Percentages: Increase, Decrease and Reverse Percentages

By Engr. Henry Akpan

Percentages Are Everywhere

"20% off all shoes." "Inflation rose to 25%." "You scored 70% in mathematics." "VAT is charged at 7.5%." Percentages are the language of prices, results, statistics and the news, and they appear in almost every WAEC, NECO, BECE and JAMB mathematics paper. Most students can find 10% of a number, but harder questions, such as finding the original price after a discount or the overall effect of two changes, catch many people out. This tutorial covers percentages step by step: converting between fractions, decimals and percentages, finding a percentage of a quantity, expressing one quantity as a percentage of another, percentage increase and decrease using multipliers, percentage change, reverse percentages, successive changes, compound growth and percentage error, with worked examples. The tutorial on commercial arithmetic applies these skills to profit, loss, interest and discount.

What Is a Percentage?

"Per cent" means "out of a hundred". So 35% means 35 out of 100, which is the fraction 35/100 or the decimal 0.35.

Converting between forms

  • Percentage to fraction: write over 100 and simplify. 45% = 45/100 = 9/20.
  • Percentage to decimal: divide by 100. 4.5% = 0.045.
  • Fraction or decimal to percentage: multiply by 100%. 3/8 = 0.375 = 37.5%.

Some useful equivalents to memorise: 50% = 1/2, 25% = 1/4, 75% = 3/4, 20% = 1/5, 10% = 1/10, 12½% = 1/8, 33⅓% = 1/3 and 66⅔% = 2/3. A percentage can also exceed 100: 125% = 1.25 = 5/4.

Finding a Percentage of a Quantity

Convert the percentage to a fraction or decimal and multiply.

Worked example 1

Find 15% of ₦24,000.

15/100 × 24,000 = ₦3,600. Or, using a decimal: 0.15 × 24,000 = ₦3,600.

Without a calculator, build it from easy pieces: 10% = ₦2,400, 5% = ₦1,200, so 15% = ₦3,600.

Expressing One Quantity as a Percentage of Another

Write the first quantity as a fraction of the second, in the same units, then multiply by 100%.

Worked example 2

(a) A student scored 42 out of 60. Find the percentage. (b) Express 45 minutes as a percentage of 2 hours.

(a) 42/60 × 100% = 70%.

(b) Convert to minutes: 45/120 × 100% = 37.5%.

Percentage Increase and Decrease Using Multipliers

The fastest method is to use a single multiplier:

  • To increase by r%, multiply by (1 + r/100). An increase of 15% means multiplying by 1.15.
  • To decrease by r%, multiply by (1 − r/100). A decrease of 12% means multiplying by 0.88.

The reason: after a 15% increase you have the original 100% plus 15%, which is 115% = 1.15 of the original.

Worked example 3

(a) Increase ₦8,000 by 15%. (b) Decrease 250 kg by 12%.

(a) 8,000 × 1.15 = ₦9,200.

(b) 250 × 0.88 = 220 kg.

Percentage Change

To find the percentage by which a quantity has changed:

percentage change = (change ÷ original value) × 100%

Always divide by the original value, not the new one.

Worked example 4

(a) The price of a bag of rice rose from ₦1,200 to ₦1,500. Find the percentage increase. (b) A goat's mass fell from 80 kg to 68 kg. Find the percentage decrease.

(a) Change = 300, so 300/1,200 × 100% = 25% increase.

(b) Change = 12, so 12/80 × 100% = 15% decrease.

Reverse Percentages: Finding the Original Value

This is the most commonly failed type of percentage question. If you know the value after a percentage change, divide by the multiplier to get back to the original.

Worked example 5

After a 20% increase, the price of a phone is ₦9,600. What was the original price?

New price = original × 1.2, so original = 9,600 ÷ 1.2 = ₦8,000.

The common mistake is to take 20% of ₦9,600 and subtract it, which gives ₦7,680. That is wrong, because the 20% was calculated on the original price, not on ₦9,600.

Worked example 6

Musa paid ₦6,800 for a shirt after a 15% discount. What was the marked price?

He paid 85% of the marked price, so marked price = 6,800 ÷ 0.85 = ₦8,000.

Worked example 7: VAT

A price including VAT at 7.5% is ₦10,750. Find the price before VAT.

Price before VAT = 10,750 ÷ 1.075 = ₦10,000.

Worked example 8: working back from a remainder

A worker spends 30% of his salary on rent and 25% on food, and saves the rest, which is ₦54,000. Find his salary.

He saves 100% − 30% − 25% = 45% of his salary. So 0.45 × salary = 54,000, and salary = 54,000 ÷ 0.45 = ₦120,000.

Successive Percentage Changes

When one percentage change follows another, multiply the multipliers. Do not add the percentages.

Worked example 9

A price is increased by 20% and then the new price is reduced by 20%. Is it back to the original?

Overall multiplier = 1.2 × 0.8 = 0.96. The final price is 96% of the original: an overall decrease of 4%. The 20% decrease was taken from a larger amount, so it removed more than the increase added.

Compound Growth and Decay

When a quantity changes by the same percentage repeatedly, such as each year, apply the multiplier once for each period:

final value = original × (1 ± r/100)ⁿ

Worked example 10

A town of 40,000 people grows by 5% each year. Find its population after 3 years.

40,000 × 1.05³ = 40,000 × 1.157625 = 46,305.

The same formula, with a multiplier less than 1, gives depreciation, such as a car losing 10% of its value every year. It is also the basis of compound interest.

Percentage Error

When a measurement or estimate differs from the true value:

percentage error = (error ÷ true value) × 100%

Worked example 11

A line of true length 5.0 cm is measured as 4.8 cm. Find the percentage error.

Error = 0.2 cm, so 0.2/5.0 × 100% = 4%.

Percentage Points

When a percentage itself changes, be careful with wording. If an interest rate rises from 10% to 12%, it has risen by 2 percentage points, but by 20% of its old value (2/10 × 100%). News reports often confuse the two.

Common Mistakes with Percentages

  • Dividing by the new value when finding a percentage change. Always divide by the original.
  • Subtracting the percentage from the new value in reverse percentage problems. Divide by the multiplier instead.
  • Adding successive percentages. A 10% rise followed by a 10% rise is 21%, not 20%.
  • Comparing quantities in different units, such as minutes and hours.

Percentages: Practice Questions

  1. Express 7/20 as a percentage.
  2. Find 8% of 450.
  3. Increase ₦45,000 by 12%.
  4. A student's mass fell from 75 kg to 63 kg. Find the percentage decrease.
  5. After a 25% discount, a bag costs ₦4,500. Find the original price.
  6. A price rises by 10% and then by another 10%. Find the overall percentage increase.
  7. A length of 5.0 m is measured as 5.1 m. Find the percentage error.

Answers: 1. 35%. 2. 36. 3. 45,000 × 1.12 = ₦50,400. 4. 12/75 × 100% = 16%. 5. 4,500 ÷ 0.75 = ₦6,000. 6. 1.1 × 1.1 = 1.21, so 21%. 7. 0.1/5.0 × 100% = 2%.

Conclusion

Percentages mean "out of 100" and convert easily to fractions and decimals. Use multipliers for every change: 1 + r/100 for an increase and 1 − r/100 for a decrease. Percentage change always divides by the original value, reverse percentages divide by the multiplier, and successive or repeated changes multiply the multipliers together. With these few rules, every percentage question, from exam marks to inflation and VAT, becomes a short calculation.

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