percentages & percent change

Percentage questions are everywhere on the SAT: tips, markups, sales, percent error, and data tables. This walkthrough covers all the classic percent setups—percent of, percent increase/decrease, percent error, and multiple-step percent problems. Every concept is explained step-by-step, and you’ll get to practice real SAT-style percent questions.

what is a percent?

A percent means “out of 100.” For example, 32% just means $\\frac{32}{100}$ or 0.32. Percentages are used to compare, increase/decrease amounts, and analyze data in tables and charts. The SAT expects you to move between percent, decimal, and fraction forms with ease.

Example:

\text{What is 15% of 60?}

0.15 \times 60 = 9

Always change percent to decimal before multiplying!

percent change, increase, & decrease

Percent change means “how much up or down, as a percent of the original.” Use the formula:

\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%

Percent increase: New value is higher.
Percent decrease: New value is lower.

Example:

The price rises from $80 to $92. What is the percent increase?

\frac{92-80}{80} \times 100 = \frac{12}{80} \times 100 = 15\%

A $12 increase on an $80 base is a 15% increase.

markup, markdown, percent error

The SAT loves “price marked up by x%,” “reduced by y%,” or “percent error” setups. Markup and markdown work like normal percent increase/decrease. Percent error is the difference between an estimate and actual value, divided by the actual value:

\text{percent error} = \frac{\left| \text{estimate} - \text{actual} \right|}{\text{actual}} \times 100\%

Example:

If a length is actually 40 cm, but you measure 37 cm, what is the percent error?

\frac{|37 - 40|}{40} \times 100 = \frac{3}{40} \times 100 = 7.5\%

Always use the actual value in the denominator for percent error.

compound percent & multiple changes

If an amount changes by x% and then y%, you can’t just add or subtract the percentages! Multiply the change factors. For a 20% increase, multiply by 1.2; for a 10% decrease, multiply by 0.9. For n increases of r%, use $(1 + \\frac{r}{100})^n$ .

Example:

A price increases by 5% two years in a row. Multiply: $1.05 \times 1.05 = 1.1025$ (10.25% total increase).

percent word problems & setup

For percent word problems, always translate “percent” to decimal, clearly define your variables, and watch for tricky “percent of what?” language. Write an equation for the relationship described and always double check the question—especially if it’s asking for percent, NOT the raw number.

Example:

After a 15% discount, a shirt costs $51. What was the original price?

0.85x = 51

x = \frac{51}{0.85} = 60

The original price was $60.

Always check if your answer makes sense—especially if a percent increase/decrease question!

Ready for real SAT percent questions? Try these!

Question 1

Jennifer bought a box of Crunchy Grain cereal. The nutrition facts on the box state that a serving size of the cereal is three fourths cup and provides 210 calories, 50 of which are calories from fat. In addition, each serving of the cereal provides 180 milligrams of potassium, which is 5% of the daily allowance for adults. If $p$ percent of an adult’s daily allowance of potassium is provided by $x$ servings of Crunchy Grain cereal per day, which of the following expresses $p$ in terms of $x$ ?

Question 2

Year US minimum wage (dollars per hour)
1960: 1.00
1970: 1.60
1980: 3.10
1990: 3.80
2000: 5.15
2010: 7.25

What was the percent increase of the minimum wage from 1960 to 1970?

Question 3

A customer’s monthly water bill was $75.74. Due to a rate increase, her monthly bill is now $79.86. To the nearest tenth of a percent, by what percent did the amount of the customer’s water bill increase?

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