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Percent is a ratio with 100 as its second part, which is why it can be found as a part.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn that percent means per hundred, so 35% is the rate 35 out of 100. You will change fractions and decimals into percents, find a percent of a number, find what percent one number is of another, and find the whole when you know a part and its percent.
You can write a fraction with 100 on the bottom, such as $\frac{35}{100}$, and you know it is the same as the decimal 0.35, or 35 hundredths. In the last two lessons you worked with ratio tables and unit rates. A percent is one more rate: it always compares a number with 100, so every tool from those lessons still works.
| Term | What it means |
|---|---|
| Percent | Per hundred. 35% means 35 out of every 100, the fraction $\frac{35}{100}$, or 0.35. |
| Whole | The full amount the percent is taken of. It is 100% of itself. |
| Part | The amount that is some percent of the whole. |
| Benchmark percent | An easy percent used to build others: 50% is half, 25% is a quarter, 10% is a tenth, 1% is a hundredth. |
| Deposit | Part of a price paid in advance, often a percent of the price. |
| Sales tax | An amount added to a price, set as a percent of the price. |
The word percent means per hundred. 35 percent, written 35%, is a rate of 35 for every 100. It can be written three ways that all mean the same number:
$$35\% = \frac{35}{100} = 0.35.$$
Because a percent is a ratio, a percent of a quantity is a part of a whole. To find 35% of 240, multiply the whole by the decimal:
$$0.35 \times 240 = 84.$$
Every percent question uses the same three numbers: the percent, the whole and the part. You are always given two of them and asked for the third.
| You know | You want | Do this |
|---|---|---|
| percent and whole | part | decimal $\times$ whole |
| part and whole | percent | part $\div$ whole, then $\times$ 100 |
| part and percent | whole | part $\div$ decimal |
The whole is always 100%. If the percent is less than 100, the part is smaller than the whole; if it is more than 100, the part is bigger.
Another way: picture
Draw a strip of paper and mark it from 0% at the left to 100% at the right. Underneath, mark the same strip from 0 to 240. The two scales line up like a double number line, so the mark for 10% sits over 24, the mark for 50% over 120, and the mark for 35% over 84. Any percent question is a question about where two marks line up.
Another way: steps
A square split into 10 rows of 10 small squares has exactly 100 small squares, so every small square is 1% of the whole. Shade 35 squares and you have shaded 35%. Shade 3 whole rows and a half row and you have shaded the same 35%, because each row is 10%.
The grid also shows why percents work for wholes that are not 100. If the whole grid stands for 240 people, then each small square stands for $240 \div 100 = 2.4$ people, and 35 squares stand for $35 \times 2.4 = 84$ people. That is the unit rate idea from Lesson 1: find what 1% is worth, then multiply by the number of percent you want. It is slower than multiplying by 0.35, but it explains why multiplying by 0.35 is right.
The grid can show more than 100% too. Two full grids and 50 more squares are 250%, which is two and a half times the whole.
A fraction, a decimal and a percent are three ways of writing one amount. To turn a fraction into a percent, make the bottom 100: $\frac{3}{4} = \frac{75}{100} = 75\%$, multiplying top and bottom by 25. To turn a decimal into a percent, multiply by 100, which moves the decimal point two places to the right: $0.6 = 60\%$ and $0.08 = 8\%$. To go back, divide by 100.
| Fraction | Decimal | Percent |
|---|---|---|
| $\frac{1}{2}$ | 0.5 | 50% |
| $\frac{1}{4}$ | 0.25 | 25% |
| $\frac{1}{5}$ | 0.2 | 20% |
| $\frac{1}{10}$ | 0.1 | 10% |
| $\frac{1}{100}$ | 0.01 | 1% |
Be careful with small decimals. 0.5 is 50%, not 5%, and 0.05 is 5%. Counting hundredths settles it: 0.05 is 5 hundredths.
Some percents are easy to find in your head, and they can be combined to make the others.
To find 35% of 240, find 10% first: $240 \div 10 = 24$. Then 30% is $3 \times 24 = 72$, and 5% is half of 24, which is 12. So 35% is $72 + 12 = 84$, the same answer as $0.35 \times 240$. This works because 30% and 5% of the same whole can be added: together they are 35% of it.
Benchmarks are also the best way to estimate. 48% of 300 is close to 50%, so the answer is close to 150. If a calculation gives 1,440 or 14.4, you know something went wrong.
The hardest percent question gives you the part and the percent. Suppose 84 students walk to school and that is 30% of the school. How many students are there?
Think of it as a ratio table with a percent column and a students column:
| Percent | Students |
|---|---|
| 30 | 84 |
| 10 | 28 |
| 100 | 280 |
Divide both columns by 3 to reach 10%, then multiply both by 10 to reach 100%. The whole school is 280 students. The same answer comes from dividing: $84 \div 0.3 = 280$, because $0.3 \times (\text{whole}) = 84$ and division undoes multiplication. The table method is often easier to do without a calculator, and it shows why the answer is right.
Step 1: Name the three numbers. Write down which of percent, part and whole you know. The whole is the amount the question says the percent is of.
Step 2: Write the percent as a decimal or as a fraction of 100. 35% is 0.35 or $\frac{35}{100}$. This is allowed because percent means per hundred.
Step 3: Choose the operation.
Step 4: Write the answer with its meaning: a number of students, a number of dollars, or a percent.
Step 5: Check.
If a check fails, the usual cause is dividing the wrong way round or using the part where the whole belongs.
Most US states charge sales tax, a percent of the price added at the register. Pennsylvania's state sales tax, for example, is 6%. A pair of sneakers priced at 65 dollars in a 6% state costs
$$0.06 \times 65 = 3.90$$
dollars in tax, so the total is $65 + 3.90 = 68.90$ dollars. A quick check with benchmarks: 1% of 65 is 0.65, and 6 of those is 3.90.
Tax works the other way too. If a receipt shows 4.20 dollars of tax at 6%, the price before tax was $4.20 \div 0.06 = 70$ dollars. That is finding the whole from a part, exactly like the survey example.
Every packaged food in the United States has a Nutrition Facts label, and many lines on it show a % Daily Value. The label compares the amount in one serving with a daily amount set by the Food and Drug Administration. For sodium, that daily value is 2,300 milligrams.
A can of soup with 690 milligrams of sodium in a serving shows $690 \div 2300 = 0.3$, which is 30% Daily Value. If you eat the whole can and it holds 2 servings, you get $2 \times 690 = 1380$ milligrams, which is 60% of the day's sodium in one meal. The FDA's own guide says 5% Daily Value or less is low and 20% or more is high, so this soup is high in sodium.
Percent is used here because it makes different foods easy to compare, even when their servings are different sizes.
A store sign says 25% off everything. A jacket has a tag price of 48 dollars. The discount is 25% of 48, and 25% is a quarter, so the discount is $48 \div 4 = 12$ dollars. The sale price is $48 - 12 = 36$ dollars.
There is a shortcut. Paying 25% less means paying $100\% - 25\% = 75\%$ of the tag price, and $0.75 \times 48 = 36$ dollars, the same answer in one step. If the store then adds 6% sales tax, the tax is taken of the sale price, not the tag price: $0.06 \times 36 = 2.16$ dollars, for a total of 38.16 dollars. Keeping track of which whole each percent is taken of is the whole skill.
Using the wrong whole. The whole is the amount the percent is of. In 18 of the 24 shots, the whole is 24, so the percent is $18 \div 24$, not $24 \div 18$.
Moving the decimal point the wrong number of places. 0.5 is 50%, and 0.05 is 5%. Say the decimal in hundredths to be sure: 0.05 is five hundredths.
Thinking a percent can never be more than 100. A price that doubles is 200% of what it was. More than 100% just means the part is bigger than the whole it is compared with.
Treating a percent as a number of things. 20% of a class is not 20 students unless the class has exactly 100 students. A percent always needs its whole.
Adding percents of different wholes. 10% of 50 and 10% of 200 are not 20% of anything useful. Percents can be added only when they are of the same whole.
Find 35% of 240. Write the percent as a fraction of 100.
$35\% = \dfrac{35}{100}$
Percent means per hundred, so 35 percent is 35 out of 100.
Write the fraction as a decimal.
$\dfrac{35}{100} = 0.35$
35 hundredths is written 0.35.
Multiply the whole by the decimal.
$0.35 \times 240 = 84$
A part of a whole is found by multiplying the whole by the part as a decimal.
Check the size.
$84 < 240$
35 percent is less than 100 percent, so the part must be less than the whole.
Check with benchmarks: 10% is 24 and 5% is 12.
$3 \times 24 + 12 = 72 + 12 = 84$
30 percent and 5 percent make 35 percent, and the two methods agree.
A player makes 18 of her 24 free throws. What percent does she make? Write the part over the whole.
$\dfrac{18}{24}$
A percent compares a part with its whole, and the whole is all 24 shots.
Simplify the fraction by dividing top and bottom by 6.
$\dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}$
A simpler fraction is easier to turn into hundredths.
Multiply top and bottom by 25 to make the bottom 100.
$\dfrac{3 \times 25}{4 \times 25} = \dfrac{75}{100}$
An equal fraction with 100 on the bottom can be read straight as a percent.
Read the hundredths as a percent.
$\dfrac{75}{100} = 75\%$
75 out of every 100 is 75 percent.
Check by dividing and multiplying by 100.
$18 \div 24 = 0.75, \qquad 0.75 \times 100 = 75$
The decimal method gives the same percent.
Check by working forward.
$0.75 \times 24 = 18$
75 percent of her 24 shots gives back the 18 she made.
84 students walk to school, and that is 30% of the school. How many students are in the school? Write the percent next to the part.
$30\% \quad\leftrightarrow\quad 84 \text{ students}$
The walkers are the part, and they match 30 out of every 100.
Divide both by 3 to find 10%.
$30 \div 3 = 10, \qquad 84 \div 3 = 28$
Dividing both parts of a ratio by the same number keeps it true.
Multiply both by 10 to reach 100%.
$10 \times 10 = 100, \qquad 28 \times 10 = 280$
100 percent is the whole school.
State the whole.
$280 \text{ students}$
The whole is the amount the percent was taken of.
Get the same answer by dividing by the decimal.
$84 \div 0.3 = 280$
$0.3 \times (\text{whole}) = 84$, and dividing by 0.3 undoes the multiplication.
Check by working forward.
$0.3 \times 280 = 84$
30 percent of the answer gives back the 84 walkers.
Check the size.
$280 > 84$
Only 30 percent walk, so the whole school must be much bigger than the number who walk.
Find 10% of the bill.
$45 \div 10 = 4.50 \text{ dollars}$
Ten percent is one tenth of the whole.
Double it to get 20%.
$4.50 \times 2 = 9.00 \text{ dollars}$
Twenty percent is two lots of ten percent.
Add the tip to the bill.
Check with 120%.
Write the fraction $2/5$ and the decimal $0.25$ as percents.
$2/5$ = a% and $0.25$ = b%
Complete the worked solution to find 25% of 260 using 10% and 5%.
Find 10% by dividing by 10.
$260 \div 10 =$ a
Ten percent is 10 out of 100, which is one tenth of the whole.
Find 5% by halving 10%.
$(10\%) \div 2 =$ b
Five percent is half of ten percent.
Multiply 10% by 2 to get 20%.
$(10\%) \times 2 =$ c
20 percent is 2 lots of 10 percent.
Add the 5% to the 20%.
$(20\%) + (5\%) =$ d
20 percent and 5 percent together make 25 percent of the same whole.
Check with the decimal method.
$0.25 \times 260 = (\text{your answer})$
Two different methods landing on the same number is a strong check.
What is 45% of 120?
45% of 120 is answer.
A basketball player made 12 of her 60 free throws this season. What percent of her free throws did she make?
She made answer% of her free throws.
20% of a number is 44. What is the number?
The number is answer.
In a school survey, 70% of the students said they walk to school. That is 112 students. How many students are in the school?
Ten percent of the school is ten students, so the school has whole students.
A store sells a bicycle for 340 dollars. It asks for a 25% deposit today and the rest when the item is delivered. How many dollars are left to pay on delivery?
The deposit is deposit dollars, and rest dollars are left to pay on delivery.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
In a school survey, 40% of the students said they walk to school. That is 56 students. How many students are in the school?
Ten percent of the school is ten students, so the school has whole students.
You can use percent as a rate out of 100 in all three directions. Without looking: how would you find 35% of 60, and how would you find the whole if 30% of it is 84?
17. Your turn: a family leaves a 20% tip on a 45-dollar meal. What do they pay in all?, step 3
$45 + 9 = 54 \text{ dollars}$
They pay the meal and the tip, which is 120 percent of the meal.
17. Your turn: a family leaves a 20% tip on a 45-dollar meal. What do they pay in all?, step 4
$1.2 \times 45 = 54$
Paying the bill plus 20 percent is the same as paying 120 percent of the bill.