Multi-Step Percent Applications
Two percent changes in a row don't add — they compound. The SAT relies on students adding them.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
Apply each factor in turn. A $10\%$ increase then a $10\%$ decrease is $\times 1.10 \times 0.90$, which is not back to the start. Watch which base each percent applies to.
How to solve one, step by step
Example: $\$100$, up $10\%$, then down $10\%$.
- Up $10\%$: $100 \times 1.10 = 110$.
- Down $10\%$: $110 \times 0.90 = 99$, not $\$100$.
The mistakes that cost points
- Adding the percents. $+10\%$ then $-10\%$ is not $0\%$ — the changes compound.
- Applying both to the same base. Each percent applies to the running amount, not the original.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A price is first increased by 10% and then increased again by 10%. What is the total percent increase from the original price?
- A$21\%$
- B$19\%$
- C$20\%$
- D$100\%$
Show solution
Answer: A, $21\%$. Compound the two increases: $1.10 \times 1.10 = 1.21$. The final price is 121% of the original, so the total increase is $21\%$.
Medium
A jacket originally costs $\$80$. A store applies a 25% discount and then charges 8% sales tax on the discounted price. What is the final price?
- A$\$66.40$
- B$\$64$
- C$\$64.80$
- D$\$62.40$
Show solution
Answer: C, $\$64.80$. Discounted price: $80 \times 0.75 = \$60$. Tax on discounted price: $60 \times 1.08 = \$64.80$.
Hard
An investor's portfolio gains 30% in year 1 and loses 30% in year 2. A second investor's portfolio loses 30% in year 1 and gains 30% in year 2. Which portfolio has a higher final value, and what is the percent loss from the original for each?
- ABoth have the same final value; each portfolio lost $9\%$ of its original value.
- BBoth portfolios break even because $+30\%$ and $-30\%$ cancel out.
- CThe portfolio that loses first ends lower because it starts from a smaller base.
- DThe portfolio that gains first ends higher because it gains on a larger base.
Show solution
Answer: A, Both have the same final value; each portfolio lost $9\%$ of its original value.. Both calculations yield $1.30 \times 0.70 = 0.91$ and $0.70 \times 1.30 = 0.91$. Multiplication is commutative, so both portfolios end at $91\%$ of the original, a loss of $9\%$. Neither restores the original value.
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