Percentages

Percentages show up on the Digital SAT in a wide range of real-world contexts: shopping discounts, sales taxes, tips at restaurants, interest on savings, and population changes. You'll need to calculate percentages, work with percent increases and decreases, and understand how percent change connects to growth factors. Expect to see roughly 2–4 questions on percentages across the math sections, with difficulty ranging from straightforward single-step calculations to trickier multi-step problems involving successive changes or growth factor expressions.

Calculating Percentages in Context

Most percentage problems boil down to three quantities: the whole (100%), the part, and the percent. If you know two, you can find the third.

Finding a percentage of a quantity: Convert the percent to a decimal and multiply.

$25\% \text{ of } 80 = 0.25 \times 80 = 20$

Finding what percent one number is of another: Divide the part by the whole, then multiply by 100.

$\frac{36}{150} \times 100 = 24\%$

Finding the whole from a part and a percent: Divide the part by the decimal form of the percent.

$\text{If } 72 \text{ is } 60\% \text{ of some number: } \frac{72}{0.60} = 120$

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Applying Percentages to Discounts, Taxes, Tips, and Interest

The SAT wraps these basic calculations in real-world scenarios. The math is the same each time, but the context changes what you're adding or subtracting.

Discounts reduce a price. A 15% discount on a $60 item:

$60 \times 0.15 = 9$ $60 - 9 = 51$

Or in one step: $60 \times 0.85 = 51$. (You pay 85% of the original price.)

Taxes add to a price. An 8.5% sales tax on a $120 purchase:

$120 \times 1.085 = 130.20$

Tips also add to a total. A 20% tip on a $45 meal:

$45 \times 1.20 = 54$

Interest (simple) adds to a balance. 4% annual interest on $2,000:

$2{,}000 \times 0.04 = 80 \text{ dollars earned in one year}$

Worked Example (SAT-style): A laptop is on sale for 30% off the original price. After the discount, a 6% sales tax is applied. If the original price is $850, what is the final cost?

Step 1: Apply the discount. $850 \times 0.70 = 595$

Step 2: Apply the tax to the discounted price. $595 \times 1.06 = 630.70$

The final cost is $\$630.70$. Notice that the tax applies to the sale price, not the original price. The SAT may try to trick you by offering an answer that applies tax first or applies tax to the original amount.

Percent Increase and Percent Decrease

Percent change measures how much a value grew or shrank relative to where it started. The formula is:

$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$

A positive result means percent increase; a negative result means percent decrease.

Example: A town's population went from 12,000 to 15,000. What is the percent increase?

$\frac{15{,}000 - 12{,}000}{12{,}000} \times 100 = \frac{3{,}000}{12{,}000} \times 100 = 25\%$

Working backward: You can also find a new value after a given percent change.

• After a percent increase: $\text{New} = \text{Original} \times (1 + \text{rate as decimal})$
• After a percent decrease: $\text{New} = \text{Original} \times (1 - \text{rate as decimal})$

A $240 item after a 35% decrease: $240 \times 0.65 = 156$

Worked Example (SAT-style): The number of students enrolled in a program decreased from 480 to 372. What is the percent decrease?

$\frac{372 - 480}{480} \times 100 = \frac{-108}{480} \times 100 = -22.5\%$

The percent decrease is $22.5\%$. Always divide by the original value, not the new one. Dividing by 372 instead of 480 is one of the most common errors on these questions.

Successive Percent Changes

Two percent changes applied one after another don't simply add. A 20% increase followed by a 20% decrease does not return you to the starting value.

Example: A stock worth $500 increases by 20%, then decreases by 20%.

After the increase: $500 \times 1.20 = 600$ After the decrease: $600 \times 0.80 = 480$

The final value is $480, not $500. That's a net 4% decrease, not 0%. To find the overall effect, multiply the individual multipliers:

$1.20 \times 0.80 = 0.96$

Since $0.96 = 1 - 0.04$, the overall change is a 4% decrease.

Growth Factors and Percentages Greater Than 100%

This is where the SAT gets more algebraic. A growth factor is the single multiplier that takes you from the original value to the new value after a percent change.

• A 5% increase has a growth factor of $1.05$
• A 12% decrease has a growth factor of $0.88$
• A 100% increase (doubling) has a growth factor of $2.00$
• A 150% increase has a growth factor of $2.50$

The relationship: growth factor = 1 + (percent change as a decimal) for increases, or growth factor = 1 − (percent change as a decimal) for decreases.

Percentages can absolutely be greater than or equal to 100%. If a quantity triples, that's a 200% increase (growth factor of 3.00), because the quantity gained 200% of its original value on top of the original.

Worked Example (SAT-style): A bacteria colony starts with 500 cells. Each hour, the number of cells increases by 40%. Which expression represents the number of cells after $h$ hours?

Each hour, the colony is multiplied by the growth factor $1.40$. After $h$ hours:

$500 \times (1.40)^h$

If the question asks "what does 1.40 represent?", the answer is that the population is 140% of its previous size each hour, or equivalently, it grows by 40% per hour.

Worked Example (harder): A car's value depreciates by 18% each year. The car was purchased for $32,000. What is the value of the car after 3 years, to the nearest dollar?

The growth factor for an 18% decrease is $1 - 0.18 = 0.82$.

$32{,}000 \times (0.82)^3 = 32{,}000 \times 0.551368 = 17{,}643.78$

The value after 3 years is approximately $\$17{,}644$.

Interpreting Growth Factors in Equations

The SAT often gives you an equation and asks you to interpret the growth factor. For instance:

$P = 2{,}400(1.07)^t$

Here, $1.07$ is the growth factor. Since $1.07 = 1 + 0.07$, the quantity increases by 7% per time period. The initial value is 2,400.

What if you see $V = 15{,}000(0.85)^t$? The growth factor is $0.85 = 1 - 0.15$, so the quantity decreases by 15% per time period.

And if you see a growth factor of $2.30$? That's $1 + 1.30$, meaning a 130% increase each period. The quantity more than doubles each time.

What to Watch For on Test Day

1. Always divide by the original value when calculating percent change. The most common wrong answer on percent increase/decrease questions comes from dividing by the new value instead.

2. Don't add successive percentages. A 10% increase then a 10% increase is not 20%. Multiply the growth factors: $1.10 \times 1.10 = 1.21$, which is a 21% total increase.

3. Know the difference between "percent of" and "percent increase." If something is 300% of the original, that's a growth factor of 3.00. But a 300% increase means the growth factor is 4.00 (the original plus 300% more).

4. Apply discounts and taxes in the right order. Read carefully whether tax is applied before or after a discount. The SAT will offer both answers as choices.

5. Use growth factors to save time. Instead of calculating a discount and then subtracting, multiply by the complement directly. A 25% discount means multiply by 0.75. This is faster and reduces arithmetic errors, especially when your calculator is right there.