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.
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.
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.
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
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.
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
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.
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.
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.
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
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%.
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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
50%
50% = 1/2
If 50% of a number is 180:
x = 360
20%
20% = 1/5
If 20% of a number is 80:
x = 400
Recognizing familiar fractions can save valuable time.
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
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
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
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
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
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.
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 |
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.
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.
