Rates, Percentages, and Proportions

Rates, Percentages, and Proportions

Almost every word problem in this part of the ACT comes down to comparing two quantities. A rate compares quantities with different units (miles per hour, dollars per pound). A percent compares a part to a whole, or a change to a starting value. A proportion says two such comparisons are equal. This chapter gives you a reliable method for each, and shows where the traps are.

Unit rates

A unit rate tells you how much of one quantity goes with exactly one unit of another. Divide the quantity you want "per" one of by the quantity that comes after "per." A car that travels 240 miles in 4 hours has a unit rate of $\frac{240}{4}=60$ miles per hour. Always keep an eye on which quantity belongs on top: $60$ miles per hour and $\frac{1}{60}$ hour per mile are both true statements about the same car, but only one answers the question you were asked.

Once you have a unit rate, scaling is just multiplication. If 5 items cost $32, one item costs $\frac{32}{5}=6.40$, so 12 items cost $6.40\times 12=76.80$ dollars.

Percent of a number, and percent change

To find a percent of a number, convert the percent to a decimal and multiply: $45\%$ of $260$ is $0.45\times 260=117$.

Percent change is the change in value divided by the original value, times 100:

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

A price that rises from $60 to $75 has changed by $\frac{75-60}{60}\times 100=25\%$. A negative result means a decrease. The most common mistake is dividing by the new value, or reporting the raw change (here, $15) as if it were the percent.

Percentage points versus percent change

A percentage point difference is plain subtraction of two percentages. A percent change compares that difference to the starting percentage. Reading which one the question asks for is the actual skill.

Worked example. A defect rate falls from $8\%$ to $5\%$. Find both the percentage-point decrease and the percent decrease relative to the original rate.

Percentage-point decrease: $8-5=3$ percentage points.

Percent decrease: $\frac{8-5}{8}\times 100=37.5\%$.

Likewise, a tax rate that rises from $10\%$ to $15\%$ has gone up $5$ percentage points, but the rate itself increased by $50\%$ relative to where it started. If a question says "by how many percentage points," subtract. If it says "by what percent," divide by the original.

Working backward from a percent change

When you know the final value and the percent change, and you need the original, divide by the growth factor. Do not apply the opposite percentage. A price increased $25\%$ to reach $150 satisfies $1.25\times \text{original}=150$, so the original is $\frac{150}{1.25}=120$. Taking $25\%$ off $150 would give $112.50, which is wrong because the $25\%$ was calculated on the original, not on $150.

The same idea works for decreases: after a $20\%$ discount the price is $0.80$ times the original, so divide the sale price by $0.80$.

Proportions: direct and inverse

A proportion states that two ratios are equal. To solve $\frac{5}{8}=\frac{x}{40}$, cross-multiply: $8x=5(40)=200$, so $x=25$. Always check that the same quantity sits in the same position on both sides of the equation (top-to-top, bottom-to-bottom).

A direct proportion keeps a fixed ratio: if one quantity doubles, so does the other. A recipe scaled up is a direct proportion. An inverse proportion keeps a fixed product: when one quantity goes up, the other goes down. Workers and machines finishing a fixed job are inverse proportions, because more workers means fewer days. Mixing these up is the most common error on this topic.

Worked example. 4 machines can complete a production run in 9 hours. How many hours would 6 machines take, at the same rate?

More machines means less time, so the quantity that stays fixed is machines times hours: $4\times 9=6\times t$, so $t=\frac{36}{6}=6$ hours. A quick reasonableness check: more machines should finish sooner, and 6 hours is less than 9.

Strategy

• Underline the thing the question asks for, and make sure your final answer is in that form (a percent, a number of percentage points, a rate with units).
• For percent change, the denominator is always the starting value.
• For "undo a percent" problems, divide by $1+r$ (or $1-r$). Never subtract the percent from the final value.
• Before solving a "workers and time" problem, decide whether the relationship is direct or inverse. Ask yourself: should the answer get bigger or smaller?
• Estimate first. If a price rises from $60 to $75, the increase is a quarter of 60, so you expect an answer near $25\%$ and can discard the choices that are far away.

A value that changes over time is often stated in words instead of as a ready-made percent. A price that drops from $40 to $32 has decreased by $\frac{8}{40}=20\%$: find the change, then divide by the original.