Percentages

Official Digital SAT skill — Problem-Solving and Data Analysis domain.

What this question tests

Percentages on the Digital SAT test whether you can interpret “out of 100” ideas and apply them to real-world quantities, including parts of a whole, percent changes, and successive percent operations. You’ll see problems that ask for a percent of a quantity, a quantity given a percent, a percent change, or a combined effect like a discount followed by tax. You may also see multi-step setups where the base changes mid-problem, so the same percent is applied to different starting amounts. This skill is tested because it blends proportional reasoning with careful reading: the math is usually straightforward, but the correct base and order of operations matter a lot. Strong percentage reasoning also helps you avoid traps that look plausible if you add, subtract, or compare the wrong quantities.

What to know

A percentage is a ratio per hundred, so converting between forms is essential: $p\% = \frac{p}{100} = 0.01p$, and you can move between percent, decimal, and fraction as needed.
To find a part from a whole, use $\text{part} = (\text{percent as decimal}) \times (\text{base})$, where the base is the quantity the percent is taken OF.
To find a percent from two quantities, use $\text{percent} = \frac{\text{part}}{\text{base}} \times 100\%$, and be sure the base matches the phrase in the problem.
Percent change compares a new value to an original value: $\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$, and the sign tells you increase or decrease.
Successive percentages multiply, not add, because each step applies to the current base: for example, increase by $a\%$ then by $b\%$ gives a factor of $(1+\frac{a}{100})(1+\frac{b}{100})$ applied to the original.
A percent point is a difference between percentages (like $40\%$ to $50\%$ is a 10 percentage point increase), which is not the same as a 10% relative increase unless specified.

How to approach it

First, rewrite the question in your own words to identify exactly what is being asked, because the target may be a percent, a quantity, or a new total after a change.
Next, mark the base for every percent phrase, since “percent of what?” determines the correct multiplication and prevents using the wrong denominator.
Then convert the percent to a decimal and set up a clean equation such as $\text{part} = r \cdot \text{base}$ or $\text{new} = \text{original}(1 \pm r)$, because a consistent setup reduces mistakes.
If the problem has multiple steps, compute in order and update the base after each step, because successive changes depend on the current value rather than the original unless stated otherwise.
After computing, label the result as a percent, a decimal, or a raw quantity to match the question, because a correct number in the wrong form can still be wrong on the test.
Finally, do a quick reasonableness check using a simpler benchmark (like 10%, 25%, or 50%), because estimates can catch arithmetic slips or impossible answers before you move on.

Common traps

Adding or subtracting percentages when you should multiply for successive steps is a common trap, because the second percent applies to a changed base; avoid it by using factors like $(1+r)$ and multiplying them.
Using the wrong base in percent change or percent-of problems happens when students grab a nearby number; avoid it by explicitly stating the base in words before you compute.
Confusing a percent with percentage points leads to wrong comparisons; avoid it by checking whether the problem asks for a relative change or a difference between two percent values.
Treating an increase by $x\%$ followed by a decrease by $x\%$ as net zero is tempting but false in general; avoid it by applying both factors to the same starting value and seeing the product is $(1+x)(1-x)=1-x^2$.
Misreading whether the answer should be a percent or a proportion causes formatting errors; avoid it by scanning the final sentence for “percent,” “decimal,” or “fraction,” and matching your output to that form.

Tips & shortcuts

Use factor form for changes: replace “increase by $r\%$” with multiply by $(1+r)$ and “decrease by $r\%$” with multiply by $(1-r)$, where $r$ is the decimal.
When multiple categories overlap (like color and pattern), think of it as a “percent of a percent” and multiply the rates to get the overall proportion.
If the numbers are messy, convert to decimals you can handle and keep four operations tidy, then estimate to see if the magnitude makes sense.
In grid-in questions, enter the numeric value requested (such as 12 for 12%) only if the prompt asks for a percent; if it asks for a proportion, enter the decimal like 0.12.

Worked example

A lab sample’s mass decreases by $20\%$ during drying, and then the remaining mass decreases by $10\%$ during testing. What percentage of the original mass remains after both decreases?

A. $80\%$
B. $72\%$ ✓ (correct answer)
C. $70\%$
D. $28\%$

Why: Successive decreases must be applied multiplicatively. After drying, $100\% - 20\% = 80\%$ remains, so the mass is multiplied by $0.80$. After testing, $100\% - 10\% = 90\%$ of the remaining mass remains, so multiply by $0.90$. The remaining fraction is $0.80 \times 0.90 = 0.72$, which is $72\%$. Therefore the correct answer is $72\%$.

Use the Practice Questions for this skill to drill it, then attempt a Timed Practice Test.