Percentages

Percent means "per hundred." The SAT tests four kinds of percent questions: finding a percent of a number, computing a percent change, working backward from a changed value to the original, and applying successive changes one after another (a markup followed by a discount, two years of growth, and so on).

• Convert a percent to a decimal by dividing by $100$: 35% is $0.35$.
• Increasing by 20% is multiplying by $1.20$. Decreasing by 20% is multiplying by $0.80$. Think in multipliers, not in "add the percent."
• Successive changes multiply: a 20% increase then a 25% decrease is $1.20 \times 0.75 = 0.90$, a 10% net decrease, not a 5% decrease.
• Percent change always uses the original value as the base.
• To undo a change, divide by the multiplier. If a price after a 15% discount is $68, the original is $68 \div 0.85 = 80$, so $80.
• In mixture questions, the amount of the ingredient (salt, juice) stays fixed when only water is added.

$\text{part} = \frac{p}{100} \times \text{whole} \qquad \text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100$

What percent of $72$ is $18$? Then: a $80 item increases in price by 15%. What is the new price?

1. $\frac{18}{72} = 0.25$, so $18$ is 25% of $72$.
2. New price: $80 \times 1.15 = 92$, so $92.

Check: 15% of $80$ is $12$, and $80 + 12 = 92$. Correct.

A store marks up the wholesale cost of a jacket by 60%. Later, it sells the jacket at 25% off the marked price, for $54. What was the wholesale cost?

1. Combine the multipliers: $1.60 \times 0.75 = 1.20$. The sale price is 120% of the cost.
2. Work backward: cost $= 54 \div 1.20 = 45$. The wholesale cost was $45.

Check forward: $45 \times 1.6 = 72$ (marked price), and $72 \times 0.75 = 54$. Correct.

• Adding or subtracting percents. A 60% markup then a 25% discount is not a 35% increase; using $1.35$ gives $40, which is wrong.
• Taking the percent of the new value when undoing a change. If $68 is after a 15% discount, adding 15% of $68 gives $78.20, not the correct $80.
• Using the wrong base for percent change. From $80 to $100 is a $\frac{20}{80} = 0.25$, or 25%, increase. Dividing by $100$ gives 20%, a trap answer.
• Answering the complement. If $18$ of $48$ students are in a club, 37.5% are in it and 62.5% are not. Read carefully to see which one the question wants.
• Percent vs. percentage points. Going from 40% to 50% is a 10-percentage-point increase but a 25% relative increase.