Percentages, Ratios, and Rates

Pentru elevii români: Problemele SAT folosesc unități americane (mile, galoane, picioare, livre, dolari) și formulări precum „percent increase”, „percent of” și „rate”, pe care trebuie să le traduci rapid în ecuații; de regulă, factorii de conversie sunt dați în enunț. Atenție la diferența dintre „percent” (procent) și „percentage points” (puncte procentuale), o distincție pe care SAT-ul o testează explicit.

What the SAT tests here

These skills belong to the Problem-Solving and Data Analysis domain. You will see:

• ratios, rates, proportional relationships, and units (including unit conversion);
• percentages: percent of a quantity, percent change, and reverse percent ("what was the original price?");
• multi-step word problems that combine these ideas, often with real-world data.

Key rules and formulas

Percent of a quantity: part $=$ (percent as a decimal) $\times$ whole. "30% of 80" is $0.30 \times 80 = 24$.

Percent change:

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

Multipliers are the fastest tool:

• increase by $r$: multiply by $(1 + r)$, so a 25% increase is $\times 1.25$;
• decrease by $r$: multiply by $(1 - r)$, so a 20% decrease is $\times 0.80$;
• successive changes: multiply the factors;
• reverse percent: original $=$ new $\div$ multiplier.

Ratios and proportions: a ratio $a : b$ splits a total into $a + b$ equal parts. Solve proportions by cross-multiplying: $\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$.

Rates: rate $=$ amount $\div$ time (distance $=$ rate $\times$ time, total cost $=$ unit price $\times$ quantity).

Unit conversion: multiply by fractions equal to 1 so that units cancel:

$66 \,\tfrac{\text{miles}}{\text{hour}} \times \frac{5{,}280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3{,}600 \text{ seconds}} = 96.8 \,\tfrac{\text{feet}}{\text{second}}$

Worked examples

Example 1 (successive percents). A jacket costs $80. The store raises the price by 25%, then puts it on sale at 20% off the new price. What is the sale price?

$80 \times 1.25 \times 0.80 = 80 \times 1 = 80$. The sale price is $80, the same as the original. A 25% increase followed by a 20% decrease cancels out exactly, because $1.25 \times 0.80 = 1$. Percent changes do not simply add or subtract.

Example 2 (reverse percent). After a 15% discount, a pair of headphones costs $102. What was the original price?

Sale price $=$ original $\times 0.85$, so original $= 102 \div 0.85 = 120$. The original price was $120. (A common wrong move is to add 15% of $102 back: $102 \times 1.15 = 117.30$.)

Example 3 (ratio with a total). In a club of 48 students, the ratio of boys to girls is $3 : 5$. How many girls are in the club?

There are $3 + 5 = 8$ equal parts, so each part is $48 \div 8 = 6$ students. Girls: $5 \times 6 = 30$ (and boys: $3 \times 6 = 18$).

Test-taking strategy

• Write units on every number. If your final units do not match what the question asks (hours versus minutes, feet versus miles), you are not done.
• Use multipliers, not two-step arithmetic. "Increase by 8%" means $\times 1.08$ in one step, which also makes successive changes easy.
• Pick 100 as a starting value when a question gives only percents. If a price starts at $100, rises 20%, and then falls 20%, it ends at $96: a 4% overall decrease.
• Desmos as a smart calculator. Type the full expression, such as 66*5280/3600, to avoid rounding mid-problem. For reverse-percent questions, you can type 0.85x=102 and read the solution from the graph, or simply compute 102/0.85.
• Estimate first. If a price drops, the original must be larger than the sale price; this alone eliminates some choices.

Common traps

• Dividing by the wrong base. Percent change always uses the original value as the denominator.
• Percent versus percentage points. If a rate rises from 20% to 25%, it rose by 5 percentage points, which is a 25% increase ($\frac{5}{20} = 0.25$).
• Adding successive percents. A 10% increase then a 10% decrease is a 1% decrease, not zero.
• "Percent of" versus "percent more than." "$A$ is 30% more than $B$" means $A = 1.3B$; "$A$ is 30% of $B$" means $A = 0.3B$.
• Part-to-part versus part-to-whole. A $3 : 5$ ratio of boys to girls means boys are $\frac{3}{8}$ of the club, not $\frac{3}{5}$.
• Unit mismatches. A rate in pages per minute cannot answer a question in hours without converting.

Summary

Turn every percent into a multiplier, always measure percent change from the original value, and undo a percent change by dividing by its multiplier. Treat ratios as equal parts of a whole and cross-multiply proportions. Track units like algebra so they cancel correctly. These habits make Problem-Solving and Data Analysis questions quick and reliable.