Sec 1 Maths · Percentage

Successive changes and percentage points questions with answers

Practice questions on successive changes and percentage points, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

A car worth $100,000 loses 30% of its value in the first year and 5% of its value in the second year. Find its value after two years.

Step 1 of 2

Write each change as a percentage of the value before it

\text{1st change: } 100\% - 30\% = 70\%

\text{2nd change: } 100\% - 5\% = 95\%

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Questions

Question 1Two changes in a row: find the final value
Siti invests $5,000. The value of her investment grows by 15% in the first year and by 20% in the second year. Find its value after two years.
Show the worked solution
1. Step 1: Write each change as a percentage of the value before it $\text{1st change: } 100\% + 15\% = 115\%$ $\text{2nd change: } 100\% + 20\% = 120\%$
2. Step 2: Multiply the original value by both, one after the other $\text{Final value} = 5000 \times \dfrac{115}{100} \times \dfrac{120}{100}$ $= 6900$ Its value after two years is $6900.
Question 2Two changes in a row: the overall percentage change
The price of an item is increased by 10% and then decreased by 5%. Find the overall percentage increase.
Show the worked solution
1. Step 1: Write each change as (100% ± the change) $\text{1st: } 110\%, \quad \text{2nd: } 95\%$
2. Step 2: Successive changes multiply $\dfrac{110}{100} \times \dfrac{95}{100} = \dfrac{104.5}{100}$
3. Step 3: Compare with the original 100% $104.5\% - 100\% = 4.5\%$ The overall percentage increase is $4.5\%$ .
Question 3The change in percentage points
The percentage of pupils in a school who cycle to school fell from 8% to 2%. Find the decrease, in percentage points.
Show the worked solution
1. Step 1: How do you find a change in percentage points?A change in percentage points is the difference between the two percentages.
2. Step 2: Work it out $8 - 2 = 6$ It decreased by $6$ percentage points.
Question 4The percentage change of a rate
The percentage of households in a town with solar panels rose from 20% to 24%. Find the percentage increase in the percentage of households with solar panels.
Show the worked solution
1. Step 1: Find the difference between the two percentages $24 - 20 = 4$
2. Step 2: How do you find a percentage change of a rate?A percentage change compares the change with the original rate.
3. Step 3: Work it out $\dfrac{4}{20} \times 100\% = 20\%$ The percentage increase is $20\%$ .
Question 5Percentage points or per cent: is the statement right?
The pass rate of a school's Mathematics examination fell from 20% to 17%. Farah says, "It fell by 15%." State whether Farah is correct.
Show the worked solution
1. Step 1: Find the change in percentage points $20 - 17 = 3$ A change of $3$ percentage points.
2. Step 2: Find the percentage change $\dfrac{3}{20} \times 100\% = 15\%$
3. Step 3: Compare with the statementIt fell by $3$ percentage points, which is $15\%$ of the original rate. So Farah is correct.
Question 6Work back one change, then the overall change
The number of visitors to a museum in 2023 was 6,720. This was 40% more than in 2022. In 2024, the number was 8,400. Find the percentage increase in the number of visitors to a museum from 2022 to 2024.
Show the worked solution
1. Step 1: Work backwards to 2022: the 2023 number is what percentage of it? $100\% + 40\% = 140\% \rightarrow 6720$
2. Step 2: Find 1%, then 100% $1\% \rightarrow \dfrac{6720}{140} = 48$ $100\% \rightarrow 4800$ In 2022 the number was $4800$ .
3. Step 3: The change from 2022 to 2024 $8400 - 4800 = 3600$
4. Step 4: Divide by the ORIGINAL value (2022), times 100% $\dfrac{3600}{4800} \times 100\% = 75\%$ The percentage increase is $75\%$ .
Question 7Two shops: discounts one after another, or cashback
A laptop has a marked price of $800 at two shops. The table shows each shop's offer. Kumar is a member of Shop A. Find how much less Kumar pays at the cheaper shop.
Shop Offer
A 10% off, then a further 10% off the reduced price for members
B 20% off, then $50 cashback
Show the worked solution
1. Step 1: Shop A (in dollars): the second discount is on the REDUCED price, so multiply one after the other $\dfrac{90}{100} \times 800 = 720$ $\dfrac{90}{100} \times 720 = 648$
2. Step 2: Shop B (in dollars): the discount first, then take off the cashback $\dfrac{80}{100} \times 800 = 640$ $640 - 50 = 590$
3. Step 3: Compare: the cheaper shop, and how much less $648 - 590 = 58$ Shop B is cheaper, by $58.00.

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Worked example

A car worth $100,000 loses 30% of its value in the first year and 5% of its value in the second year. Find its value after two years.

Step 1 of 2

Write each change as a percentage of the value before it

\text{1st change: } 100\% - 30\% = 70\%

\text{2nd change: } 100\% - 5\% = 95\%

Questions

Question 1Two changes in a row: find the final value

Siti invests $5,000. The value of her investment grows by 15% in the first year and by 20% in the second year. Find its value after two years.

Show the worked solution

Step 1: Write each change as a percentage of the value before it $\text{1st change: } 100\% + 15\% = 115\%$ $\text{2nd change: } 100\% + 20\% = 120\%$
Step 2: Multiply the original value by both, one after the other $\text{Final value} = 5000 \times \dfrac{115}{100} \times \dfrac{120}{100}$ $= 6900$ Its value after two years is $6900.

Question 2Two changes in a row: the overall percentage change

The price of an item is increased by 10% and then decreased by 5%. Find the overall percentage increase.

Show the worked solution

Step 1: Write each change as (100% ± the change) $\text{1st: } 110\%, \quad \text{2nd: } 95\%$
Step 2: Successive changes multiply $\dfrac{110}{100} \times \dfrac{95}{100} = \dfrac{104.5}{100}$
Step 3: Compare with the original 100% $104.5\% - 100\% = 4.5\%$ The overall percentage increase is $4.5\%$ .

Question 3The change in percentage points

The percentage of pupils in a school who cycle to school fell from 8% to 2%. Find the decrease, in percentage points.

Show the worked solution

Step 1: How do you find a change in percentage points?A change in percentage points is the difference between the two percentages.
Step 2: Work it out $8 - 2 = 6$ It decreased by $6$ percentage points.

Question 4The percentage change of a rate

The percentage of households in a town with solar panels rose from 20% to 24%. Find the percentage increase in the percentage of households with solar panels.

Show the worked solution

Step 1: Find the difference between the two percentages $24 - 20 = 4$
Step 2: How do you find a percentage change of a rate?A percentage change compares the change with the original rate.
Step 3: Work it out $\dfrac{4}{20} \times 100\% = 20\%$ The percentage increase is $20\%$ .

Question 5Percentage points or per cent: is the statement right?

The pass rate of a school's Mathematics examination fell from 20% to 17%. Farah says, "It fell by 15%." State whether Farah is correct.

Show the worked solution

Step 1: Find the change in percentage points $20 - 17 = 3$ A change of $3$ percentage points.
Step 2: Find the percentage change $\dfrac{3}{20} \times 100\% = 15\%$
Step 3: Compare with the statementIt fell by $3$ percentage points, which is $15\%$ of the original rate. So Farah is correct.

Question 6Work back one change, then the overall change

The number of visitors to a museum in 2023 was 6,720. This was 40% more than in 2022. In 2024, the number was 8,400. Find the percentage increase in the number of visitors to a museum from 2022 to 2024.

Show the worked solution

Step 1: Work backwards to 2022: the 2023 number is what percentage of it? $100\% + 40\% = 140\% \rightarrow 6720$
Step 2: Find 1%, then 100% $1\% \rightarrow \dfrac{6720}{140} = 48$ $100\% \rightarrow 4800$ In 2022 the number was $4800$ .
Step 3: The change from 2022 to 2024 $8400 - 4800 = 3600$
Step 4: Divide by the ORIGINAL value (2022), times 100% $\dfrac{3600}{4800} \times 100\% = 75\%$ The percentage increase is $75\%$ .

Question 7Two shops: discounts one after another, or cashback

A laptop has a marked price of $800 at two shops. The table shows each shop's offer. Kumar is a member of Shop A. Find how much less Kumar pays at the cheaper shop.

Shop	Offer
A	10% off, then a further 10% off the reduced price for members
B	20% off, then $50 cashback

Show the worked solution

Step 1: Shop A (in dollars): the second discount is on the REDUCED price, so multiply one after the other $\dfrac{90}{100} \times 800 = 720$ $\dfrac{90}{100} \times 720 = 648$
Step 2: Shop B (in dollars): the discount first, then take off the cashback $\dfrac{80}{100} \times 800 = 640$ $640 - 50 = 590$
Step 3: Compare: the cheaper shop, and how much less $648 - 590 = 58$ Shop B is cheaper, by $58.00.