Percentages

Unit 3 · Problem-Solving & Data Analysis

What the SAT asks here

A percent is a fraction out of 100. The Digital SAT pushes on three skills: computing percent change off the correct base, applying sequential percent changes as multiplicative factors, and solving percent-of-a-percent problems. Expect 1–2 problems per test.

Percent change

From original A to new B

\text{change}\% = \frac{B - A}{A} \cdot 100\%

Sequential percent changes multiply

20% up then 20% down ≠ 0%

1.20 \cdot 0.80 = 0.96 \;\Rightarrow\; 4\% \text{ net decrease}

SAT Tip: Translate each percent into its multiplicative factor first. 50% markup is ×1.50; 20% discount is ×0.80. Multiplying factors is much faster than computing intermediate dollar amounts.

Caution: Percent change is always relative to the ORIGINAL value. Dividing by the new value is the most common SAT mistake on this topic.

Type 1

Percent change

Use (new − old) / old · 100%. Watch the sign: positive for increase, negative for decrease.

Example 1

A stock price rose from

80 to

92. What was the percent increase?

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Type 2

Markup and discount (multi-step)

Multiply the factors. If the final price after a 50% markup and 20% discount is known, divide by 1.50 · 0.80 to recover the original.

Example 2

A shirt is marked up 50% from its wholesale cost, then discounted 20% at sale. If the sale price is $48, what was the wholesale cost?

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Type 3

Percent of a percent

Recover the original number from one percentage statement, then apply a different percentage.

Example 3

30% of a number is 12. What is 75% of the same number?

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