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CAT 2025 Lesson : Percentages - Successive Changes & %age Points

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4.7 Successive changes

If there are successive changes of p%p\%, q%q\% and r%r\% in three stages, the effective percentage change is

((1+p100)(1+q100)(1+r100)−1)×100\left( \left( 1 + \dfrac{p}{100} \right) \left( 1 + \dfrac{q}{100} \right) \left( 1 + \dfrac{r}{100} \right) - 1 \right) \times 100

The same can be extended to any number of successive increase or decrease percentages.

Example 15

The arable area of a certain field was 100000100000 sq.ft. at the end of the year 16591659. During the years 16601660 and 16611661, the arable area increased by 10%10\% and decreased by 18.18%18.18\% respectively. What was the arable area (in sq.ft.) at the end of 16611661?

Solution

10%=11010\% = \dfrac{1}{10} and 18.18%∼21118.18\% \sim \dfrac{2}{11}

=100000×(1+110)×(1−211)= 100000 \times \left( 1 + \dfrac{1}{10} \right) \times \left( 1 - \dfrac{2}{11} \right)

=100000×(1110)×(911)= 100000 \times \left( \dfrac{11}{10} \right) \times \left( \dfrac{9}{11} \right)

= 9000090000

Answer: 9000090000

4.8 Percentage vs Percentage Points

A percentage point or percent point is the unit for the arithmetic difference of two percentages.

Let's understand this with an example.

Wheat forms 8%8\% of total food grain production in Year I and 12%12\% of total food grain production in Year II.

Percentage increase of wheat's share = 12−88×100\dfrac{12 - 8}{8} \times 100 = 50%

Percentage point increase of wheat's share = 12%−8%12\% - 8\% = 4% points

Note that percentage points change is just the difference in the percentages.

Example 16

Population of a state was 200,000200,000 in the year 20202020. The annual growth in the population was 20%20\% in 20212021. There was a 1010 percentage point increase in the annual growth rate in 20222022. What was the population in 20232023, if its annual growth rate increased by 10%10\%?

Solution

Growth rate in 20212021 = 20%

Growth rate in 20222022 increased by 1010 percentage points
∴\therefore Growth rate in 2022=20%+10%2022 = 20\% + 10\% = 30%

Growth rate in 20232023 increased by 10%10\%
∴ Growth rate in 2023=30%×1.12023 = 30\% \times 1.1 = 33%

Population in 2020=200,0002020 = 200,000
Population in 2021=200,000×1.2=240,0002021 = 200,000 \times 1.2 = 240,000
Population in 2022=240,000×1.3=312,0002022 = 240,000 \times 1.3 = 312,000
Population in 2023=312,000×1.33=414,9602023 = 312,000 \times 1.33 = 414,960

Answer: 414960414960

Example 17

When Anil brought in additional capital, his stake in the company increased from 40%40\% to 50%50\%. What is the difference in percentage increase and percentage point increase of his stake?

Solution

Percentage Increase = 50−4040×100%=25%\dfrac{50 - 40}{40} \times 100\% = 25\%

Percentage Point Increase = 50%−40%=10%50\% - 40\% = 10\%

Difference = 25%−10%=15%25\% - 10\% = 15\%

Answer: 15%15\%

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