Percentages Pattern - Multi Step

Performing two or more percentage calculations in sequence

Multi-step percentage problems chain two or more percentage changes together. The result of the first calculation becomes the base for the second. The key insight: you cannot simply add or subtract the percentages — each step uses a different base value.

The Core Pattern

Given an initial value and two successive percentage changes:

Step 1: Apply the first change to the initial value → get an intermediate value.
Step 2: Apply the second change to the intermediate value (not the original).

Example — Sequential Increase and Decrease

The wholesale price of a laptop is $500. The retail price $R$ is 120% greater than the wholesale price. During a sale, the final price $S$ is 40% less than $R$. What is the sale price $S$?

A) $660
B) $440
C) $400
D) $240

Step 1 — Find $R$: "120% greater than" means $100\% + 120\% = 220\%$ of wholesale.
$R = 500 \times 2.20 = 1{,}100$

Step 2 — Find $S$: "40% less than $R$" means $100\% - 40\% = 60\%$ of $R$.
$S = 1{,}100 \times 0.60 = 660$
Answer: A

Trap 1: Subtracting Percentages Directly

Choice C ($400) comes from: $120\% - 40\% = 80\%$, then $500 \times 0.80 = 400$. This is wrong because the two percentages apply to different base values. The 120% applies to the wholesale price, and the 40% applies to the retail price — you can't just subtract them.

Trap 2: Confusing "Greater Than" with "Of"

Choice B ($440) calculates $R$ correctly as $1,100 but then computes "40% of $R$" ($440) instead of "40% less than $R$" ($660). "40% less" means you keep 60%.

Example — Successive Growth Rates

An investment increased by $8\%$ from 2015 to 2016, then by $5\%$ from 2016 to 2017. If the 2017 value was $k$ times the 2015 value, what is $k$?

A) 1.1300
B) 1.1340
C) 1.0040
D) 0.0040

Multiply successive growth factors:
$k = 1.08 \times 1.05 = 1.1340$
Answer: B

Gotcha — Adding the Percents: $8\% + 5\% = 13\%$ gives a multiplier of $1.13$, but the true combined multiplier is $1.08 \times 1.05 = 1.134$. The extra $0.004$ comes from the "interest on interest" effect — the 5% applies to a value that already includes the 8% growth.

Example — Extreme Percentages

City A has a population of 40,000. Town B is 95% less than City A. Village C is 450% greater than Town B. What is Village C's population?

A) 9,000
B) 11,000
C) 142,000
D) 171,000

Step 1 — Town B: 95% less → keep $5\%$: $40{,}000 \times 0.05 = 2{,}000$
Step 2 — Village C: 450% greater → $100\% + 450\% = 550\%$: $2{,}000 \times 5.50 = 11{,}000$
Answer: B

Gotcha: "95% less" means only 5% remains. "450% greater" means the new value is 550% of the base, not 450%.

What to Do on Test Day

• Never add or subtract successive percentages. Each change applies to a different base.
• Translate carefully: "X% greater than" → multiply by $1 + \dfrac{X}{100}$. "X% less than" → multiply by $1 - \dfrac{X}{100}$. "X% of" → multiply by $\dfrac{X}{100}$.
• For combined growth factors, multiply: $(1 + r_1)(1 + r_2) \neq 1 + r_1 + r_2$
• Work step by step. Compute the intermediate value first, then use it as the base for the next calculation.
• Check your answer against the original. If a series of increases gives a final value smaller than the original, something went wrong.