SAT Math Percentages, Ratios, Proportions

Percent change, ratios, unit conversion — reliable techniques

Percentages and proportions test middle-school skills dressed up in adult contexts: discounts, taxes, mixtures, speeds. Expect 4–5 questions per section, mostly medium difficulty — quick wins if you handle percentages confidently.

Speed unlock: always translate "percent of" into multiplication. "20% of 80" is $0.2 \cdot 80 = 16$ . "Increase by 15%" is $\times 1.15$ . "Decrease by 25%" is $\times 0.75$ . Those three substitutions cover 80% of the topic.

What the SAT actually tests

Computing $p$ of $N$ = $\frac{p}{100} \cdot N$
Percent change: increase ( $\times (1 + r)$ ) and decrease ( $\times (1 - r)$ )
Compound interest
Ratios in $a : b$ form and converting to fractions
Unit conversion (km/h ↔ m/s, inches ↔ cm, etc.)
Map scales and geometric proportions

Key concepts

Percent as fraction

$p$ . "30% of 50" is $\frac{30}{100} \cdot 50 = 15$ . Percent is ALWAYS multiplication after converting to a fraction.

Change multiplier

Increase by $r$ = multiply by $1 + \frac{r}{100}$ . Decrease by $r$ = multiply by $1 - \frac{r}{100}$ . The fundamental shortcut.

Ratio vs fraction

A $3:5$ ratio means 3 parts out of 8 and 5 parts out of 8. The first object is $\frac{3}{8}$ of the total, not $\frac{3}{5}$ .

Compound interest

After $t$ periods, value grows from $P$ to $P \cdot (1 + r)^{t}$ . Classic SAT problem — savings with annual interest.

Worked examples

Example 1

A shoe cost 80 dollars. The store applied a 25% discount, then added an 8% tax. What's the final price?

Solution

After discount: $80 \cdot 0.75 = 60$ . After tax: $60 \cdot 1.08 = 64.80$ .

💡 Discount → multiply by $0.75$ . Tax → multiply by $1.08$ . Order of percentage operations doesn't change the final number (commutative under multiplication).

Example 2

The ratio of girls to boys in a class is $3:2$, and there are 30 students. How many boys?

Solution

Sum of ratio parts: $3 + 2 = 5$ . One part: $30/5 = 6$ . Boys = 2 parts: $2 \cdot 6 = 12$ .

💡 Ratio $a : b$ + total → divide total by $a + b$ to find one part.

Common pitfalls

Confusing "increase by 50%" with "increase to 50%". "By" adds, "to" replaces.
Adding percentages instead of multiplying multipliers. "15% off + 10% off" is NOT 25% off — it's $0.85 \cdot 0.9 = 0.765$ , i.e., 23.5% off.
Reading a ratio as a fraction. $3:2$ is not $\frac{3}{2}$ of the total; it's 3 of 5 parts.
Unit conversion direction errors. Always check that units cancel in the result.

Exam strategy

Translate percents to multipliers — faster and safer than the "percent change = (new − old)/old × 100%" formula. For sequential changes, multiply the multipliers, never add the percents. For ratios, sum the parts and divide the total by the sum to find one part. For unit conversion, write the chain: "60 km/h × 1000 m/km × 1/3600 h/s = 16.67 m/s" — units must cancel.

Frequently asked questions

How do I compute a percent of a number?

Convert the percent to a decimal and multiply. "30% of 80" is 0.30 × 80 = 24. Percent is always multiplication after conversion.

How do I compute a percent increase?

Increase by r% = multiply by (1 + r/100). "Price rose 15%" means new = old × 1.15. Decrease by r% = multiply by (1 − r/100).

Is a 3:5 ratio the same as the fraction 3/5?

No. A 3:5 ratio means 3 parts out of 8 and 5 parts out of 8. The first object is 3/8 of the total, not 3/5.

How does compound interest work?

Formula: P × (1 + r)^t. P = starting amount, r = rate as decimal (0.05 for 5%), t = periods. Example: $1000 at 5% annual for 3 years = $1000 \times 1.0 5^{3} =$ 1157.63.

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