Percentages on the Digital SAT: Percent Change, Growth & Word Problems
Master SAT percentage questions: percent of a number, percent increase/decrease, reverse percentages, multipliers, repeated change, traps, and original practice.
Percentages are an official skill in the SAT Problem-Solving and Data Analysis domain.
These questions are often more about translation than arithmetic. The most important habit is knowing what quantity is the original/base.
Three percentage structures
1. Percent of a number
“What is 18% of 250?”
[ 0.18(250)=45 ]
2. What percent is one number of another?
“45 is what percent of 250?”
[ \frac{45}{250}\times100=18% ]
3. Percent change
A price rises from 250 to 295.
Change:
[ 295-250=45 ]
Divide by original:
[ \frac{45}{250}=0.18 ]
Increase = 18%.
The denominator is the original value
This causes many SAT errors.
If a population grows from 800 to 920:
[ \frac{120}{800}=0.15 ]
15% increase.
Do not divide by 920.
Use multipliers
A percentage increase/decrease is often fastest as a multiplier.
Increase by 12%
Multiply by:
[ 1.12 ]
Decrease by 12%
Multiply by:
[ 0.88 ]
Example 1
A jacket costs $80 and increases by 15%.
[ 80(1.15)=92 ]
New price = $92.
Reverse percentages
Example 2
After a 20% discount, a device costs $240. What was the original price?
After 20% off, 80% remains:
[ 0.80P=240 ]
[ P=300 ]
Do not calculate 20% of 240 and add it. The 20% was based on the original price, not the discounted price.
Repeated percentage change
A 10% increase followed by a 10% decrease does not return to the original.
Start with 100:
Increase 10%:
[ 100(1.10)=110 ]
Then decrease 10%:
[ 110(0.90)=99 ]
Net change = 1% decrease.
Why? The second 10% uses a different base.
General repeated-growth formula
If a quantity grows by rate (r) each period:
[ A=P(1+r)^t ]
For decline:
[ A=P(1-r)^t ]
Rates must be written as decimals. This is the same compounding formula used in exponential growth and decay problems.
Percentage points vs percent
Suppose a rate goes from 40% to 50%.
Difference: 10 percentage points.
Percent increase relative to original:
[ \frac{50-40}{40}=0.25 ]
= 25% increase.
Those are not the same.
Part-to-whole questions
Example 3
In a survey, 84 of 240 students choose option A.
Percentage:
[ \frac{84}{240}\times100=35% ]
If the test asks for the fraction or ratio, do not automatically convert to a percentage.
When a question asks a percentage to stay within a range rather than hit an exact value, that's really a linear inequalities problem wearing a percentage disguise.
Tax, tips, markup, discount
Use multipliers.
8% tax
[ price(1.08) ]
18% tip
[ bill(1.18) ]
25% discount
[ price(0.75) ]
40% markup
[ cost(1.40) ]
Successive discounts
30% off, then another 20% off:
[ 0.70\times0.80=0.56 ]
Final price = 56% of original.
Total discount = 44%, not 50%.
Mixture with percentages
Example 4
A 500 mL solution is 12% salt by volume. How much salt is present?
[ 0.12(500)=60 ]
60 mL.
If 100 mL of pure water is added, the amount of salt stays 60 mL but total volume changes to 600 mL.
New concentration:
[ \frac{60}{600}=10% ]
Percent equations with variables
Example 5
(x) is 30% greater than (y).
That means:
[ x=1.30y ]
Not (x=0.30y).
If (x) is 30% of (y):
[ x=0.30y ]
Tiny wording difference, completely different equation. This kind of equation is just a specific case of the linear equations skill applied to percentages.
“A is p% more than B”
[ A=\left(1+\frac p{100}\right)B ]
“A is p% less than B”
[ A=\left(1-\frac p{100}\right)B ]
Can Desmos help?
Yes, but most percentage questions are faster without graphing.
Desmos helps if:
- the percentage relation is embedded in a more complex equation;
- repeated growth is involved;
- you want to solve an exponential equation numerically.
For direct percent change, mental/algebraic calculation is usually faster.
Common traps
Trap 1: wrong base
Percent change uses original value.
Trap 2: percent vs percentage points
40% → 50% = +10 points, +25%.
Trap 3: adding repeated percentages
Use multiplication.
Trap 4: “30% greater” interpreted as 30%
130%, not 30%.
Trap 5: reverse discount
Divide by remaining multiplier.
Original mini-practice
1
A price rises from $160 to $184. What is the percent increase?
2
After a 25% discount, a coat costs $90. What was the original price?
3
A population increases by 8% in one year and 8% the next year. What multiplier relates the final population to the original?
A. 1.16
B. 1.1664
C. 1.64
D. 0.84
4
A rate rises from 24% to 30%. By what percent did the rate itself increase?
5
Quantity (A) is 40% less than quantity (B). If (B=250), what is (A)?
Answers
-
15%
Change = 24. (24/160=0.15). -
$120
(0.75P=90). -
B. 1.1664
(1.08^2=1.1664). -
25%
Increase = 6 points; (6/24=0.25). -
150
(250(0.60)=150).
What to learn next
- ratios and rates;
- exponential growth;
- probability;
- unit conversions.
See the rest of our SAT Math guides.
Official references
- College Board, Problem-Solving and Data Analysis: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving
Trademark note: SAT® is a registered trademark of College Board. ACT® is a registered trademark of ACT Education Corp. MastaPrep is not affiliated with, endorsed by, or sponsored by College Board or ACT Education Corp.
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