Percent

Study Sheet

Percent

Per hundred

Percent Means Out of 100

Concept
The Big Idea
35% is 35 of the 100 small squares: 0.35 of the whole.

“Percent” literally means per hundred, so p%p\% is the fraction p100\dfrac{p}{100}. Finding a percent of a number is just multiplying by that fraction.

Percent means "per hundred": 35%35\% is 35100\tfrac{35}{100}, or 0.350.35. Finding a percent of a number is multiplying by that decimal; finding what percent one number is of another is dividing and then multiplying by 100100. Every percent problem is one of three questions: find the part, find the whole, or find the percent.

Example
Finding 25%25\% of 8080
00%2025%4050%6075%80100%amountpercent
80 is 100%; a quarter of the way along is 25%, which is 20.

25100×80=14×80=20\dfrac{25}{100} \times 80 = \dfrac{1}{4} \times 80 = 20.

Example
Three ways to write it

40%=40100=25=0.440\% = \dfrac{40}{100} = \dfrac25 = 0.4. The fraction and the decimal are the same amount written differently, so use whichever makes the arithmetic easiest.

Side note
Where it goes wrong

Percent change divided by the wrong base. A rise from 4040 to 5050 is 1040=25%\tfrac{10}{40} = 25\%, not 1050=20%\tfrac{10}{50} = 20\%. Always divide by the ORIGINAL amount.

Try it
Try it: two percents

Find 20%20\% of 4545 and 60%60\% of 3030.

Show the answer

20%=1520\% = \tfrac15, and 15×45=9\tfrac15 \times 45 = \mathbf{9}. 60%=0.660\% = 0.6, and 0.6×30=180.6 \times 30 = \mathbf{18}.

In Plain Terms & More Examples

Concept
In plain terms

A percent is a fraction out of 100100. To find a percent of a number, turn it into that fraction and multiply.

Example
Finding 10%10\% of 5050

10100×50=110×50=5\dfrac{10}{100} \times 50 = \dfrac{1}{10} \times 50 = 5.

Example
Finding 75%75\% of 4040
00%1025%2050%3075%40100%amountpercent
Each quarter of 40 is 10, so 75% is 30.

75100×40=34×40=30\dfrac{75}{100} \times 40 = \dfrac{3}{4} \times 40 = 30.

Example
A multi-step problem

A $24\$24 shirt is 25%25\% off, and you also have a $2\$2 coupon. Step 1: 25%25\% of 2424 is 66, so the sale price is 24−6=1824 - 6 = 18. Step 2: the coupon brings it to 18−2=$1618 - 2 = \$16.

Side note
Where it goes wrong

Subtracting instead of taking a part. "10%10\% of 5050" is not 50−10=4050 - 10 = 40; it is one tenth of 5050, which is 55. The percent tells you what fraction of the whole to take.

Try it
Try it: build from 10%

Find 30%30\% of 7070 and 5%5\% of 6060.

Show the answer

10%10\% of 7070 is 77, so 30%30\% is 3×7=213 \times 7 = \mathbf{21}. 10%10\% of 6060 is 66, so 5%5\% is half of that, 3\mathbf{3}.

Finding the Whole and the Percent

Concept
Three questions, one equation

part=percent×whole\text{part} = \text{percent}\times\text{whole}. Cover the unknown: whole =part÷percent= \text{part} \div \text{percent}; percent =part÷whole= \text{part} \div \text{whole} (then ×100\times100).

Example
What is the whole?
0%410%1230%?100%amountpercent
30% is 12, so each 10% is 4, and 100% is 10 of those.

30%30\% of a number is 1212. Whole =12÷0.30=40= 12 \div 0.30 = 40.

Example
What percent?

What percent of 2424 is 1818? Divide the part by the whole: 18÷24=0.7518 \div 24 = 0.75. Multiply by 100100: 75%75\%.

Side note
Where it goes wrong

Dividing the wrong way. For "3434 out of 4040", working out 40÷3440 \div 34 gives about 1.181.18, or 118%118\%, more than the whole quiz. The part goes on top: 34÷40=0.85=85%34 \div 40 = 0.85 = 85\%.

Try it
Try it: what percent?

Sam got 3434 out of 4040 on a quiz. What percent is that? And 15%15\% of what number is 99?

Show the answer

34÷40=0.85=85%34 \div 40 = 0.85 = \mathbf{85\%}. 9÷0.15=609 \div 0.15 = \mathbf{60}.

Word Problems

Concept
Discounts, tax and tips

A discount is a percent of the original price, taken off. Tax and tips are a percent of the price, added on. You can also multiply once: 25%25\% off leaves 75%75\% of the price; 8%8\% tax makes 108%108\%.

Taking 25%25\% off means paying for the other 75%75\%, so the sale price is 0.750.75 times the original in one step. Adding 8%8\% tax means paying 100%+8%=108%100\% + 8\% = 108\% of the price, which is 1.081.08 times it.

Example
A discount
$12 off$12$12$12$48
Four quarters of $48; one quarter comes off.

A $48\$48 jacket is 25%25\% off. Discount 0.25×48=$120.25\times48 = \$12, sale price $36\$36 — or directly 0.75×48=360.75\times48 = 36.

Example
A tip

A 15%15\% tip on a $40\$40 meal: 10%10\% of 4040 is 44, and 5%5\% is half of that, 22. So 15%15\% is 4+2=$64 + 2 = \$6.

Side note
Where it goes wrong

Confusing the discount with the sale price. "25%25\% off a $48\$48 jacket" means you save $12\$12 and pay $36\$36. Answering "$12\$12" to "what does it cost?" gives the saving, not the price.

Try it
Try it: tax

A meal costs $32\$32 and tax is 8%8\%. What is the total?

Show the answer

1.08×32=$34.561.08\times32 = \mathbf{\$34.56}.

Benchmark Percents

Concept
Easy pieces

A few percents are easy: 50%50\% is half, 25%25\% is a quarter, 10%10\% is dividing by 1010, and 1%1\% is dividing by 100100. Build any other percent from these pieces.

"Percent of" is multiplication, so it splits the way multiplication does: 35%35\% of 8080 is 30%30\% of 8080 plus 5%5\% of 8080. Choosing pieces you can find in your head turns most percent problems into mental arithmetic.

Example
Finding 35%35\% of 8080
30%: 245%: 4100%: 80
30% and 5% of the bar, side by side.

10%10\% of 8080 is 88, so 30%30\% is 2424. 5%5\% is half of 10%10\%: 44. Together: 24+4=2824 + 4 = 28.

Example
Finding 3%3\% of 450450

1%1\% of 450450 is 450÷100=4.5450 \div 100 = 4.5. So 3%3\% is 3×4.5=13.53 \times 4.5 = 13.5.

Example
Finding 150%150\% of 2020

100%100\% of 2020 is 2020, and 50%50\% is 1010, so 150%150\% is 3030. A percent above 100100 means more than the whole.

Side note
Where it goes wrong

Moving the point the wrong way. 10%10\% of 4545 is 4.54.5, not 0.450.45 (that is 1%1\%). Sense check: 10%10\% is a tenth, and a tenth of 4545 is about 55.

Try it
Try it: 45%

Find 45%45\% of 6060 using 10%10\% and 5%5\%.

Show the answer

10%10\% of 6060 is 66, so 40%40\% is 2424. 5%5\% is 33. Together: 24+3=2724 + 3 = \mathbf{27}.

Why It Works

Percents are a common denominator. Comparing 720\tfrac{7}{20} with 925\tfrac{9}{25} is awkward; comparing 35%35\% with 36%36\% is instant, because both have been rewritten with denominator 100100. So "p%p\% of NN" =p100×N= \tfrac{p}{100}\times N is nothing more than multiplying by a fraction, and every percent rule is a fraction rule with 100100 on the bottom.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.