Percentages Pattern - Percent Change

Calculating percent change between values or finding originals/new values after a change

Percent change questions come in two directions: forward (given an original and a percent, find the new value) and backward (given a new value and a percent, find the original). The SAT also tests whether you can extract the percent change from a multiplier expression like $1.08P$.

Forward: Finding a New Value

"X% of the original" means multiply the original by $\dfrac{X}{100}$.

A factory produced $300$ widgets last month. This month, production was $125\%$ of last month's production. How many widgets did the factory produce this month?

A) 75
B) 325
C) 375
D) 425

$125\%$ of $300 = 1.25 \times 300 = 375$
Answer: C

Critical distinction — "of" vs. "greater than":

• "$125\%$ of" the original → multiply by $1.25$
• "$25\%$ greater than" the original → also multiply by $1.25$ (because $100\% + 25\% = 125\%$)

Both give the same result, but the language differs.

Trap: Choice A (75) is just the increase ($300 \times 0.25$), not the new total. Choice B (325) adds the percent digits directly: $300 + 25$.

Backward: Finding the Original Value

The market value of a collector's item is now $75. This represents a 200% increase from its original purchase price. What was the original purchase price?

A) $37.50
B) $25.00
C) $50.00
D) $150.00

A 200% increase means the new price is $100\% + 200\% = 300\%$ of the original.
$3.00 \times p = 75$
$p = \dfrac{75}{3} = 25$
Answer: B

Gotcha — Forgetting the Original 100%: The most common error is dividing by 2 instead of 3, giving $37.50. A "200% increase" does NOT mean the new value is 200% of the old — it means 200% was added to the original 100%, making it 300% of the original.

Interpreting a Multiplier Expression

The expression $1.08P$ represents a town's population after one year. This represents an increase by what percent?

A) 0.8%
B) 8%
C) 10.8%
D) 108%

The multiplier is $1.08 = 1 + 0.08$.
The $1$ represents the original 100%. The $0.08$ represents the increase.
$0.08 \times 100 = 8\%$ increase.
Answer: B

For a decrease: $0.82C$ means $1 - 0.18 = 0.82$, so the decrease is $18\%$.

Gotcha — Confusing the multiplier with the percent: Choice D (108%) is the multiplier converted straight to a percent. But $1.08$ means the population is $108\%$ of the original — the increase is only $8\%$.

What to Do on Test Day

• "X% of" → multiply by $\dfrac{X}{100}$. "X% greater than" → multiply by $1 + \dfrac{X}{100}$. "X% less than" → multiply by $1 - \dfrac{X}{100}$.
• Going backward? Set up $\text{multiplier} \times \text{original} = \text{new}$ and solve for the original.
• For multiplier expressions: Subtract 1 from the multiplier to find the percent change (as a decimal). If the multiplier is less than 1, the change is a decrease.
• "200% increase" ≠ "200% of." A 200% increase means the new value is $300\%$ of the original.