Reverse Percentages

Finding the original value from a value after a percentage increase or decrease, or from a part and its percentage: find 1% or 10% and then 100%, or divide by the multiplier, and check by working forward.

Before you start

To find a percentage of a quantity, write the percentage as a decimal (or a fraction) and multiply: 35% of $240 is $0.35 \times 240 = 84$ , so $84. The quantity that the percentage is taken of is the base, and the base is 100%.

After an increase of 20%, the new value is $100\% + 20\% = 120\%$ of the original, so the new value is the original $\times\ 1.2$ . After a decrease of 15%, the new value is $100\% - 15\% = 85\%$ of the original, so it is the original $\times\ 0.85$ . The numbers 1.2 and 0.85 are the multipliers.

When one percentage change follows another, each change is on the value it starts from, so the value is multiplied by each multiplier in turn. A rise of 20% and then a fall of 15% multiply the value by $1.2 \times 0.85 = 1.02$ , and the order of the two changes does not change the final value.

In a bar model a quantity is drawn as a bar of equal units. If 4 units stand for 36 students, then 1 unit stands for $36 \div 4 = 9$ students.

What a Reverse Percentage Is

In the lesson on increasing and decreasing by a percentage, the original value was known and the new value was found from it. For example: a bowl of laksa costs $5.50 and the price goes up by 20%; what is the new price? In a reverse percentage problem the question runs the other way. A value after the change is known (or a part of a whole is known), together with the percentage, and the original value is to be found: the price after a 20% rise is $6.60, so what was the price before?

Every reverse percentage problem turns on one question: the value I know is what percentage of the original? The original is always the base, 100%.

30% of the students in a cohort walk to school, and 54 students walk. The 54 students are 30% of all the students.

A price rose by 20% to $6.60. The rise was 20% of the old price, so $6.60 is $100\% + 20\% =$ 120% of the old price.

After a 15% discount a pair of shoes costs $102. The discount was 15% of the price before the discount, so $102 is $100\% - 15\% =$ 85% of that price.

Once that percentage is known, there are two ways to find the original, and they always give the same answer.

Rule: finding the original value

First decide what percentage of the original value the value you know is. The original value is 100%.

The unitary method. Write down what the known value is: "120% of the original → $6.60". Divide to find 1% of the original (or 10%, or any percentage that divides evenly), then multiply to find 100%.

Dividing by the multiplier. Write that percentage as a decimal and divide: original value = known value ÷ multiplier. After a 20% increase, the original price is $6.60 \div 1.2$ dollars.

Check by working forward. Apply the percentage change to your answer. It must give the value you were told.

Why dividing by the multiplier works

Working forward, the original price is multiplied by 1.2 to give the new price: original price $\times\ 1.2$ = $6.60. Dividing undoes multiplying, so dividing $6.60 by 1.2 takes us back to the original price.

The unitary method does the same thing in two steps. Dividing by 120 (to find 1%) and then multiplying by 100 (to find 100%) is the same as dividing by $\frac{120}{100}$ , which is 1.2.

Finding the Whole from a Part

The simplest case is a part of a whole. The whole is 100% and the part is a percentage of it that is less than 100%. The part and its percentage are known, and the whole is to be found.

Example 1: Students who walk to school

In a Secondary 1 cohort, 30% of the students walk to school. 54 students walk to school. How many students are there in the cohort?

Solution

The whole cohort is 100% and the students who walk are 30% of it. Draw the cohort as a bar of 10 equal units, so that each unit is 10% of the cohort. The students who walk are 3 of these units.

30% of the cohort → 54 students

10% of the cohort → 18 students, since $54 \div 3 = 18$ (3 units are 54 students, so 1 unit is 18 students)

100% of the cohort → 180 students, since $18 \times 10 = 180$

Dividing by the multiplier: 30% is 0.3, and $54 \div 0.3 = 180$ .

Here 10% is a better step than 1%, because 1% of the cohort would be $54 \div 30 = 1.8$ students. That is only a step in the working, so it still gives the right answer ( $1.8 \times 100 = 180$ ), but whole numbers are easier to check as you go.

Check by working forward: 30% of 180 is $0.3 \times 180 = 54$ . This is the number of students who walk.

There are 180 students in the cohort.

Example 2: A number from a percentage of it

35% of a number is 140. Find the number.

Solution

35% of the number → 140

1% of the number → 4, since $140 \div 35 = 4$

100% of the number → 400, since $4 \times 100 = 400$

Dividing by the multiplier: 35% is 0.35, and $140 \div 0.35 = 400$ .

Check by working forward: $0.35 \times 400 = 140$ .

The number is 400.

The Original Value after an Increase

After an increase, the new value is more than the original, so it is more than 100% of the original. After a 20% increase, the original is 100% and the increase is 20% of the original, so the new value is 120% of the original. In a bar model, the original is 10 units of 10% each and the new value is 12 of the same units.

Example 3: A price after an increase

The price of a bowl of laksa at a hawker stall goes up by 20%. It now costs $6.60. What was the price before the increase?

Solution

The old price is 100%. The increase is 20% of the old price, so the new price is $100\% + 20\% = 120\%$ of the old price. Draw the old price as 10 units and the new price as 12 of the same units.

120% of the old price → $6.60

10% of the old price → $0.55, since $6.60 \div 12 = 0.55$

100% of the old price → $5.50, since $0.55 \times 10 = 5.5$

Dividing by the multiplier: the new price is the old price $\times\ 1.2$ , so the old price is $6.60 \div 1.2 = 5.5$ dollars, which is $5.50.

Check by working forward: the increase is 20% of $5.50, which is $0.2 \times 5.50 = 1.10$ , so $1.10, and $5.50 + 1.10 = 6.60$ . This is the price we were told.

The price before the increase was $5.50.

Rule: the original value after an increase

After an increase, the new value is (100% + the percentage increase) of the original value. After a 20% increase it is 120% of the original; after a 35% increase it is 135% of the original.

$\text{Original value} = \text{new value} \div (100\% + \text{percentage increase})$

So after a 20% increase, divide the new value by 120%, which is 1.2.

Example 4: A value that is more than 100% of the original

Amar is saving for a badminton racket that costs $80. The racket costs 125% of the amount he has saved so far. How much has he saved so far?

Solution

The amount he has saved is the base, 100%, and the $80 is 125% of it. Both 100 and 125 divide by 25, so take 25% as the unit: the savings are 4 units and the price of the racket is 5 units.

125% of the savings (5 units) → $80

25% of the savings (1 unit) → $16, since $80 \div 5 = 16$

100% of the savings (4 units) → $64, since $16 \times 4 = 64$

Dividing by the multiplier: 125% is 1.25, and $80 \div 1.25 = 64$ .

Check by working forward: 125% of $64 is $1.25 \times 64 = 80$ , so $80, the price of the racket.

Amar has saved $64 so far.

The Original Value after a Decrease

After a decrease, the new value is less than the original. The decrease is a percentage of the original, so after a 30% decrease the new value is $100\% - 30\% = 70\%$ of the original: 7 units of a bar of 10.

Example 5: Rainfall below the average

The rainfall in March was 30% below the average March rainfall. 175 mm of rain fell in March. Find the average March rainfall.

Solution

The average is the base, 100%. March was 30% below it, so the March rainfall is $100\% - 30\% = 70\%$ of the average: 7 units of a bar of 10.

70% of the average → 175 mm

10% of the average → 25 mm, since $175 \div 7 = 25$

100% of the average → 250 mm, since $25 \times 10 = 250$

Dividing by the multiplier: 70% is 0.7, and $175 \div 0.7 = 250$ .

Check by working forward: 30% of 250 mm is $0.3 \times 250 = 75$ , so 75 mm, and $250 - 75 = 175$ . This is the March rainfall.

The average March rainfall is 250 mm.

Rule: the original value after a decrease

After a decrease, the new value is (100% − the percentage decrease) of the original value. After a 30% decrease it is 70% of the original; after a 15% discount the price paid is 85% of the price before the discount.

$\text{Original value} = \text{new value} \div (100\% - \text{percentage decrease})$

So after a 30% decrease, divide the new value by 70%, which is 0.7.

Example 6: A price before a discount

In a sale at a mall, a pair of running shoes is sold at a 15% discount. Mei Ling pays $102 for them. What was the price before the discount?

Solution

The price before the discount is 100%. The discount is 15% of it, so Mei Ling pays $100\% - 15\% = 85\%$ of the price before the discount.

85% of the price before the discount → $102

1% → $1.20, since $102 \div 85 = 1.2$

100% → $120, since $1.2 \times 100 = 120$

With a bar model, 5% is a good unit, because 85 and 100 both divide by 5: the sale price is 17 units and the price before the discount is 20 units. 17 units → $102, so 1 unit → $6, since $102 \div 17 = 6$ , and 20 units → $120, since $6 \times 20 = 120$ .

Dividing by the multiplier: 85% is 0.85, and $102 \div 0.85 = 120$ .

Check by working forward: the discount is 15% of $120, which is $0.15 \times 120 = 18$ , so $18, and $120 - 18 = 102$ . This is what Mei Ling pays.

The price before the discount was $120.

Why the Percentage Must Not Be Taken of the New Value

The commonest mistake in reverse percentages is to take the percentage of the value you know, and to add it or take it away. That fails every time, because the percentage was a percentage of the original value, and the original value is the one that is not known yet. Here is one question worked both ways.

Example 7: The same question worked both ways

After a 20% increase, the price of a bicycle helmet is $120. What was the price before the increase?

Solution

The wrong way: "take 20% off $120". 20% of $120 is $0.2 \times 120 = 24$ , so $24, and $120 - 24 = 96$ . This gives $96.

Check $96 by working forward: a 20% increase on $96 gives $96 \times 1.2 = 115.2$ , so $115.20, not $120. So $96 is wrong.

The right way. The old price is 100% and the new price is 120% of it: 12 units of 10%, against 10 units for the old price.

120% (12 units) → $120, so 10% (1 unit) → $10, since $120 \div 12 = 10$ , and 100% (10 units) → $100. Dividing by the multiplier gives the same: $120 \div 1.2 = 100$ .

Check by working forward: 20% of $100 is $20, and $100 + 20 = 120$ . This is the price we were told.

Why the wrong way fails. The increase was 20% of the old price, which is 2 units of the bar. "20% of $120" is 20% of all 12 units, which is 2.4 units, so taking it away removes more than was added. The base of the 20% is the old price, not the new one.

The price before the increase was $100, not $96. Taking 20% off $120 does not undo a 20% rise to $120.

The same mistake after a decrease

After a 30% discount a bag costs $238. Adding 30% of $238 back gives $238 \times 1.3 = 309.4$ , so $309.40. Working forward, a 30% discount on $309.40 gives $309.40 \times 0.7 = 216.58$ , so $216.58, not $238.

The right way: $238 is 70% of the price before the discount, so that price is $238 \div 0.7 = 340$ , which is $340. Check: 30% of $340 is $102, and $340 - 102 = 238$ .

The Price before GST

In Singapore, a business registered for Goods and Services Tax (GST) adds 9% GST to the price of most things it sells. A price with GST is the price before GST together with 9% of the price before GST, so it is 109% of the price before GST. Finding the price before GST is a reverse percentage after an increase of 9%. The lesson on GST and service charge has more of these.

Example 8: Taking GST off a price

A pair of earphones costs $65.40, including 9% GST. Find the price before GST.

Solution

The price before GST is 100%. The GST is 9% of it, so the price with GST is $100\% + 9\% = 109\%$ of the price before GST.

109% of the price before GST → $65.40

1% → $0.60, since $65.40 \div 109 = 0.6$

100% → $60, since $0.6 \times 100 = 60$

Dividing by the multiplier: 109% is 1.09, and $65.40 \div 1.09 = 60$ .

Check by working forward: the GST is 9% of $60, which is $0.09 \times 60 = 5.4$ , so $5.40, and $60 + 5.40 = 65.40$ . This is the price we were told.

Taking 9% of $65.40 away from $65.40 would give $65.40 - 5.886 = 59.514$ , about $59.51, which is wrong: the GST was 9% of the price before GST, not 9% of $65.40.

The price before GST is $60.

Undoing Two Successive Changes

Recall from the lesson on successive percentage changes: when one percentage change follows another, each change is on the value it starts from, and the value is multiplied by each multiplier in turn.

To find the original value, undo each change by dividing by its multiplier. The changes can be undone one at a time, in either order, or all at once by dividing by the product of the multipliers. All three ways give the same original value.

Example 9: A price after a rise and then a discount

The price of a school bag went up by 20%. Later, in a sale, the new price was reduced by 15%. The bag now costs $183.60. What was its price before the two changes?

Solution

Working forward, the original price was multiplied by 1.2 (a 20% rise, to 120%) and then by 0.85 (a 15% fall, to 85% of the higher price). To go back, divide by the same multipliers.

One change at a time, last change first. Undo the 15% fall: the price before the sale was $183.60 \div 0.85 = 216$ dollars, so $216.

Undo the 20% rise: the original price was $216 \div 1.2 = 180$ dollars, so $180.

One change at a time, in the other order. $183.60 \div 1.2 = 153$ and $153 \div 0.85 = 180$ . The original price is $180 again, because dividing by 1.2 and by 0.85 can be done in either order. Only the value in between differs: $216 was the price of the bag after the rise, but $153 is not a price the bag ever had. So undo the last change first when you also want the value between the changes.

All at once. The two changes together multiply the price by $1.2 \times 0.85 = 1.02$ , so the price now is 102% of the original price. The original price is $183.60 \div 1.02 = 180$ dollars, so $180.

Check by working forward: $180 \times 1.2 = 216$ and $216 \times 0.85 = 183.6$ , which is $183.60. This is the price now.

The price of the bag before the two changes was $180.

Rule: the original value before successive changes

Divide the value you know by the multiplier of each change, one at a time, in either order; or divide it once by the product of all the multipliers.

Undoing the last change first also gives the value between the changes.

Do not add or take away the percentages: a rise of 20% and then a fall of 15% is a rise of 2%, not 5%, because the 15% is of the higher price.

Putting the Cases Together

Every example in this lesson came down to the same question: the value you know is what percentage of the original? Here are the answers side by side.

What you are told	The value you know, as a percentage of the original	Divide by
54 students are 30% of the cohort	30%	0.3
after a 20% rise, a price is $6.60	120%	1.2
a racket costs 125% of the savings	125%	1.25
March rainfall, 30% below the average, is 175 mm	70%	0.7
after a 15% discount, a price is $102	85%	0.85
with 9% GST, a price is $65.40	109%	1.09
after a 20% rise and then a 15% fall, a price is $183.60	102%, since $1.2 \times 0.85 = 1.02$	1.02

A multiplier greater than 1 means that the value you know is more than the original (it went up, or, like the racket, it is more than the base), so the original value is smaller than the value you know. A multiplier less than 1 means that the value went down (or that the value you know is a part), so the original value is larger. Use this as a quick test of every answer, before the full check by working forward.

Common Mistakes

Taking the percentage of the value you know. After a 20% rise to $120, the price before was not $120 - 24 = 96$ dollars. The 20% was of the old price, so divide: $120 \div 1.2 = 100$ , which is $100.
Taking GST off the price with GST. If a price with 9% GST is $109, the price before GST is not $109 - 9.81 = 99.19$ dollars. It is $109 \div 1.09 = 100$ , which is $100.
Dividing by the percentage of the change. After a 30% discount, $238 is 70% of the price before the discount, not 30% of it. $238 \div 0.3 \approx 793.33$ is far too big; $238 \div 0.7 = 340$ is right.
Adding or taking away the percentages of successive changes. A rise of 20% and then a fall of 15% is not a rise of 5%. Dividing $183.60 by 1.05 gives about $174.86, and working forward, $174.86 \times 1.2 \times 0.85 \approx 178.36$ , not 183.60. Divide by $1.2 \times 0.85 = 1.02$ instead.
Thinking the changes must be undone in a fixed order. They can be undone in either order and the original value is the same. Only the value in between depends on the order.
Leaving out the check. Work forward from your answer. After an increase the original value must be smaller than the value you know, and after a decrease it must be larger.

Practice

Work each question on paper before you open its solution. Decide first what percentage of the original value the value you know is, and check every answer by working forward.

Question 1

40% of a number is 56. Find the number.

ShowHide solution

40% of the number → 56

10% of the number → 14, since $56 \div 4 = 14$

100% of the number → 140, since $14 \times 10 = 140$

Dividing by the multiplier: $56 \div 0.4 = 140$ .

Check: $0.4 \times 140 = 56$ .

The number is 140.

Question 2

A school library has 2400 fiction books, and these are 60% of all its books.
(a) How many books does the library have?
(b) How many of its books are not fiction?

ShowHide solution

(a) All the books are 100%, and the fiction books are 60% of them.

60% of the books → 2400

10% of the books → 400, since $2400 \div 6 = 400$

100% of the books → 4000, since $400 \times 10 = 4000$

Dividing by the multiplier: $2400 \div 0.6 = 4000$ .

Check: $0.6 \times 4000 = 2400$ .

(b) $4000 - 2400 = 1600$ . Or: the books that are not fiction are $100\% - 60\% = 40\%$ of all the books, and 40% → $4 \times 400 = 1600$ .

(a) The library has 4000 books.
(b) 1600 of its books are not fiction.

Question 3

Mrs Goh's monthly salary went up by 8%. Her new salary is $3888 a month. What was her monthly salary before the increase?

ShowHide solution

The old salary is 100%. The increase is 8% of it, so the new salary is $100\% + 8\% = 108\%$ of the old salary.

108% of the old salary → $3888

1% → $36, since $3888 \div 108 = 36$

100% → $3600, since $36 \times 100 = 3600$

Dividing by the multiplier: $3888 \div 1.08 = 3600$ .

Check: 8% of $3600 is $0.08 \times 3600 = 288$ , so $288, and $3600 + 288 = 3888$ .

Her monthly salary before the increase was $3600.

Question 4

In a sale, a watch is sold at 25% off. Ravi pays $96 for it. What was the price before the discount?

ShowHide solution

The price before the discount is 100%. Ravi pays $100\% - 25\% = 75\%$ of it. Take 25% as the unit: the price before the discount is 4 units and the sale price is 3 units.

75% (3 units) → $96

25% (1 unit) → $32, since $96 \div 3 = 32$

100% (4 units) → $128, since $32 \times 4 = 128$

Dividing by the multiplier: $96 \div 0.75 = 128$ .

Check: the discount is 25% of $128, which is $32, and $128 - 32 = 96$ .

The price before the discount was $128.

Question 5

Siti's score in the final round of a quiz was 40% higher than her score in the first round. She scored 420 points in the final round. How many points did she score in the first round?

ShowHide solution

The first-round score is the base, 100%. The final-round score is $100\% + 40\% = 140\%$ of it.

140% of the first-round score → 420 points

10% → 30 points, since $420 \div 14 = 30$

100% → 300 points, since $30 \times 10 = 300$

Dividing by the multiplier: $420 \div 1.4 = 300$ .

Check: 40% of 300 is $0.4 \times 300 = 120$ , and $300 + 120 = 420$ .

Siti scored 300 points in the first round.

Question 6

A 10 kg bag of rice costs $25, including 9% GST. Find the price before GST, correct to the nearest cent.

ShowHide solution

The price with GST is $100\% + 9\% = 109\%$ of the price before GST.

Price before GST $= 25 \div 1.09 = 22.935\ldots$ dollars

Rounded to the nearest cent, this is $22.94.

By the unitary method: 1% → $25 \div 109 = 0.2293\ldots$ and 100% → $0.2293\ldots \times 100 = 22.935\ldots$ , the same. Keep all the digits until the end and round only once.

Check: $22.94 \times 1.09 = 25.0046$ , which is $25.00 to the nearest cent. The check is not exact because $22.94 is a rounded value.

The price before GST is $22.94, correct to the nearest cent.

Question 7

The price of a T-shirt went up by 25% to $30. Ali says that the price before the increase was $30 − 25% of $30, which is $22.50.
(a) Check Ali's answer by working forward.
(b) Find the correct price before the increase.

ShowHide solution

(a) A 25% increase on $22.50 gives $22.50 \times 1.25 = 28.125$ dollars, which is not $30. So Ali is wrong. He took 25% of the new price, but the 25% was of the old price.

(b) The new price is $100\% + 25\% = 125\%$ of the old price. With 25% as the unit, the new price is 5 units and the old price is 4 units.

5 units → $30, so 1 unit → $6, since $30 \div 5 = 6$ , and 4 units → $24, since $6 \times 4 = 24$ .

Dividing by the multiplier: $30 \div 1.25 = 24$ .

Check: 25% of $24 is $6, and $24 + 6 = 30$ .

(a) A 25% increase on $22.50 does not give $30, so Ali is wrong.
(b) The price before the increase was $24.

Question 8

After two months of training, Ravi's time for the 2.4 km run fell by 12%. His new time is 11 minutes. What was his time before the training? Give your answer in minutes and seconds.

ShowHide solution

Write the new time in seconds: 11 min $= 11 \times 60 = 660$ s.

The old time is 100%. The new time is $100\% - 12\% = 88\%$ of it.

88% of the old time → 660 s

1% → 7.5 s, since $660 \div 88 = 7.5$

100% → 750 s, since $7.5 \times 100 = 750$

Dividing by the multiplier: $660 \div 0.88 = 750$ .

750 s = 12 min 30 s, since $750 = 12 \times 60 + 30$ .

Check: 12% of 750 s is $0.12 \times 750 = 90$ s, and $750 - 90 = 660$ s, which is 11 minutes.

His time before the training was 12 minutes 30 seconds.

Question 9

In a school, 45% of the Secondary 1 students are boys and the rest are girls. There are 132 girls.
(a) How many Secondary 1 students are there?
(b) How many of them are boys?

ShowHide solution

(a) The 132 girls are not 45% of the students: 45% are the boys. The girls are $100\% - 45\% = 55\%$ of the students.

Both 55 and 100 divide by 5, so take 5% as the unit: the girls are 11 units and all the students are 20 units.

55% (11 units) → 132 students, so 5% (1 unit) → 12 students, since $132 \div 11 = 12$

100% (20 units) → 240 students, since $12 \times 20 = 240$

Dividing by the multiplier: $132 \div 0.55 = 240$ .

(b) Boys: $240 - 132 = 108$ . Or: the boys are 9 units, and $9 \times 12 = 108$ .

Check: 45% of 240 is $0.45 \times 240 = 108$ , and $108 + 132 = 240$ .

(a) There are 240 Secondary 1 students.
(b) 108 of them are boys.

Question 10

The number of members of a school's robotics club rose by 25% from 2024 to 2025, and then by 20% from 2025 to 2026. In 2026 the club has 60 members.
(a) How many members did it have in 2025?
(b) How many members did it have in 2024?

ShowHide solution

(a) Undo the last change first. The 2026 number is 120% of the 2025 number, so the 2025 number is $60 \div 1.2 = 50$ .

(b) The 2025 number is 125% of the 2024 number, so the 2024 number is $50 \div 1.25 = 40$ .

All at once: the two changes multiply the number by $1.25 \times 1.2 = 1.5$ , and $60 \div 1.5 = 40$ , the same.

Check by working forward: $40 \times 1.25 = 50$ and $50 \times 1.2 = 60$ .

(a) The club had 50 members in 2025.
(b) It had 40 members in 2024.

Question 11

A mobile phone loses 20% of its value in its first year. In its second year it loses 10% of the value it had at the start of that year. After two years it is worth $504.
(a) What was it worth when it was new?
(b) What was it worth after one year?

ShowHide solution

(a) The value is multiplied by 0.8 in the first year and by 0.9 in the second year. Together the two changes multiply it by $0.8 \times 0.9 = 0.72$ , so $504 is 72% of the value when new.

72% → $504, so 1% → $7, since $504 \div 72 = 7$ , and 100% → $700.

Dividing by the multiplier: $504 \div 0.72 = 700$ .

(b) Undo only the second year: $504 \div 0.9 = 560$ , so $560. Or work forward from (a): $700 \times 0.8 = 560$ .

Check by working forward: $700 \times 0.8 = 560$ and $560 \times 0.9 = 504$ .

(a) The phone was worth $700 when it was new.
(b) It was worth $560 after one year.

Question 12

A shop raises the price of a fan by 10%. A month later it gives 10% off the new price, and Hui Min pays $49.50 for the fan.
(a) Find the price of the fan before the two changes.
(b) Hui Min thinks the price before the changes was $49.50, because "10% up and 10% down cancel out". Explain why she is wrong.

ShowHide solution

(a) The two changes multiply the price by $1.1 \times 0.9 = 0.99$ , so $49.50 is 99% of the original price.

99% → $49.50, so 1% → $0.50, since $49.50 \div 99 = 0.5$ , and 100% → $50.

One change at a time, last first: $49.50 \div 0.9 = 55$ , so the price after the rise was $55; and $55 \div 1.1 = 50$ , so the original price was $50.

Check by working forward: $50 \times 1.1 = 55$ and $55 \times 0.9 = 49.5$ .

(b) The two 10%s have different bases. The rise is 10% of $50, which is $5. The discount is 10% of the higher price, $55, which is $5.50. The discount is 50 cents more than the rise, so the price ends 50 cents below where it started: $49.50 is 99% of the original price, not 100% of it.

(a) The price before the two changes was $50.
(b) The 10% discount is taken of the higher price, $55, so it is $5.50, more than the $5 rise. The two changes do not cancel out.

Key Points

In a reverse percentage problem the original value is not known. The original value is the base, 100%.
First ask: the value I know is what percentage of the original? A part is its own percentage; after a 20% increase the new value is 120% of the original; after a 15% decrease it is 85%; a price with 9% GST is 109% of the price before GST.
Unitary method: from the percentage you know, find 1% (or 10%, or another percentage that divides evenly), then 100%.
Multiplier: original value = known value ÷ multiplier, such as ÷ 1.2 after a 20% increase and ÷ 0.85 after a 15% decrease. Dividing undoes the multiplying.
Never take the percentage of the value you know: the percentage was of the original value.
After successive changes, divide by each multiplier in turn, in either order, or once by their product. Undoing the last change first also gives the values in between.
Check every answer by working forward. After an increase the original value is smaller than the value you know; after a decrease it is larger.
Give money to the nearest cent and other values that are not exact to 3 significant figures, rounding only at the end.