How to solve SAT percentage questions

Percentage questions on the Digital SAT ask you to find a percent of a number, calculate percent increase or decrease, work backward from a final amount to an original amount, and combine several percent changes. Many come wrapped in shopping, population, or survey contexts. Problem-Solving and Data Analysis is about 15% of the Math section, or 5 to 7 of the 44 questions. Those questions are split across seven skills, so expect about one percentage question per test. Percents also appear inside exponential growth and data questions.

What the questions look like

Percent of, and what percent. You find 35% of a value, or decide what percent one number is of another. These are often based on a table.

Percent change. You compute the percent increase or decrease from an old value to a new one, or apply a change to a starting value.

Reverse and successive percents. You know the price after a discount and tax and must find the original price, or you combine two changes such as "increased by 20%, then decreased by 20%." Questions that chain several percent statements together, like "$p$ is 150% of $q$," also fit here.

Step-by-step method

1. Turn every percent change into a multiplier.
   • An increase of $r\%$ means multiply by $1 + \frac{r}{100}$. A 20% increase is $\times 1.20$.
   • A decrease of $r\%$ means multiply by $1 - \frac{r}{100}$. A 20% decrease is $\times 0.80$.
2. Translate "of" and "is." "$p$% of $x$" means $\frac{p}{100} \cdot x$. "$a$ is what percent of $b$" means $\frac{a}{b} \times 100$.
3. Find percent change from the original.
   $$\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100$$
4. Chain changes by multiplying the multipliers. Never add the percents. A 20% increase followed by a 20% decrease is $1.20 \times 0.80 = 0.96$, a net 4% decrease.
5. Work backward by dividing. If the final value is the original times a multiplier, the original is the final value divided by that multiplier.

Worked example

Example 1. A jacket is discounted by 20%, and then an 8% sales tax is applied to the discounted price. The final cost is $97.20. What was the original price of the jacket?

Write the two changes as multipliers: the discount is $\times 0.80$ and the tax is $\times 1.08$. If $x$ is the original price,

$$x(0.80)(1.08) = 97.20$$
$$0.864x = 97.20$$
$$x = \frac{97.20}{0.864} = 112.50$$

The original price was $112.50. Check by going forward: 20% off $112.50 is $90.00, and 8% tax on $90.00 is $7.20, for a total of $97.20.

Example 2. The number $p$ is 150% of $q$, and $q$ is 40% less than $r$. The number $p$ is what percent of $r$?

Translate each statement into a multiplier:

$$q = 0.60r \qquad p = 1.50q$$

Substitute the first into the second:

$$p = 1.50(0.60r) = 0.90r$$

So $p$ is 90% of $r$. Picking a number works too. If $r = 100$, then $q = 60$ and $p = 90$, which is 90% of 100.

Common traps

• Dividing by the wrong value for percent change. Percent change uses the original value as the denominator. Going from 40 to 50 is a 25% increase, but going from 50 to 40 is a 20% decrease.
• Adding successive percents. A 30% increase followed by a 10% increase is $1.30 \times 1.10 = 1.43$, a 43% increase, not 40%.
• Taking the percent off the final amount. In Example 1, subtracting 8% and then adding 20% to $97.20 does not recover the original price, because each percent was applied to a different base. Divide by the multipliers instead.
• Confusing "150% of" with "150% more than." "150% of $q$" is $1.5q$. "150% more than $q$" is $q + 1.5q = 2.5q$. Read the exact wording.
• Mixing up percent and percentage points. If a rate goes from 10% to 15%, it rose by 5 percentage points, which is a 50% increase. Questions sometimes offer both numbers as choices.

Desmos tip

For reverse-percent questions, you can skip the rearranging. Type the equation exactly as you would write it, such as x*0.8*1.08 = 97.2, and Desmos draws a vertical line at the solution. Click where it crosses the $x$-axis to read $x = 112.5$. For a quick forward check, type 112.5*0.8*1.08 on a new line and confirm it returns 97.2. This catches the most common error, which is applying the percents to the wrong base.