SAT Reverse Percentage Problems: The Trick That Makes Them Easy

 

SAT Reverse Percentage Problems: The Hidden “Original Amount” Trick

Percentage questions on the SAT often look easier than they really are.

A question may give you a final price, final population, final salary, or final quantity and ask you to determine what the number was before a percentage change.

That is where many students make the same mistake:

They see a percentage and immediately multiply.

The better question to ask is:

“What percentage of the original amount does the final amount represent?”

Once you identify that percentage, the problem usually becomes a simple equation.

This guide focuses entirely on that idea.

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1. The Secret Behind Reverse Percentage Questions

Suppose an amount starts at x.

If it increases by 30%, the new amount is:

x + 0.30x

Therefore:

1.30x

So a 30% increase means the final amount is 130% of the original.

Now suppose the amount decreases by 30%.

You lose 30%, so 70% remains:

x − 0.30x = 0.70x

Therefore:

A 30% decrease leaves 70% of the original amount.

This gives us the basic pattern:

Increase by r% → multiply by 1 + r/100

Decrease by r% → multiply by 1 − r/100

When working backward, divide instead of multiply.

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2. The Reverse Percentage Rule

If the final amount is known:

After an increase:

Original = Final ÷ (1 + r/100)

After a decrease:

Original = Final ÷ (1 − r/100)

You do not necessarily need to memorize these formulas.

Instead, remember:

Find the percentage that the final amount represents, then divide by that percentage written as a decimal.

That idea is often easier to remember under SAT time pressure.

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3. The 100% Method

One of the easiest ways to understand reverse percentage problems is to think in terms of 100%.

Suppose a price is reduced by 25%.

The original price is:

100%

The discount is:

25%

The amount left is:

75%

So if the question tells you that the final price is $90, you know:

75% = $90

You want:

100% = ?

Therefore:

Original = 90 ÷ 0.75

Original = 120

The original price was:

$120

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4. SAT Example: The Discount That Hides the Original Price

A bookstore reduces the price of a calculator by 20%. The discounted price is $56. What was the price before the discount?

Step 1: Identify what remains

A 20% discount leaves:

100% − 20% = 80%

Step 2: Write the equation

80% of original = 56

Therefore:

0.80x = 56

Step 3: Solve

x = 56 ÷ 0.80

x = 70

Answer:

$70

Quick check

20% of $70 is:

$14

Subtract:

$70 − $14 = $56

Correct.

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5. Why Adding the Discount Back Does Not Work

Suppose a product costs $72 after a 20% discount.

It is tempting to calculate:

20% of $72 = $14.40

and then:

$72 + $14.40 = $86.40

But $86.40 is not the original price.

Why?

Because the original 20% discount was calculated using the original price, not the discounted price.

The correct calculation is:

72 ÷ 0.80 = 90

So the original price was:

$90

Check:

20% of $90 = $18

$90 − $18 = $72

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6. Reverse Percentage Increase

Reverse percentage questions can also work with increases.

Suppose a school's enrollment increases by 25% and becomes 1,500 students.

What was the enrollment before the increase?

A 25% increase means the final enrollment represents:

125% of the original

Therefore:

1.25x = 1,500

Divide:

x = 1,500 ÷ 1.25

x = 1,200

Answer:

1,200 students

Check:

25% of 1,200 is:

300

Then:

1,200 + 300 = 1,500

Correct.

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7. A Useful Translation Trick

SAT questions often use words instead of directly saying “multiply by 1.25.”

Learn to translate them.

“Increased by 15%”

means:

115% of the original

or:

1.15 × original

“Decreased by 15%”

means:

85% of the original

or:

0.85 × original

“Increased by 40%”

means:

140% of the original

or:

1.40 × original

“Decreased by 40%”

means:

60% of the original

or:

0.60 × original

This translation step is one of the most useful habits you can develop for SAT percentage problems.

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8. A Fast Way to Spot the Direction

Ask:

Is the final amount larger or smaller than the original?

If the amount increased, the final amount must represent more than 100%.

If the amount decreased, the final amount must represent less than 100%.

For example:

35% increase → 135%

35% decrease → 65%

This simple observation can prevent many calculator mistakes.

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9. SAT Example: A Salary Increase

A worker receives a 12% increase in annual salary. After the increase, the salary is $67,200. What was the salary before the increase?

A 12% increase means:

112% of the original = $67,200

So:

1.12x = 67,200

Therefore:

x = 67,200 ÷ 1.12

x = 60,000

Answer:

$60,000

Check:

12% of $60,000 is:

$7,200

Therefore:

$60,000 + $7,200 = $67,200

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10. SAT Example: Population Decrease

A town's population decreases by 16%. After the decrease, the population is 42,000. What was the population before the decrease?

A 16% decrease leaves:

84%

Therefore:

0.84x = 42,000

So:

x = 42,000 ÷ 0.84

x = 50,000

Answer:

50,000

The population decreased by:

50,000 − 42,000 = 8,000

And:

8,000 ÷ 50,000 = 0.16

So the decrease really was 16%.

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11. The “Final Is Not the Base” Rule

This is perhaps the most important idea in this entire guide.

When a question says:

“After a 20% decrease, the value is 240.”

The 20% was calculated using the original value.

It was not calculated using 240.

So do not automatically calculate:

20% of 240

Instead, write:

80% of original = 240

Then:

Original = 240 ÷ 0.80

Original = 300

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12. Reverse Percentage With Tax

A computer is sold for $1,080 after a 20% tax is added. What was the price before tax?

The original price represents:

100%

The tax adds:

20%

Therefore, the final price represents:

120%

So:

1.20x = 1,080

Therefore:

x = 1,080 ÷ 1.20

x = 900

Answer:

$900

Check:

20% of $900 = $180

$900 + $180 = $1,080

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13. Reverse Percentage With a Commission

A salesperson receives a commission equal to 10% of sales. If the commission is $450, what was the total sales amount?

Here the $450 represents 10% of the sales.

Therefore:

0.10x = 450

So:

x = 450 ÷ 0.10

x = 4,500

Answer:

$4,500

This is technically a reverse percentage problem even though the question may not use the words “increase” or “decrease.”

That is why identifying the relationship is more important than memorizing a particular question format.

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14. The Difference Between “Percent” and “Percentage Points”

This distinction can appear in more advanced questions.

Suppose an interest rate changes from:

5% to 7%

The increase is:

7% − 5% = 2 percentage points

But the percent increase relative to the original 5% rate is:

2 ÷ 5 = 0.40

So the rate increased by:

40%

Therefore:

2 percentage points ≠ 2% increase

This distinction is worth remembering.

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15. The Famous 20% Increase and 20% Decrease Trap

Suppose a quantity starts at:

100

Increase it by 20%:

100 × 1.20 = 120

Now decrease the result by 20%:

120 × 0.80 = 96

The final value is:

96

So the overall change is:

4% decrease

Not zero.

Why?

Because the first 20% was based on 100.

The second 20% was based on 120.

The base changed.

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16. Percentage Changes Are Multipliers

A useful way to think about percentage changes is through multipliers.

Increase by 10%

× 1.10

Increase by 20%

× 1.20

Increase by 50%

× 1.50

Decrease by 10%

× 0.90

Decrease by 20%

× 0.80

Decrease by 50%

× 0.50

This makes multiple-change questions much easier.

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17. A Two-Step Reverse Problem

A quantity is increased by 20% and then decreased by 10%. The final value is 540. What was the original value?

Let the original value be x.

First:

x × 1.20

Then:

× 0.90

Therefore:

1.20 × 0.90 × x = 540

Calculate the combined multiplier:

1.20 × 0.90 = 1.08

So:

1.08x = 540

Therefore:

x = 540 ÷ 1.08

x = 500

Answer:

500

Check:

20% increase:

500 × 1.20 = 600

10% decrease:

600 × 0.90 = 540

Correct.

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18. Working Backward Through Multiple Changes

Suppose the final value is known and several percentage changes occurred.

The safest strategy is to work backward one change at a time.

Suppose:

A price was increased by 25% and then decreased by 20%. The final price was $300.

Start with the final price:

$300

Undo the 20% decrease:

300 ÷ 0.80 = 375

Now undo the 25% increase:

375 ÷ 1.25 = 300

Therefore:

Original price = $300

This example has an interesting result: the two changes cancel.

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19. When Percentage Changes Cancel

A 25% increase corresponds to:

× 1.25

A 20% decrease corresponds to:

× 0.80

Multiply:

1.25 × 0.80 = 1

So the overall multiplier is 1.

Therefore, the final value equals the original value.

This is a useful pattern to recognize quickly.

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20. The Algebra Method

If you prefer equations, let the original amount be x.

Then translate the percentage change.

For a 35% increase:

1.35x

For a 35% decrease:

0.65x

For example:

A number is decreased by 35% and becomes 260. Find the original number.

Write:

0.65x = 260

Then:

x = 260 ÷ 0.65

x = 400

Answer:

400

This method is reliable because it forces you to identify exactly what the percentage applies to.

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21. Reverse Percentage Questions With Fractions

Some percentage values are particularly friendly to mental math.

25%

25% = 1/4

Therefore:

75% = 3/4

If 75% of a number is 150:

3/4 × x = 150

Therefore:

x = 200

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50%

50% = 1/2

If 50% of a number is 180:

x = 360

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20%

20% = 1/5

If 20% of a number is 80:

x = 400

Recognizing familiar fractions can save valuable time.

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22. Practice Question 1

After a 30% discount, a pair of headphones costs $63. What was the original price?

A) $81
B) $84
C) $90
D) $93

Solution

A 30% discount leaves:

70%

Therefore:

0.70x = 63

x = 63 ÷ 0.70

x = 90

Answer:

C) $90

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23. Practice Question 2

A quantity is increased by 15% and becomes 460. What was the original quantity?

A) 391
B) 400
C) 410
D) 425

Solution

A 15% increase means:

115% = 460

Therefore:

1.15x = 460

x = 460 ÷ 1.15

x = 400

Answer:

B) 400

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24. Practice Question 3

After a 40% decrease, the number of visitors to a museum is 7,200. How many visitors were there before the decrease?

A) 10,800
B) 11,200
C) 12,000
D) 12,400

Solution

A 40% decrease leaves:

60%

Therefore:

0.60x = 7,200

x = 7,200 ÷ 0.60

x = 12,000

Answer:

C) 12,000

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25. Practice Question 4

A price is increased by 20% and then decreased by 25%. The final price is $360. What was the original price?

A) $360
B) $375
C) $400
D) $450

Solution

First increase:

× 1.20

Then decrease:

× 0.75

Combined:

1.20 × 0.75 = 0.90

Therefore:

0.90x = 360

So:

x = 360 ÷ 0.90

x = 400

Answer:

C) $400

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26. Practice Question 5: The Trickier One

A company's number of customers increases by 25% during one month and then decreases by 20% during the next month. At the end of the second month, there are 3,000 customers. How many customers were there originally?

Let the original number be x.

First change:

1.25x

Second change:

0.80(1.25x)

Therefore:

1.25 × 0.80 × x = 3,000

Since:

1.25 × 0.80 = 1

we get:

x = 3,000

Answer:

3,000 customers

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27. The Five-Second SAT Test

When you see a reverse percentage question, mentally ask:

① What is the original?

Call it x.

② What happened?

Increase or decrease?

③ What percentage remains?

For example:

20% decrease → 80% remains

④ Turn it into a multiplier.

80% → 0.80

⑤ Solve.

Final ÷ 0.80

That is the entire process.

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28. Reverse Percentage Cheat Sheet

Percentage Change	Final Represents	To Find Original
10% increase	110%	Final ÷ 1.10
15% increase	115%	Final ÷ 1.15
20% increase	120%	Final ÷ 1.20
25% increase	125%	Final ÷ 1.25
30% increase	130%	Final ÷ 1.30
10% decrease	90%	Final ÷ 0.90
15% decrease	85%	Final ÷ 0.85
20% decrease	80%	Final ÷ 0.80
25% decrease	75%	Final ÷ 0.75
30% decrease	70%	Final ÷ 0.70
40% decrease	60%	Final ÷ 0.60
50% decrease	50%	Final ÷ 0.50

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29. The Biggest SAT Lesson

Do not ask:

“What is the percentage of the final number?”

Ask:

“What percentage of the original number is the final number?”

That tiny change in thinking can completely change how you approach the question.

For a 20% decrease:

Final = 80% of Original

For a 20% increase:

Final = 120% of Original

For a 35% decrease:

Final = 65% of Original

For a 35% increase:

Final = 135% of Original

Once you see that relationship, the calculation becomes straightforward.

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30. Final SAT Strategy

Whenever the SAT asks you to find an amount before a percentage change:

Step 1: Find the percentage represented by the final amount.

Step 2: Convert it to a decimal.

Step 3: Divide the final amount by that decimal.

Remember:

Increase → final is MORE than 100%

Decrease → final is LESS than 100%

And the most important rule:

Never calculate the percentage from the final amount unless the question specifically tells you to do so.

The percentage change normally uses the original amount as its base.

One-line memory trick:

“Find what remains, turn it into a multiplier, then divide.”

That is the reverse percentage technique you want to have ready when the SAT gives you an original-amount problem.