Fractions in Disguise · Lesson 6 of 12
Percentage Increase or Decrease
“Calculate percentage changes, use final-value multipliers and recover original quantities.”
• Calculate an absolute change and its percentage relative to the original amount. • Distinguish a final percentage from a percentage increase or decrease. • Use multipliers to find a changed quantity. • Recover an original quantity from its value after a change.
A kilogram of tomatoes once cost ₹30 and later costs ₹42. The increase is ₹12. A different item also becomes ₹12 dearer, rising from ₹300 to ₹312. The rupee increase is identical, but it feels much larger for the tomatoes because ₹12 is a much larger share of ₹30 than of ₹300. Percentage change captures this relationship between a change and the amount from which you started.
The starting amount is the reference, or base. A percentage increase compares the amount added with that base; a percentage decrease compares the amount removed with that base. Keeping the base fixed during each comparison is the central habit in this lesson.
We will also reverse these calculations. If you know the final value and the percentage change, you can recover the starting value by undoing the multiplier.
For the tomatoes, the amount added is 42 − 30 = ₹12. Compare ₹12 with the original ₹30: 12/30 = 0.4 = 40%. We say that the price increased by 40%. The final price is the original 100% plus an additional 40%, so it is 140% of the original.
For the second item, the ₹12 increase is 12/300 = 0.04 = 4%. A percentage change explains the relative size of the change. It does not replace the actual amount: ₹12 remains ₹12 in both situations.
Always read the direction of time or comparison. The question “How much did the price rise from ₹30 to ₹42?” uses ₹30 as its base. A question about falling from ₹42 back to ₹30 would use ₹42 as the base and produce a different percentage.
The increase or decrease expressed as a percentage of the original nonzero quantity. The original quantity is the reference base.
Problem
A price rises from ₹30 to ₹42. Find the increase and the percentage increase.
- 1.Original price = ₹30; new price = ₹42.
- 2.Increase = 42 − 30 = ₹12.
- 3.Percentage increase = (12/30) × 100 = 40%.
- 4.Check: 40% of ₹30 is ₹12, and ₹30 + ₹12 = ₹42.
- 5.The final ₹42 is 140% of ₹30. Do not call the increase 140%.
Problem
The average attendance at a venue falls from 160 people to 100. Find the percentage decrease.
- 1.Original attendance = 160; new attendance = 100.
- 2.Decrease = 160 − 100 = 60.
- 3.Percentage decrease = (60/160) × 100 = 37.5%.
- 4.The remaining proportion is 100% − 37.5% = 62.5%. Check: 62.5% of 160 is 100.
- 5.Dividing 60 by 100 would use the new attendance as the base and answer a different comparison.
A percentage change can also be described with a multiplier. If an amount grows by 20%, it keeps its original 100% and gains another 20%. The final amount is therefore 120% = 1.2 times the original. If it decreases by 20%, it keeps 80% = 0.8 of the original.
The multiplier must describe the final amount, not just the amount of change. Multiplying by 0.2 finds the 20% added or removed. Multiplying by 1.2 or 0.8 finds the amount remaining after the corresponding change.
Problem
A population becomes 165% of its earlier value. If the earlier value was 20,000, find the new population and the percentage increase.
- 1.165% of the original means a multiplier of 1.65.
- 2.New population = 1.65 × 20,000 = 33,000.
- 3.Increase = 33,000 − 20,000 = 13,000.
- 4.Percentage increase = (13,000/20,000) × 100 = 65%.
- 5.Thus “165% of the original” and “increased by 65%” express the same change.
A large decrease needs equally careful wording. If demand falls by 85%, the remaining demand is 15% of the original, not 85%. In an equation, the new demand is 0.15 times the old demand.
A fall to zero is a 100% decrease from a positive starting value. A percentage change from an original value of zero is not defined by these formulas because division by zero is impossible. For a rise from zero, report the actual increase instead of forcing it into this formula.
Problem
A number increases by 20% and becomes 90. Find the original number.
- 1.Let the original number be W. After a 20% increase, the final number is 1.2W.
- 2.The given relationship is 1.2W = 90, so W = 90 ÷ 1.2 = 75.
- 3.Check: 20% of 75 is 15; 75 + 15 = 90.
- 4.Subtracting 20% of 90 gives 72, which is wrong because the original increase was based on 75, not on 90.
Problem
After falling by 15%, a quantity is 170. What was it originally?
- 1.A 15% decrease leaves 85% of the original, so 0.85W = 170.
- 2.Divide by the remaining proportion: W = 170 ÷ 0.85 = 200.
- 3.Check: 15% of 200 is 30; 200 − 30 = 170.
- 4.The original must be greater than 170, which our answer satisfies.
To solve a reverse-change problem, describe the final value as a percentage of the original first. A 12% increase means the final value is 112% of the original. A 12% decrease means it is 88%. Then divide the final value by 1.12 or 0.88 respectively.
The base also explains why reversing a change does not usually use the same percentage. If a price rises from ₹100 to ₹120, the increase is 20%. To fall back by ₹20, compare that reduction with the new starting price of ₹120. The required decrease is 20/120 × 100 = 16⅔%.
Problem
A model price rises from ₹38 to ₹42 over a year. Calculate its percentage increase to two decimal places.
- 1.Find the actual increase: ₹42 − ₹38 = ₹4.
- 2.Compare the increase with the original price: (4/38) × 100 = 10.526315…%.
- 3.To two decimal places, the increase is 10.53%.
- 4.Using ₹42 in the denominator would measure the change relative to the final price, which is not the requested annual increase from ₹38.
For 1.05, 0.85, 1.65 and 0.15, write a sentence about how an original quantity changed. Answers: increased by 5%; decreased by 15%; increased by 65%; decreased by 85%. Test each statement with an original amount of 200 to obtain 210, 170, 330 and 30.
Quiz
A price rises from ₹80 to ₹100. What is the percentage increase?
A demand falls by 85%. What fraction of the original demand remains?
Which multiplier represents an increase of 5%?
After a 20% increase, a value is 90. Which operation recovers the original?
Why do changes from 100 to 120 and from 120 to 100 have different percentage sizes?
Practice Problems
- A price rises from ₹60 to ₹100. Find the percentage increase exactly and approximately. Solution: The increase is ₹40. Percentage increase = 40/60 × 100 = 66⅔%, approximately 66.67%.
- A count falls from 240 to 180. Find its percentage decrease and its remaining percentage. Solution: Decrease = 60. The decrease percentage is 60/240 × 100 = 25%. The remaining share is 75%.
- A population p increases by 5%. Write the new population in terms of p and calculate it for p = 1600. Solution: The new population is 1.05p because 100% + 5% = 105%. For p = 1600, it is 1.05 × 1600 = 1680.
- After decreasing by 30%, a length is 84 cm. Find the original length and verify your answer. Solution: The final length is 70% of the original. Original = 84 ÷ 0.70 = 120 cm. Thirty per cent of 120 cm is 36 cm, and 120 − 36 = 84 cm.
- A price rises from ₹200 to ₹250 and later returns to ₹200. Find the increase percentage and the later decrease percentage. Explain the difference. Solution: Increase = 50/200 × 100 = 25%. Later decrease = 50/250 × 100 = 20%. The absolute change is ₹50 in both directions, but the bases are ₹200 and ₹250 respectively.
Key Takeaways
• Measure a percentage change against the original amount. • Find the difference first, then divide by the original and multiply by 100. • An increase of p% produces a final share of (100 + p)%. • A decrease of p% leaves (100 − p)% of the original. • Reverse a change by dividing by its final-value multiplier. • The same absolute change can have different percentages when the reference changes.