1.1 = 55,000, and 55,000 × 1.2 = 66,000 ✓.

8. Percentages in Mixtures and Solutions

In a mixture the concentration is a percent of the total. The key idea is that the amount of pure substance (salt, acid, alcohol, copper) is conserved when mixing:

Amount of substance = (percent ÷ 100) × total quantity
(c₁ × v₁) + (c₂ × v₂) = c₃ × (v₁ + v₂)

where c₁, c₂ are the concentrations being mixed, v₁, v₂ their quantities, and c₃ the final concentration. Adding pure water means adding a 0% solution; adding pure substance means adding a 100% solution.

Understanding the concept

Percentages of different totals cannot be added directly (20% of 6 L and 40% of 4 L do not give 60%). Convert each part to an actual amount of substance, add the amounts, and divide by the new total volume. Setting up the one equation above, with x as the unknown quantity, solves nearly every mixture question.

Example 1How many litres of pure water must be added to 10 L of a 30% solution to make a 20% solution?

Solute in 10 L = 0.30 × 10 = 3 L. This stays the same. The new total volume V satisfies 20% of V = 3.

V = 3 ÷ 0.20 = 15 L

Water to add = 15 − 10 = 5 L.

Example 24 L of a 10% acid solution is mixed with 6 L of a 20% solution. Find the concentration of the mixture.
Acid = 0.10 × 4 + 0.20 × 6 = 0.4 + 1.2 = 1.6 L
Concentration = 1.610 × 100% = 16%
Example 3IBA levelHow many litres of a 50% solution must be added to 8 L of a 20% solution to get a 30% solution?

Let x = litres of the 50% solution.

0.50x + 0.20(8) = 0.30(8 + x)
0.5x + 1.6 = 2.4 + 0.3x → 0.2x = 0.8 → x = 4

Add 4 L.

Example 4IBA levelA 20 kg alloy contains 30% copper. How much pure copper must be added to make an alloy that is 50% copper?

Copper now = 0.30 × 20 = 6 kg. Pure copper is 100% copper. Let x kg be added.

6 + x = 0.50(20 + x)
6 + x = 10 + 0.5x → 0.5x = 4 → x = 8

Add 8 kg. Check: (6 + 8) ÷ 28 = 50% ✓.

Aptitude-Test Shortcuts

  • Of means multiply; is means equals. Identify the whole (after “of”) before anything else.
  • Percent change: always divide by the original value.
  • Increase by p%: multiply by 1 + p/100. Decrease by p%: multiply by 1 − p/100. “Increased by 300%” gives a multiplier of 4.
  • Successive changes: multiply the multipliers; never add the percents.
  • Reverse percentages: divide by the multiplier.
  • Mixtures: track the amount of pure substance, not the percents.
  • Use fractions for speed: 25% = 1/4, 12.5% = 1/8, 33⅓% = 1/3, 66⅔% = 2/3.
  • Read the wording: “20% of 500” and “500 increased by 20%” are different.

Practice Questions: Test Yourself

Choose an answer to see instantly whether it is right or wrong. If you pick a wrong option, the correct one is highlighted. Then press Show Solution to review the steps.

1. In the first quarter of the year, an entrepreneur sold 36 percent of the 75 crafts she put up for sale online. In the second quarter, she sold one-fourth of the remaining crafts. What was the percent decrease from the first quarter to the second in the number of crafts she sold?
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Correct answer: C. 5559%

First quarter: 36% of 75 = 0.36 × 75 = 27 crafts sold.

Remaining: 75 − 27 = 48 crafts.

Second quarter: 14 of 48 = 12 crafts sold.

Decrease = 27 − 12 = 15, and the original (first quarter) value is 27.

1527 × 100% = 59 × 100% = 5559%.

2. Alma bought a laptop at a store that gave a 20% discount off its original price. The total she paid was p dollars, including an 8% sales tax on the discounted price. Which of the following represents the original price in terms of p?
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Correct answer: D. p(0.8)(1.08)

Let the original price be P.

After the 20% discount: 0.8P. After adding 8% tax: (0.8P)(1.08).

So p = (0.8)(1.08)P.

Divide by the combined multiplier: P = p(0.8)(1.08).

Options 0.88p and p/0.88 wrongly combine the percentages as 20% − 8% (they have different bases).

3. How many litres of a 40% saline solution must be added to 6 litres of a 20% saline solution to obtain a 25% saline solution?
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Correct answer: B. 2 litres

Let x be the litres of 40% solution added. Salt is conserved:

0.40x + 0.20(6) = 0.25(6 + x)

0.4x + 1.2 = 1.5 + 0.25x

0.15x = 0.3, so x = 2.

Check: salt = 0.8 + 1.2 = 2.0 L in 8 L, which is 25% ✓.

4. The result of increasing the quantity x by 400% is 60. What is the value of x?
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Correct answer: D. 12

Increasing by 400% means the multiplier is 1 + 4 = 5.

5x = 60, so x = 12.

(The trap: 400% of x is 4x, but “increased by” adds that to x.)

5. 13 is 3313% of what number?
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Correct answer: D. 39

3313% = 13.

13 × N = 13, so N = 13 × 3 = 39.

6. What is 30% of 35 of 50?
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Correct answer: B. 9

Inner layer: 35 of 50 = 30.

Outer layer: 30% of 30 = 0.3 × 30 = 9.

7. The fraction 0.45 + 0.150.80 is equivalent to what percent?
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Correct answer: A. 75%

Numerator: 0.45 + 0.15 = 0.60.

0.600.80 = 0.75 = 75%.

8. In a cinema, 360 of the 450 seats were occupied. What percentage of the seats were NOT occupied?
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Correct answer: C. 20%

Unoccupied seats = 450 − 360 = 90.

90450 × 100% = 20%.

Check: 80% occupied (360 ÷ 450) and 20% empty add to 100% ✓. The option 80% is the occupied percentage, a common trap.

9. Sara works for $20 per hour. She is promised a raise of 90 cents per hour. What will be her percent increase in hourly pay?
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Correct answer: A. 4.5%

Raise = $0.90 on an original pay of $20.

0.9020 × 100% = 4.5%.

Remember to divide by the original value and to use dollars (0.90) not cents.

10. If the result of increasing a by 150% is b, then a is what percent of b?
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Correct answer: A. 40%

Increase by 150% means multiplier 1 + 1.5 = 2.5, so b = 2.5a.

ab = a2.5a = 12.5 = 0.4 = 40%.

11. Successive discounts of 30% and 20% are given on a price. What single discount is equivalent?
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Correct answer: B. 44%

Multipliers: 0.70 and 0.80.

0.70 × 0.80 = 0.56, so the customer pays 56% of the price.

Equivalent discount = 100% − 56% = 44% (not 50%, which would just add the percents).

12. The population of a town increases by 20% in one year and then decreases by 20% the following year. What is the net change over the two years?
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Correct answer: B. 4% decrease

Multipliers: 1.20 and 0.80.

1.20 × 0.80 = 0.96, so the population is 96% of the original.

Net change: 100% − 96% = a 4% decrease. The second change applies to a larger base, so it removes more than the first added.