Percentages

A percent is a ratio out of one hundred, and most percent questions are one multiplication. The work that earns points is reading which number plays which role, and, often, running the change backward.

A percent is a ratio per 100: p percent means p out of 100. That makes every percent a special case of the proportional reasoning from the last section, and it means percent questions are usually a single multiplication once you have read them correctly. The points are earned not in the arithmetic but in the setup, deciding which quantity is the whole, whether you are finding a part or working back to the whole, and whether a change should be applied or undone. We build from the plain calculation toward the two situations that catch careful readers: reversing a change, and stacking two changes in a row.

▸ Interpret percent as a rate per 100

Because a percent is a ratio over 100, you can move freely between three forms (percent, decimal, and fraction) and choosing the easiest form for a given calculation is half the skill. To turn a percent into a decimal, divide by 100 (move the point two places left); to turn a decimal into a percent, do the reverse.

Fractions, decimals, and percents

35 percent = 0.35 = 35/100 = 7/20.

0.6 = 60 percent = 3/5.

3/4 = 0.75 = 75 percent. A fraction becomes a percent by writing it over 100.

▸ Compute a percent of a quantity

The most basic percent calculation is finding a percent of a quantity, and the word “of” is the instruction: it means multiply. To find p percent of a number, convert the percent to a decimal and multiply.

WORKED Compute a percent of a quantity

What is 18 percent of 250? Convert and multiply:

0.18 × 250 = 45.

Going the other way, to ask what percent one number is of another, divide and convert: 30 is 30/120 = 0.25 = 25 percent of 120.

One quiet fact saves time: a percent of a number is symmetric. Twenty-five percent of 80 equals eighty percent of 25, both are 20, because multiplication does not care about order. When one direction looks ugly, the other is sometimes clean.

A harder version compares two percents of the same base rather than a percent of a plain number, and the base itself never needs a value, since it cancels.

WORKED Compare two percents of a shared base

A is 40 percent of B. C is 25 percent of B. A is what percent of C?

Write both in terms of the shared base B, then divide; B cancels out completely.

A = 0.40B, C = 0.25B → A/C = 0.40B / 0.25B = 0.40/0.25 = 1.6

A is 160 percent of C. The value of B was never needed, only the ratio between the two percents; assigning B a convenient number, such as 100, reaches the same answer for anyone who prefers concrete numbers to letters.

▸ Compute percent change using the original value as the base

A percent change measures how much a quantity grew or shrank, always relative to where it started. The rule is to put the amount of change over the original value, then convert to a percent. The single most common error is dividing by the new value instead of the original, the percent change is always measured against the starting amount.

WORKED Compute a percent change

A price rises from 40 to 50. The change is 10, and the original is 40:

percent change = 10 / 40 = 0.25 = 25 percent increase.

A quantity that falls from 50 to 40 has change -10 over the original 50: -10/50 = -20 percent, a 20 percent decrease.

Notice the two are not the same percent, even though the amounts moved by the same 10, because the originals differ.

▸ Translate percent comparisons into multiplicative factors

Phrases like “30 percent more than” and “15 percent less than” are best handled as a single factor. Increasing by p percent multiplies by (1 + p/100); decreasing by p percent multiplies by (1 - p/100). Turning the words into one factor avoids the two-step “find the part, then add it” and is far less error-prone.

WORKED Apply a more-or-less change

A number that is 30 percent more than 60 is 60 multiplied by 1.30:

60 × 1.30 = 78.

A number that is 15 percent less than 200 is 200 multiplied by 0.85: 200 × 0.85 = 170.

The factor does the addition or subtraction for you in one stroke.

▸ Recover an original quantity from a percent change

Here is the situation that separates careful readers from quick ones. Often you are told the amount after a percent change and asked for the original: “after a discount, the price is sixty dollars; what was it before?” The mistake is to take the percent of the final amount. The correct move is to recognize the final amount as the result of multiplying the original by a factor, and then divide by that factor to undo it.

FIGURE 1 To undo a percent change, divide by the factor that produced it, not take the percent of the final amount.

WORKED Recover the original

After a 20 percent discount, a coat costs 60 dollars. The sale price is 80 percent of the original, so:

0.80 × (original) = 60 → original = 60 / 0.80 = 75 dollars.

The same logic recovers a number after a markup: if a number increased by 25 percent equals 90, then 1.25 × n = 90, so n = 90 / 1.25 = 72.

Taking 20 percent of 60 would give the wrong answer, because the discount was 20 percent of the larger original, not of the sale price.

▸ Combine successive percent changes multiplicatively

When two percent changes happen one after another, you cannot simply add the percents, because the second change acts on the new amount, not the original. Instead you multiply the two factors together. The classic case shows why it matters: a 20 percent increase followed by a 20 percent decrease does not return to the start.

FIGURE 2 Two changes in a row multiply their factors. A 20% rise then a 20% fall lands at 96, not back at 100.

WORKED Stack two changes

Start at 100, increase by 20 percent, then decrease the result by 20 percent:

100 × 1.20 × 0.80 = 96.

The result is 96, not 100, because the 20 percent decrease was taken on 120, not on 100.

A markup then a discount works the same way: a 50-dollar item marked up 40 percent then discounted 25 percent is 50 × 1.40 × 0.75 = 52.50 dollars.

▸ Distinguish percentage-point change from percent change

One distinction the test prizes: a change measured in percentage points is not the same as a percent change. If an interest rate goes from 8 percent to 12 percent, it rose by 4 percentage points, the plain difference. But as a percent change, it rose by 4/8 = 50 percent, because the 4-point increase is measured against the original 8. The two answers describe the same shift, but they are different numbers, and the question's wording tells you which it wants.

▸ Model percent relationships in context

The everyday uses are all the same factor reasoning. A tax of 8 percent multiplies a bill by 1.08, so an 80-dollar purchase costs 80 × 1.08 = 86.40. A 20 percent tip on a 60-dollar bill makes the total 60 × 1.20 = 72. A discount multiplies by less than one; a markup, by more than one. And when changes repeat over many periods rather than just once or twice, the factor is raised to a power, the exponential growth you met in the previous domain. For one or two changes, multiply the factors directly; for many, reach for the exponent.

▸ Practice: percentages

1. What is 18 percent of 250? (grid-in)

2. The number 30 is what percent of 120? Enter the percent. (grid-in)

3. A price rises from 40 to 50. What is the percent increase? (grid-in)

4. What number is 30 percent more than 60? (grid-in)

5. After a 20 percent discount, a coat costs 60 dollars. What was the original price, in dollars? (grid-in)

6. A number increased by 25 percent equals 90. What is the number? (grid-in)

7. The value 100 is increased by 20 percent, and then that result is decreased by 20 percent. What is the final value? (grid-in)

Question 8

Which operation increases a quantity by 15 percent?

1. AA. multiply by 0.15
2. BB. multiply by 1.15
3. CC. add 15
4. DD. multiply by 0.85

9. A 50-dollar item is marked up 40 percent, then discounted 25 percent. What is the final price, in dollars? (grid-in)

10. An interest rate rises from 8 percent to 12 percent. By how many percentage points did it rise? (grid-in)

11. For the same rise from 8 percent to 12 percent, what is the percent increase in the rate? (grid-in)

12. A restaurant bill is 60 dollars, and a 20 percent tip is added. What is the total, in dollars? (grid-in)

Question 13

Which is larger?

1. AA. 25 percent of 80
2. BB. 80 percent of 25
3. CC. they are equal
4. DD. cannot be determined

14. A shirt costs 32 dollars after a 20 percent discount. What was its original price, in dollars? (grid-in)

▸ Solutions: percentages

1: 45. “Of” means multiply: 0.18 × 250 = 45. Converting the percent to a decimal first keeps the place value straight.

2: 25. Divide the part by the whole, then convert: 30/120 = 0.25 = 25 percent. The whole here is 120, the number after “of.”

3: 25. Percent change is the change over the original: (50 - 40)/40 = 10/40 = 25 percent. Dividing by 50 instead of 40 is the common error.

4: 78. “30 percent more than” multiplies by 1.30: 60 × 1.30 = 78. Equivalently, 60 + 0.30(60) = 60 + 18 = 78.

5: 75. The sale price is 80 percent of the original: 0.80 × (original) = 60, so the original is 60 / 0.80 = 75. Dividing undoes the discount; taking 20 percent of 60 would be wrong.

6: 72. A 25 percent increase means 1.25 × n = 90, so n = 90 / 1.25 = 72. The final amount is divided by the factor to recover the start.

7: 96. Multiply the factors: 100 × 1.20 × 0.80 = 96. The percents do not cancel, because the decrease acts on 120, not on 100.

Show the answers and explanations

8: B. Increasing by 15 percent multiplies by 1 + 0.15 = 1.15. Choice A finds only the 15 percent part; choice D would decrease by 15 percent.

9: 52.50. Multiply both factors in turn: 50 × 1.40 × 0.75 = 52.50 dollars. Adding 40 percent and subtracting 25 percent as if they cancel would give the wrong total.

10: 4. In percentage points, the rise is the plain difference: 12 - 8 = 4 points.

11: 50. As a percent change, the 4-point rise is measured against the original 8: 4/8 = 0.5 = 50 percent. The same shift, a different measure from the previous question.

12: 72. A 20 percent tip multiplies by 1.20: 60 × 1.20 = 72 dollars.

Show the answers and explanations

13: C. Both equal 20, since 0.25 × 80 = 20 and 0.80 × 25 = 20: a percent of a number is symmetric, as multiplication does not care about order.

14: 40. The discounted price is 80 percent of the original: 0.80 × (original) = 32, so the original is 32 / 0.80 = 40 dollars.

▸ Percentages: chapter summary

A percent is a ratio out of one hundred, so “p percent of” is just multiplication by p/100, and percent change is always the change measured against the original. Turn “more than” and “less than” into a single factor, (1 + p/100) or (1 - p/100); to recover an original from a final amount, divide by that factor rather than taking a percent of the result; and stack successive changes by multiplying their factors, never by adding the percents. Keep percentage points and percent change distinct. The arithmetic is light; the care is all in the setup.

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