SAT percentages: use multipliers and skip the traps

Percent means “per hundred”

A percent is a number out of 100, so 35% is 0.35. “35% of 80” means $\displaystyle 0.35 \times 80 = 28$. Read “of” as multiplication, and you can translate most percent sentences directly into an equation.

One trick makes some questions easy to do in your head: p% of q equals q% of p. For example, 8% of 50 is the same as 50% of 8, which is 4.

Think in multipliers

A percent increase or decrease is one multiplication. Increasing by r% multiplies by $\displaystyle 1 + \frac{r}{100}$. Decreasing by r% multiplies by $\displaystyle 1 - \frac{r}{100}$. That single number replaces the two-step “find the percent, then add or subtract” routine.

See the skill

Translate the words into one multiplier

Increase by 30%

× 1.30

$40 becomes 40 × 1.30 = $52.

Decrease by 15%

× 0.85

A 15% discount means you pay 85%.

Grow 5% per year for t years

× 1.05ᵗ

Repeated growth multiplies the factor t times.

Is 125% of the original

× 1.25

Same as a 25% increase.

Use the visual as a reasoning aid: tie each choice to the relationship, grammar, or evidence the passage actually gives.

Up 20%, down 20% is not a round trip

Start with $100. A 20% increase gives $120. A 20% decrease takes 20% of $120, which is $24, leaving $96. The second percent applies to a bigger number, so the decrease is larger than the increase.

Multipliers show it at once: $\displaystyle 1.20 \times 0.80 = 0.96$. Move the slider to see that the round trip always ends below the start, and the loss grows quickly as the percent gets larger.

Try it yourself · interactive

Raise it, then cut it by the same percent

Percent change, up and then down20%

Start

Up 20%

Down 20%

Net change

-4.00%

One multiplier does it all

1.20 × 0.80 = 0.9600

For any r, (1 + r)(1 − r) = 1 − r², which is always less than 1. At 20% the net change is −4%; at 50% it is −25%.

Percent change: always divide by the original

Percent change is $\displaystyle \frac{\text{new} - \text{original}}{\text{original}} \times 100$. Going from 40 to 50 is a 25% increase, since $\displaystyle \frac{10}{40} = 0.25$. Going back from 50 to 40 is only a 20% decrease, since $\displaystyle \frac{10}{50} = 0.2$. The same difference of 10 gives different percents because the starting values differ.

Work backward by dividing

A jacket costs $150 after a 25% price increase. What was the original price? The trap is to take 25% off $150, which gives $112.50. Instead, write the equation: $\displaystyle \text{original} \times 1.25 = 150$, so the original price is $\displaystyle 150 \div 1.25 = 120$. Check: $120 increased by 25% is $150.

Your turn

Quick practice

A price is increased by 10% and then the new price is decreased by 10%. How does the final price compare with the original price?

Hint: Multiply the two multipliers instead of adding the two percents.

It is the same as the original price.It is 1% less than the original price.It is 1% more than the original price.It is 10% less than the original price.

Practice: try these before checking the answers

Write a multiplier first, then calculate.

1. What is 35% of 80?
2. A value goes from 64 to 80. What is the percent increase?
3. After a 20% discount, a shirt costs $56. What was the original price?
4. A town of 2,000 people grows by 5% each year. What is the population after 2 years?

SAT percentages: use multipliers and skip the traps

What you'll take away

Percent means “per hundred”

Think in multipliers

Translate the words into one multiplier

Up 20%, down 20% is not a round trip

Raise it, then cut it by the same percent

Percent change: always divide by the original

Work backward by dividing

Quick practice

Practice: try these before checking the answers

Sources & further reading

Put the idea to work.

More from the journal

SAT math circle equations: find the center and radius

SAT math linear functions: understand slope, intercepts, and graphs