Successive Percentage Changes

One percentage change after another: each change is taken of the value just before it, the multipliers are multiplied, and the result can be written as one percentage change of the original value.

Before you start

To increase a quantity by 15%, the new value is $100\% + 15\% = 115\%$ of the original value, so multiply by 1.15. To decrease it by 15%, the new value is $100\% - 15\% = 85\%$ of the original value, so multiply by 0.85. The number you multiply by is the multiplier. For example, $80 increased by 15% is $80 \times 1.15 = 92$ , which is $92, and $80 decreased by 15% is $80 \times 0.85 = 68$ , which is $68.

A percentage increase or decrease is the change as a percentage of the original value: from $80 to $92 is an increase of $\frac{12}{80} \times 100\% = 15\%$ .

In a calculation with more than one step, a value used in the next step is not rounded. Only the final answer is rounded: money to the nearest cent, and any other answer that is not exact correct to 3 significant figures, unless the question says otherwise.

One Change after Another

Prices, populations and heights often change more than once: a price rises and later rises again, or falls in a sale and then falls further. Changes that come one after another are called successive changes. The question to ask at each change is: a percentage of what? Each change is taken of the value as it is when that change happens.

Example 1: Two rises of 10%

A bowl of laksa at a hawker stall costs $5.00. In January the price rises by 10%. In July the new price rises by another 10%. Find the price after the second rise. Is it the same as a single rise of 20%?

Solution

The first rise is 10% of $5.00, the price it starts from: $\frac{10}{100} \times 5 = 0.5$ . The rise is 50 cents and the price becomes $5 + 0.5 = 5.5$ , which is $5.50.

The second rise is taken of the price just before it, $5.50, not of $5.00: $\frac{10}{100} \times 5.5 = 0.55$ . The second rise is 55 cents, which is more than the first.

The price after the second rise is $5.5 + 0.55 = 6.05$ , which is $6.05.

The same with multipliers: each rise of 10% makes the price 110% of the price before it, so multiply by 1.1 each time: $5 \times 1.1 = 5.5$ , then $5.5 \times 1.1 = 6.05$ .

A single rise of 20% would give $5 \times 1.2 = 6$ , which is $6.00. The two rises give 5 cents more, because the second 10% is taken of a bigger price.

After the second rise the laksa costs $6.05. This is not the same as a single rise of 20%, which would make it $6.00.

Rule: successive percentage changes

When a quantity changes by one percentage and then by another, each change is a percentage of the value just before that change, not of the original value.

Multiplying the Multipliers

Each change multiplies the value by its multiplier. Two changes multiply the original value by the first multiplier and then by the second. So the two multipliers can be multiplied together first, and one multiplication does the work of both changes.

Rule: the overall multiplier

New value = original value × first multiplier × second multiplier

An increase of 8% has the multiplier $108\% = 1.08$ , and a decrease of 4% has the multiplier $96\% = 0.96$ .

The product of the multipliers is the overall multiplier. For three or more changes, multiply all their multipliers.

Example 2: A rise and then a fall

A secondary school has 1250 students in 2024. The enrolment rises by 8% in 2025 and then falls by 4% in 2026. Find the enrolment in 2026
(a) one change at a time,
(b) in one step, with the overall multiplier.

Solution

(a) 2025: the rise of 8% makes the enrolment 108% of 1250: $1250 \times 1.08 = 1350$ students.

2026: the fall of 4% is 4% of 1350, the enrolment just before it. The enrolment becomes 96% of 1350: $1350 \times 0.96 = 1296$ students.

(b) Overall multiplier: $1.08 \times 0.96 = 1.0368$

$1250 \times 1.0368 = 1296$ students

Both ways give 1296. Way (a) also gives the enrolment in 2025; way (b) is quicker when only the final value is wanted.

The enrolment in 2026 is 1296 students.

A Rise and a Fall of the Same Percentage

A price of $100 rises by 10%, and later the new price falls by 10%. It is tempting to think that the price comes back to $100. It does not. The bar model shows why: the start is 10 units of $10, and each 10% is a tenth of the bar as it is at that moment.

The rise is 10% of $100, which is $10. The price becomes $110.

The fall is 10% of $110, the price just before the fall, which is $11. The price becomes $110 - 11 = 99$ , which is $99.

With the multipliers: $100 \times 1.1 \times 0.9 = 100 \times 0.99 = 99$ .

Why the price does not come back

The rise and the fall are both 10%, but not 10% of the same amount. The rise is 10% of the smaller price, $100, and the fall is 10% of the larger price, $110, so the fall takes away $1 more than the rise added. In the bar model, the rise adds 1 unit, and the fall takes away a tenth of 11 units, which is 1.1 units, so 9.9 units are left: 99% of the start. The overall multiplier, $1.1 \times 0.9 = 0.99$ , says the same for any starting price: after a 10% rise and a 10% fall, a value is 99% of what it was.

Example 3: A rise and a fall of 20%

A bag at a shop costs $45. The shop raises its price by 20%, and a month later reduces the new price by 20%.
(a) Find the final price.
(b) The final price is what percentage of the original price?

Solution

(a) After the rise of 20%: $45 \times 1.2 = 54$ , so the price is $54.

The reduction is 20% of $54, the price just before it: $\frac{20}{100} \times 54 = 10.8$ , so $10.80. The price becomes $54 - 10.8 = 43.2$ , which is $43.20. With the multiplier: $54 \times 0.8 = 43.2$ .

(b) Overall multiplier: $1.2 \times 0.8 = 0.96$ , which is 96%. So the final price is 96% of the original price.

Check: $\frac{43.2}{45} \times 100\% = 96\%$ .

The price ends $1.80 below where it started, which is 4% of the original price: $\frac{1.8}{45} \times 100\% = 4\%$ .

(a) The final price is $43.20.
(b) It is 96% of the original price, which is 4% less than the price the bag started at.

The Order of the Changes Does Not Matter

Suppose the 10% fall comes first and the 10% rise second. From $100 the fall gives $90, and the rise, 10% of $90, gives $90 + 9 = 99$ : $99 again. This is no accident. The two changes multiply the price by 0.9 and by 1.1, and two numbers can be multiplied in either order: $1.1 \times 0.9 = 0.9 \times 1.1 = 0.99$ .

Rule: the order of successive changes

The final value after successive percentage changes is the same whatever the order of the changes, because their multipliers can be multiplied in any order. It is sometimes said that the order matters. For the final value, it does not.

What does depend on the order is the value in between, and the size of each change in dollars (or students, or centimetres).

Example 4: Two shops, two orders

A phone costs $800. Shop P raises the price by 15% and later gives a discount of 20% on the new price. Shop Q gives a discount of 20% first and later raises the discounted price by 15%. Which shop's final price is lower?

Solution

Shop P: $800 \times 1.15 = 920$ , then $920 \times 0.8 = 736$ . The price goes from $800 to $920 to $736.

Shop Q: $800 \times 0.8 = 640$ , then $640 \times 1.15 = 736$ . The price goes from $800 to $640 to $736.

Both shops end at $736. The overall multiplier is the same in either order: $1.15 \times 0.8 = 0.8 \times 1.15 = 0.92$ , and $800 \times 0.92 = 736$ .

The prices in between are different, $920 and $640, and so are the changes in dollars. At Shop P the rise is $120 and the discount is $184. At Shop Q the discount is $160 and the rise is $96. Only the final price is the same.

Neither: both shops end at $736. The order of the two changes does not change the final price.

The Single Equivalent Change

Successive changes can be replaced by one change that has the same effect. It is called the single equivalent change, and its multiplier is the overall multiplier. It is written as a percentage increase or decrease of the original value.

Rule: from the overall multiplier to one percentage change

Multiply the multipliers, and write the overall multiplier as a percentage: $1.0368 = 103.68\%$ .
If it is more than 100%, the single equivalent change is an increase of (overall multiplier − 100%) of the original value: $103.68\% - 100\% = 3.68\%$ .
If it is less than 100%, the single equivalent change is a decrease of (100% − overall multiplier) of the original value: $0.96 = 96\%$ , and $100\% - 96\% = 4\%$ .
If it is exactly 100%, the value ends where it started.

Example 5: Finding the single equivalent change

Find the single percentage change of the original value that is equivalent to
(a) an increase of 8% followed by a decrease of 4%,
(b) a decrease of 25% followed by a decrease of 10%,
(c) an increase of 50% followed by a decrease of 40%.

Solution

(a) Overall multiplier: $1.08 \times 0.96 = 1.0368 = 103.68\%$ . This is more than 100%, so it is an increase: $103.68\% - 100\% = 3.68\%$ .

Check with Example 2: the enrolment went from 1250 to 1296, a change of 46 students, and $\frac{46}{1250} \times 100\% = 3.68\%$ . It is not $8\% - 4\% = 4\%$ .

(b) Overall multiplier: $0.75 \times 0.9 = 0.675 = 67.5\%$ . This is less than 100%, so it is a decrease: $100\% - 67.5\% = 32.5\%$ . The value keeps 67.5% of itself and loses 32.5%, not $25\% + 10\% = 35\%$ .

(c) Overall multiplier: $1.5 \times 0.6 = 0.9 = 90\%$ , a decrease of $100\% - 90\% = 10\%$ . An increase of 50% and a decrease of 40% may look like an overall increase, but the 40% is taken of a value that is already 50% bigger.

(a) An increase of 3.68% (b) a decrease of 32.5% (c) a decrease of 10%, each a percentage of the original value.

Example 6: Two discounts of 20%

In a sale at a mall, a pair of shoes marked $150 is sold at 20% off. On the last day of the sale, a further 20% is taken off the sale price. Ravi says that this is the same as 40% off.
(a) Find the price on the last day.
(b) Is Ravi right? Find the single discount, as a percentage of the marked price, that gives the same price.

Solution

(a) After the first discount: $150 \times 0.8 = 120$ , so $120. The second discount is 20% of $120, the sale price: $120 \times 0.8 = 96$ , so $96.

(b) A discount of 40% would leave 60% of the marked price: $150 \times 0.6 = 90$ , which is $90. The shoes cost $96, not $90, so Ravi is not right.

Overall multiplier: $0.8 \times 0.8 = 0.64 = 64\%$ . The single discount is $100\% - 64\% = 36\%$ of the marked price.

Check: 36% of $150 is $0.36 \times 150 = 54$ , and $150 - 54 = 96$ .

The second 20% is taken of $120, not of $150, so it takes off only $24, not $30.

(a) The price on the last day is $96.
(b) No. Two discounts of 20% are the same as a single discount of 36% of the marked price, not 40%.

Changes over Several Years

A population, a price or the height of a plant often changes by a percentage each year, or each month. Each year's change is a percentage of the value the year before, not of the value in the first year. When the percentage is the same every year, the multiplier is the same every year, and index notation shortens the product: $1.05 \times 1.05 \times 1.05 = 1.05^{3}$ .

Example 7: A population over three years

The population of a town is $24\,000$ at the start of 2023. Each year it grows by 5% of the population at the start of that year.
(a) Find the population at the start of 2026.
(b) Find the overall percentage increase over the three years.

Solution

(a) Each year the population becomes 105% of the population the year before, so the multiplier for each year is 1.05.

Start of 2024: $24\,000 \times 1.05 = 25\,200$

Start of 2025: $25\,200 \times 1.05 = 26\,460$

Start of 2026: $26\,460 \times 1.05 = 27\,783$

In one step: $24\,000 \times 1.05^{3} = 24\,000 \times 1.157625 = 27\,783$

(b) The overall multiplier is $1.05^{3} = 1.157625 = 115.7625\%$ , so the overall increase is $115.7625\% - 100\% = 15.7625\% \approx 15.8\%$ (correct to 3 significant figures) of the population at the start of 2023.

This is more than $3 \times 5\% = 15\%$ , because each year's 5% is taken of a bigger population than the year before: the town grows by 1200, then by 1260, then by 1323 people.

(a) The population at the start of 2026 is $27\,783$ .
(b) The overall increase is about 15.8% of the population at the start of 2023.

Round Only at the End

In successive changes each value is used in the next step. If a value in the middle is rounded, its rounding error is carried into every step after it, and it can change the answer. Keep every digit the calculator gives until the last step, and round only the final answer.

Example 8: A plant over three months

A plant is 50 cm tall at the start of February. It grows by 7% in February, by 6.5% in March and by 5% in April, each month's growth a percentage of its height at the start of that month. Find its height at the end of April, correct to 3 significant figures.

Solution

End of February: $50 \times 1.07 = 53.5$ cm

End of March: $53.5 \times 1.065 = 56.9775$ cm. This is not the answer, so keep all its digits.

End of April: $56.9775 \times 1.05 = 59.826375$ cm

In one step: $50 \times 1.07 \times 1.065 \times 1.05 = 59.826375$

Round only now: $59.826375 \approx 59.8$ cm (correct to 3 significant figures).

What rounding in the middle does. Suppose the height at the end of March is rounded to 57.0 cm and then used: $57.0 \times 1.05 = 59.85$ , which gives 59.9 cm correct to 3 significant figures. That is wrong in the third significant figure. The rounding added $57.0 - 56.9775 = 0.0225$ cm, and the last step carried it into the answer as $0.0225 \times 1.05 = 0.023625$ cm: $59.85 - 59.826375 = 0.023625$ . That small error was enough to change the answer.

Adding the percentages is wrong too: $7\% + 6.5\% + 5\% = 18.5\%$ would give $50 \times 1.185 = 59.25$ cm, too short. The overall multiplier is $1.07 \times 1.065 \times 1.05 = 1.1965275$ , an increase of about 19.7%.

At the end of April the plant is about 59.8 cm tall (correct to 3 significant figures).

Rule: round only at the end

Do not round the values in the middle of successive changes. Keep the full calculator value (or use the overall multiplier in one step), and round once, at the end.

Common Mistakes

Adding the percentages. A rise of 10% followed by another rise of 10% is a rise of 21%, not 20%, since $1.1 \times 1.1 = 1.21$ . Two discounts of 20% are a discount of 36%, not 40%.
Subtracting the percentages. A rise of 8% followed by a fall of 4% is not a rise of 4%: $1.08 \times 0.96 = 1.0368$ , a rise of 3.68%.
Taking the second change of the original value. The second change is a percentage of the value just before it. After a rise from $100 to $110, a fall of 10% takes off $11, not $10.
Expecting a rise and a fall of the same percentage to cancel. A rise of 10% and a fall of 10%, in either order, leave 99% of the starting value.
Reading the overall multiplier as the change. An overall multiplier of 0.64 means the value is now 64% of the original, so the decrease is 36%, not 64%. An overall multiplier of 1.0368 is an increase of 3.68%, not 103.68%.
Rounding in the middle. In Example 8, rounding 56.9775 to 57.0 before the last step turns the right answer, 59.8 cm, into 59.9 cm.

Practice

Work each question on paper before you open its solution. Give answers that are not exact correct to 3 significant figures, and money to the nearest cent.

Question 1

Increase 400 by 20%, and then increase the result by 10%. Is the answer the same as increasing 400 by 30%?

ShowHide solution

After the first increase: $400 \times 1.2 = 480$

The second increase is 10% of 480, not of 400: $480 \times 1.1 = 528$

An increase of 30%: $400 \times 1.3 = 520$

The answers differ by 8, because the second 10% is 48 (10% of 480), not 40 (10% of 400).

The answer is 528. It is not the same as increasing 400 by 30%, which gives 520.

Question 2

A price of $250 is decreased by 10%, and then the new price is increased by 10%.
(a) Find the final price.
(b) Find the overall percentage change, as a percentage of the original price.

ShowHide solution

(a) After the decrease: $250 \times 0.9 = 225$ , so $225.

The increase is 10% of $225: $225 \times 1.1 = 247.5$ , so $247.50.

(b) Overall multiplier: $0.9 \times 1.1 = 0.99 = 99\%$ . This is less than 100%, so it is a decrease of $100\% - 99\% = 1\%$ .

Check: the price fell by $250 - 247.5 = 2.5$ , which is $2.50, and $\frac{2.5}{250} \times 100\% = 1\%$ .

(a) The final price is $247.50.
(b) A decrease of 1% of the original price.

Question 3

Find the overall multiplier and the single equivalent change, as a percentage of the original value, for
(a) an increase of 30% followed by an increase of 10%,
(b) a decrease of 40% followed by an increase of 50%,
(c) an increase of 25% followed by a decrease of 20%.

ShowHide solution

(a) $1.3 \times 1.1 = 1.43 = 143\%$ . This is an increase of $143\% - 100\% = 43\%$ .

(b) $0.6 \times 1.5 = 0.9 = 90\%$ . This is a decrease of $100\% - 90\% = 10\%$ .

In (c) the fall of 20% is taken of a value that is 125% of the start, and 20% of 125% is 25%, so the fall takes away exactly what the rise added.

(a) 1.43, an increase of 43%
(b) 0.9, a decrease of 10%
(c) 1, no change

Question 4

At a jewellery shop a necklace is marked at $3400. In a sale it is sold at 30% off, and members get a further 20% off the sale price.
(a) How much does a member pay?
(b) Find the single discount, as a percentage of the marked price, that a member gets.

ShowHide solution

(a) Sale price: $3400 \times 0.7 = 2380$ , so $2380.

A member gets 20% off $2380: $2380 \times 0.8 = 1904$ , so $1904.

(b) Overall multiplier: $0.7 \times 0.8 = 0.56 = 56\%$ . A member pays 56% of the marked price, so the discount is $100\% - 56\% = 44\%$ .

Check: a member saves $3400 - 1904 = 1496$ , which is $1496, and $\frac{1496}{3400} \times 100\% = 44\%$ .

(a) A member pays $1904.
(b) A discount of 44% of the marked price, not $30\% + 20\% = 50\%$ .

Question 5

Siti can buy the same $60 shirt at one of two shops. Shop A gives a discount of 15% and then a further 15% off the discounted price. Shop B gives a single discount of 30%. At which shop is the shirt cheaper, and by how much?

ShowHide solution

Shop A: $60 \times 0.85 = 51$ , then $51 \times 0.85 = 43.35$ , so $43.35.

Shop B: $60 \times 0.7 = 42$ , so $42.

Difference: $43.35 - 42 = 1.35$ , which is $1.35.

At Shop A the overall multiplier is $0.85 \times 0.85 = 0.7225 = 72.25\%$ , a single discount of $100\% - 72.25\% = 27.75\%$ . That is less than 30%.

The shirt is cheaper at Shop B, by $1.35.

Question 6

The Lim family's electricity bill was $150 in May. It rose by 12% in June, and the June bill then fell by 8% in July.
(a) Find the bill in July.
(b) Find the overall percentage change from May to July. Is it a rise of $12\% - 8\% = 4\%$ ?

ShowHide solution

(a) June: $150 \times 1.12 = 168$ , so $168.

July: the fall is 8% of the June bill: $168 \times 0.92 = 154.56$ , so $154.56.

(b) Overall multiplier: $1.12 \times 0.92 = 1.0304 = 103.04\%$ . This is an increase of $103.04\% - 100\% = 3.04\%$ .

Check: $154.56 - 150 = 4.56$ and $\frac{4.56}{150} \times 100\% = 3.04\%$ .

It is not 4%, because the fall of 8% is taken of $168, which is more than $150.

(a) The bill in July is $154.56.
(b) An increase of 3.04% of the May bill, not 4%.

Question 7

Ali says: "Increasing a price by 20% and then decreasing it by 15% gives a different price from decreasing it by 15% first and then increasing it by 20%." Test his claim on a price of $150, and explain the result with the multipliers.

ShowHide solution

Increase first: $150 \times 1.2 = 180$ , then $180 \times 0.85 = 153$ , so $153.

Decrease first: $150 \times 0.85 = 127.5$ , then $127.5 \times 1.2 = 153$ , so $153.

Both orders give $153. The two changes multiply the price by 1.2 and by 0.85, and $1.2 \times 0.85 = 0.85 \times 1.2 = 1.02$ . So the final price is 102% of the original price, whatever the order.

Only the price in between is different: $180 in one order and $127.50 in the other.

Ali is wrong. Both orders give $153, an overall increase of 2% of the original price.

Question 8

A new MRT station is used by $18\,000$ people a day in its first year. The number grows by 10% each year, each year's growth a percentage of the number the year before.
(a) How many people a day use the station three years later?
(b) Find the overall percentage increase over the three years.

ShowHide solution

(a) The multiplier for each year is $100\% + 10\% = 110\% = 1.1$ .

After 1 year: $18\,000 \times 1.1 = 19\,800$

After 2 years: $19\,800 \times 1.1 = 21\,780$

After 3 years: $21\,780 \times 1.1 = 23\,958$

In one step: $18\,000 \times 1.1^{3} = 18\,000 \times 1.331 = 23\,958$

(b) The overall multiplier is $1.1^{3} = 1.331 = 133.1\%$ , an increase of $133.1\% - 100\% = 33.1\%$ . It is not $3 \times 10\% = 30\%$ .

(a) $23\,958$ people a day
(b) An increase of 33.1% of the number in the first year.

Question 9

Mr Tan buys a car for $ $90\,000$ . Its value falls by 12% each year, each year's fall a percentage of its value at the start of that year.
(a) Find its value after 2 years.
(b) Find the overall percentage decrease in its value over the 2 years.

ShowHide solution

(a) The multiplier for each year is $100\% - 12\% = 88\% = 0.88$ .

After 1 year: $90\,000 \times 0.88 = 79\,200$

After 2 years: $79\,200 \times 0.88 = 69\,696$

In one step: $90\,000 \times 0.88^{2} = 90\,000 \times 0.7744 = 69\,696$

(b) The overall multiplier is $0.88^{2} = 0.7744 = 77.44\%$ , so the decrease is $100\% - 77.44\% = 22.56\% \approx 22.6\%$ (correct to 3 significant figures).

Check: $90\,000 - 69\,696 = 20\,304$ , and $\frac{20\,304}{90\,000} \times 100\% = 22.56\%$ .

(a) After 2 years the car is worth $ $69\,696$ .
(b) A decrease of about 22.6% of the price Mr Tan paid, not $2 \times 12\% = 24\%$ .

Question 10

A sunflower is 40 cm tall. It grows by 5% in the first week, by 18% in the second week and by 7% in the third week, each week's growth a percentage of its height at the start of that week.
(a) Find its height after three weeks, correct to 3 significant figures.
(b) Hui Min rounds the height after two weeks to 3 significant figures before the last step. What answer does she get, and why is it wrong?

ShowHide solution

(a) After week 1: $40 \times 1.05 = 42$ cm

After week 2: $42 \times 1.18 = 49.56$ cm. Keep every digit: this is not the answer.

After week 3: $49.56 \times 1.07 = 53.0292$ cm

In one step: $40 \times 1.05 \times 1.18 \times 1.07 = 53.0292$

Round only now: $53.0292 \approx 53.0$ cm (correct to 3 significant figures).

(b) Hui Min rounds 49.56 to 49.6 and works out $49.6 \times 1.07 = 53.072$ , which gives 53.1 cm correct to 3 significant figures.

Her rounding added $49.6 - 49.56 = 0.04$ cm in the middle, and the last step carried it into her answer as $0.04 \times 1.07 = 0.0428$ cm. That was enough to change the third significant figure. She should have kept 49.56 and rounded only the final answer.

(a) About 53.0 cm
(b) She gets 53.1 cm. It is wrong because she rounded a value in the middle of the working; the height is 53.0 cm correct to 3 significant figures.

Question 11

A bubble tea shop's sales were $ $20\,000$ in 2023. They rose by 15% in 2024, fell by 15% in 2025 and rose by 12% in 2026, each change a percentage of the sales the year before.
(a) Find the sales in 2026.
(b) Find the overall percentage change in the sales from 2023 to 2026.
(c) Would the sales in 2026 be different if the three changes had come in another order?

ShowHide solution

(a) 2024: $20\,000 \times 1.15 = 23\,000$

2025: $23\,000 \times 0.85 = 19\,550$

2026: $19\,550 \times 1.12 = 21\,896$

(b) Overall multiplier: $1.15 \times 0.85 \times 1.12 = 1.0948 = 109.48\%$ . This is an increase of $109.48\% - 100\% = 9.48\%$ .

Check: $21\,896 - 20\,000 = 1896$ , and $\frac{1896}{20\,000} \times 100\% = 9.48\%$ .

The rise of 15% and the fall of 15% together multiply the sales by $1.15 \times 0.85 = 0.9775$ , a fall of 2.25%. That is why the overall increase is less than 12%.

(c) No. The three multipliers can be multiplied in any order, so the sales in 2026 would still be $ $21\,896$ . Only the sales in 2024 and 2025 would be different.

(a) The sales in 2026 were $ $21\,896$ .
(b) An increase of 9.48% of the sales in 2023.
(c) No. Only the sales in the years in between would change.

Question 12

A concert ticket costs $50. Its price is raised by 20%.
(a) Find the new price.
(b) By what percentage must the new price be reduced to bring it back to $50?
(c) Explain why the answer to (b) is not 20%.

ShowHide solution

(a) $50 \times 1.2 = 60$ , so the new price is $60.

(b) The reduction needed is $60 - 50 = 10$ , which is $10. A reduction is a percentage of the price just before it, $60: $\frac{10}{60} \times 100\% = 16.666\ldots\% \approx 16.7\%$ .

Check with the multipliers: a reduction of $16\frac{2}{3}\%$ leaves $83\frac{1}{3}\% = \frac{5}{6}$ of the price, and $1.2 \times \frac{5}{6} = 1$ , so the price ends where it started.

(c) A reduction of 20% would be taken of $60: $\frac{20}{100} \times 60 = 12$ , so it would take off $12 and leave $48. The rise of $10 was 20% of the smaller price, $50; the same $10 is a smaller percentage of the larger price, $60.

(a) $60
(b) About 16.7% (exactly $16\frac{2}{3}\%$ )
(c) The reduction is taken of $60, not $50. A reduction of 20% of $60 is $12, which is more than the $10 rise, so the price would fall to $48.

Key Points

In successive percentage changes, each change is a percentage of the value just before it, not of the original value.
An increase of 8% multiplies by 1.08 and a decrease of 4% multiplies by 0.96. Multiply by the multiplier of each change in turn.
The product of the multipliers is the overall multiplier: new value = original value × overall multiplier.
The percentages of successive changes are not added or subtracted: two rises of 10% are a rise of 21%, and two discounts of 20% are a discount of 36%.
A rise and a fall of the same percentage do not cancel: a 10% rise and a 10% fall leave 99% of the start.
The order of the changes does not change the final value, because multipliers can be multiplied in any order. Only the values in between change.
The single equivalent change is a percentage of the original value: an overall multiplier of 1.0368 is an increase of 3.68%, and 0.64 is a decrease of 36%.
For the same percentage change each year, use index notation: three years of growth of 5% multiply by $1.05^{3}$ .
Round only at the end. A value rounded in the middle carries its error into the answer.