A percent is not a number you can calculate with. Convert it to a decimal and it becomes one, and then every percent question is a single multiplication.
A price tag in Riverside says $48 and the register adds 7.75% sales tax. Nobody adds 7.75 to 48. The percent is an instruction about the size of a share, and it has to be turned into a number before it can do any work.
That conversion is the whole skill. Once the percent is a decimal, the arithmetic is Skill 0.02 again: one multiplication, nothing more.
1. Percent means per hundred
The word percent is Latin for per hundred. Writing 15% is another way of writing 15 hundredths, which as a decimal is 0.15. Shading a hundred square grid makes the two notations the same picture.
Figure 1. The grid holds exactly 100 squares, so the count of shaded squares and the percent are the same number by construction.
2. Convert first, then multiply
Dividing by 100 moves the decimal point two places left. After that, the word of tells you to multiply. Fifteen percent of 60 becomes 0.15 times 60, which is 9.
Figure 2. The conversion happens once, before any arithmetic. Doing it last, or not at all, is where the answers go wrong.
Key idea
Divide the percent by 100 to get a decimal, then multiply. The word of between a percent and an amount always means multiply.
3. The mistake: skipping the conversion, or adding instead
Two errors account for almost all of them. The first keeps the 15 and multiplies, producing 900 from an amount of only 60. The second adds 15 to 60 and reports 75, which answers no question that was asked.
Watch out: a part of 60 cannot be larger than 60
Notice that the right column contains two different correct answers, for two different questions.
Sense check: a percent below 100 must produce an answer smaller than the original amount. If yours is bigger, the conversion was skipped.
4. One multiplier for the whole change
An increase or a decrease does not need two steps. To take 20% off, multiply by 0.80, because 80% of the price is what remains. To add 8% tax, multiply by 1.08. The multiplier itself tells you which direction the amount moved.
Figure 3. A multiplier of exactly 1 leaves an amount untouched, which is why every increase sits above it and every decrease below.
5. Reading percents as decimals
The conversion is mechanical, including the cases that look unusual. A percent above 100 gives a decimal above 1, and a percent with its own decimal point still just moves two places.
Figure 4. Only the last row exceeds 1, and it is the only row whose result is bigger than the amount you started with.
8% of 25 is 2. Sense check: 8% is a small slice, and 2 is a small slice of 25.
Example B
A jacket costs $35 and is marked 20% off. Find the sale price.
Taking 20% off leaves 80%, so multiply by 0.80 in one step. You could also find 0.20 × 35 = 7 and subtract, which gives the same result.
The sale price is $28. Sense check: $28 is less than $35, which any discount must be.
Example C
Riverside County sales tax is 7.75%. Find the total on a $48 purchase.
The tax alone is 0.0775 × 48 = 3.72. Because tax is added, the single-step multiplier is 1.0775.
The total is $51.72. Sense check: $3.72 on $48 is a little under a tenth, which matches a rate near 8%.
Words to know
Percent
A number of parts per hundred.
Decimal multiplier
The percent divided by 100, ready to multiply with.
Of
The word that signals multiplication in a percent question.
Discount
An amount removed, giving a multiplier below 1.
Markup
An amount added, giving a multiplier above 1.
7. A look ahead
Coming next: Skill 0.12, which axis is which, and what a point on a graph means
Skill 0.12 returns to the coordinate plane from Skill 0.07, this time asking what a plotted point says about a real situation. The trap there is reading a y-intercept off the horizontal axis, which reports the wrong number entirely.
Check your understanding
Explain why 15% is written as 0.15 and not as 15.
Explain why 15 × 60 cannot be the answer to 15% of 60.
Describe the difference between 15% of 60 and 15% more than 60.
Explain why taking 20% off can be done by multiplying by 0.80.
State what a multiplier above 1 tells you before you compute anything.
What to carry away
Percent means per hundred, so a percent is a count of hundredths.
Divide by 100 to turn a percent into a decimal you can multiply with.
Convert before you calculate, never after.
The word of between a percent and an amount means multiply.
A percent below 100 must give an answer smaller than the original amount.
Adding the percent to the amount answers no question that was asked.
A decrease and an increase each need only one multiplier, such as 0.80 or 1.08.
A multiplier below 1 shrinks an amount; above 1 grows it; exactly 1 leaves it alone.