Percent problems become easier when you stop treating each wording pattern as a new trick. Translate the percentage into a multiplier, identify whether the question wants the change or the final value, and check the direction.
The multiplier language
| Phrase | Multiplier | Equation |
|---|---|---|
| Increase by 20% | 1+0.20=1.20 | new=1.20(original) |
| Decrease by 20% | 1-0.20=0.80 | new=0.80(original) |
| Is 20% of | 0.20 | part=0.20(whole) |
| Is 20% more than | 1.20 | larger=1.20(smaller) |
| Is 20% less than | 0.80 | smaller=0.80(larger) |
Change amount versus final value
A $75 guide is discounted 20%.
The discount amount is 75(0.20)=15. The final price is 75(0.80)=60. Both calculations are valid; only one answers “What is the sale price?”
Underline the requested quantity before calculating:
- amount of increase or decrease;
- new total;
- original total;
- percent change.
Reverse percentage: the final value is given
After a 25% increase, a value is 100. Let the original value be x.
1.25x=100, so x=80.
The common error is subtracting 25% of 100. The increase was based on the unknown original, not on the final value. Multiplication created the final value, so division reverses it.
Find the percent change from two values
A quantity rises from 60 to 78.
- Find the change: 78-60=18.
- Divide by the original: 18/60=0.30.
- Convert to a percentage: 30% increase.
The denominator answers “change compared with what?” For percent change, that comparison is the starting value.
Successive changes multiply
A price increases 20% and then decreases 20%. The changes do not cancel.
Starting from 100:
100(1.20)(0.80)=96.
The final value is 4% below the original. The decrease acts on 120, not on the original 100.
For several changes, multiply all factors:
final=original(1+r_1)(1+r_2)×…
Use a factor below 1 for a decrease.
Percentage points are not percent change
A survey result rises from 40% to 50%.
- The increase is 10 percentage points.
- The percent increase relative to the original 40% is (50-40)/40=0.25, or 25%.
The SAT may distinguish these. Check whether the prompt asks for “percentage points” or “percent increase.”
Repeated growth and decay
A population starts at 500 and grows 8% per year. After t years:
P=500(1.08)^t.
The exponent counts how many times the multiplier is applied. After 3 years, use 500(1.08)^3, not 500(1+0.08×3). The latter models simple, not compounded, growth.
For 8% annual decay, the factor is 0.92.
Translate a word problem before touching the calculator
A store marks a $120 item up by 15% and then applies a 10% discount. What is the final price?
- Markup factor: 1.15.
- Discount factor: 0.90.
- Final price: 120(1.15)(0.90)=124.20.
The final price is above $120 because the combined multiplier 1.15(0.90)=1.035 is greater than 1.
Use answer choices as a reasonableness tool
- A 12% change should not usually create a result several times larger.
- A discount answer larger than the original signals the wrong factor.
- If reverse percentage choices are far apart, estimate before dividing.
- For successive changes, compare the combined factor with 1.
Back-solving can be useful when the original is unknown: test a choice by applying the stated multiplier and checking whether it reaches the given final value.
Common translation traps
- Using 20 instead of 0.20
- Adding the change amount when the problem asks for the final value
- Dividing by the final value instead of the original in percent change
- Assuming equal increases and decreases cancel
- Confusing “20% of” with “20% more than”
- Applying repeated growth linearly
- Ignoring whether the answer needs dollars, people, or a percentage
Mixed practice
- A value of 240 decreases by 15%. Find the new value.
- After a 12% discount, a price is $88. Find the original price.
- Enrollment rises from 320 to 400. Find the percent increase.
- A rate rises from 30% to 36%. Give both the percentage-point increase and percent increase.
- An amount grows 5% annually for two years. Write, but do not simplify, the expression for a starting value of 600.
Show answers
- 240(0.85)=204.
- 0.88x=88, so x=100.
- (400-320)/320=0.25, or 25%.
- 6 percentage points; 6/30=0.20, or 20%.
- 600(1.05)^2.
Retry these ideas later in a mixed set. A percentage method is ready for the SAT when you can select it without a heading announcing the type.
