This page is about three connected ideas: ratio, proportion, and percent. It starts from the beginning. It assumes you can multiply and divide, including with a calculator, and that you know what a fraction such as 3/4 means. If fractions feel shaky, the Fractions family page is a good place to go first.
Each part uses only what the parts before it have explained, so it is best read in order. Keep paper and a pencil beside you. When a worked example asks a question, try it yourself before you read the answer.
It is a long page. Two or three sittings is a good pace. Parts 1 to 4 are about ratios, Parts 5 to 8 about rates and proportions, and Parts 9 to 11 about percent. There are places to practice along the way, and a check at the end with the reasoning for every answer.
Two ways to compare two amounts
At a corner store, a small coffee costs $2 and a large coffee costs $4. There are two ways to compare those two prices.
The first way subtracts. It answers the question “how much more?” The answer is called the difference: the large costs $2 more than the small.
The second way divides. It answers the question “how many times as much?” The small price fits into the large price 2 times, so the large costs 2 times as much, or twice as much.
Both answers are true. With these two prices they even come out to the same number, 2, which can hide the fact that they are different kinds of answer. The next example keeps them apart.
Two tenants each get a rent increase of $150 a month. Rosa was paying $1,500. Kwame was paying $3,000. Is the increase the same for both of them?
Compare by subtracting, and the increase is the same: each pays $150 more than before.
Now compare by dividing. For Rosa, how many times does $150 fit into $1,500? 1,500 ÷ 150 = 10. So the increase is one tenth of what she was paying. For Kwame, 3,000 ÷ 150 = 20, so his increase is one twentieth of what he was paying.
Measured against what each of them was already paying, the same $150 is a bigger increase for Rosa: a tenth is twice as big as a twentieth. Comparing by dividing takes account of the size of the amount you started with. Comparing by subtracting does not.
Everything on this page is about the second kind of comparison, the kind made by multiplying and dividing. A ratio is the name for that kind of comparison. A proportion says that two ratios are the same. A percent is a ratio in which the second number is always 100. The rest of the page explains each of these in turn.
What a ratio is
A ratio compares two amounts by division. It tells you how much there is of one thing for a certain amount of another.
Take an evening class with 12 women and 8 men. The ratio of women to men is 12 to 8.
Three ways to write it
The same ratio can be written in three ways, and all three are read aloud the same way, “12 to 8”:
- with the word to: 12 to 8
- with a colon, the two dots: 12 : 8
- as a fraction: 128
The GED test uses all three, so it is worth being comfortable with each.
The order matters
The ratio of women to men is 12 : 8. The ratio of men to women is 8 : 12. The words tell you which number comes first: the amount named first in the words is the number written first. Reading a ratio backward is one of the most common mistakes on the whole topic, so it is worth checking the words every time.
Simplifying a ratio
A ratio can be made simpler in the same way a fraction can: divide both numbers by the same number. Both 12 and 8 can be divided by 4. 12 ÷ 4 = 3 and 8 ÷ 4 = 2, so the ratio 12 : 8 is the same as 3 : 2.
The picture shows what that means. The 12 women and 8 men can be sorted into 4 equal groups, and each group has 3 women and 2 men. So the ratio 3 : 2 says: for every 3 women in the class, there are 2 men.
A ratio is in simplest form when no whole number bigger than 1 divides both of its numbers.
Part to part, and part to whole
The ratio 12 : 8 compares one part of the class with another part: women with men. It does not say anything directly about the whole class.
The whole class is 12 + 8 = 20 students. So the ratio of women to the whole class is 12 : 20, which simplifies to 3 : 5. That means 3/5 of the class are women. It is tempting to read 3 : 2 as “3/2 of the class,” but that cannot be right, since 3/2 is more than the whole class.
When a ratio compares two parts, add the two numbers to get the whole. With a ratio of 3 : 2, the whole is 3 + 2 = 5 shares, so the first part is 3/5 of the whole and the second part is 2/5.
Practice: ratios
Type a ratio with a colon, like 3 : 2. Type a fraction with a slash, like 3/5.
A soccer team has 9 players on the field and 6 on the bench. What is the ratio of players on the field to players on the bench, in simplest form?
The ratio is 9 : 6. Both numbers divide by 3, so it is 3 : 2. The field is named first, so 9 comes first.
On the same team, what is the ratio of players on the bench to players on the field?
Now the bench is named first: 6 : 9, which simplifies to 2 : 3.
What fraction of the whole team is on the field?
The whole team is 9 + 6 = 15 players. 9 of the 15 are on the field: 9/15, which simplifies to 3/5.
A bag holds 10 red marbles and 15 green marbles. What is the ratio of red to green, in simplest form?
10 : 15. Both divide by 5, giving 2 : 3.
A drink is made with 1 cup of juice for every 4 cups of seltzer. What fraction of the drink is juice?
The whole drink is 1 + 4 = 5 cups for every batch. 1 of the 5 is juice, so the fraction is 1/5. Not 1/4: the 4 counts only the seltzer.
Equivalent ratios: more of the same mix
A recipe for a pot of rice and beans uses 3 cups of rice for every 2 cans of beans. The ratio of rice to beans is 3 : 2.
For a party you want to make twice as much. You use 6 cups of rice and 4 cans of beans. For three times as much, 9 cups of rice and 6 cans of beans. Each time, you have made more food, but the mix is the same: it tastes the same, because there is the same amount of rice for each can of beans.
Ratios that describe the same mix are called equivalent ratios. The ratios 3 : 2, 6 : 4, 9 : 6 and 12 : 8 are all equivalent. Each one simplifies to 3 : 2.
A ratio table
A ratio table lists equivalent ratios side by side. Each column is one size of the recipe.
| Batches | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Cups of rice | 3 | 6 | 9 | 12 | 15 |
| Cans of beans | 2 | 4 | 6 | 8 | 10 |
Look at how each column is made from the first one. For 4 batches, both numbers in the first column are multiplied by 4: 3 × 4 = 12 cups of rice, and 2 × 4 = 8 cans of beans. The number that both amounts are multiplied by is called the scale factor.
To make an equivalent ratio, multiply both numbers by the same number, or divide both numbers by the same number. Whatever you do to one, do to the other.
Try it: make more batches
Press the buttons to make more or fewer batches. Watch the two amounts grow together: each new batch adds 3 cups of rice and 2 cans of beans, so both amounts are always the same number of batches of the recipe.
Working backward, and in halves
You have 15 cups of rice. How many cans of beans do you need?
How many batches is 15 cups of rice? One batch uses 3 cups, and 15 ÷ 3 = 5. So this is 5 batches: the scale factor is 5.
Beans: 2 × 5 = 10 cans.
You have only 1 can of beans. How much rice goes with it?
1 can is half of 2 cans, so this is half a batch. The scale factor is 1/2, or 0.5.
Rice: 3 × 0.5 = 1.5 cups, which is 1½ cups. A scale factor does not have to be a whole number.
Why adding does not keep a ratio
There is a tempting wrong way to make more of a mix: add the same amount to both numbers. It is worth looking at closely, because it feels reasonable and gives the wrong answer.
Start with the recipe, 3 cups of rice and 2 cans of beans. Suppose you add 3 to each number, to get 6 cups of rice and 5 cans of beans. Is that the same mix?
Compare it with a real double batch, which is 6 cups of rice and 4 cans. Adding 3 to each gave 6 cups and 5 cans: one can too many for that much rice. The mix has changed. There is more bean in every spoonful.
The bigger the amount you add, the further off it gets. Add 10 to each and you have 13 cups of rice and 12 cans of beans: almost one can for every cup, which is nothing like the recipe.
A double batch: 6 and 4
Adding 3 to each: 6 and 5
Part 1 explains why. A ratio is a comparison by dividing, so it is kept the same by multiplying or dividing. Adding the same amount to both numbers keeps the difference the same (here, one more cup of rice than cans of beans), and the difference is the other kind of comparison.
3 pounds of apples cost $6. What do 5 pounds cost?
The tempting answer: 5 pounds is 2 more pounds than 3, so add $2 and get $8. That is adding, and it is wrong.
Instead, find what 1 pound costs: $6 ÷ 3 = $2. Then 5 pounds cost 5 × $2 = $10.
A quick check: 5 pounds is more than one and a half times 3 pounds, because one and a half times 3 is only 4.5. So the price must be more than one and a half times $6, which is $9. $8 is too little. $10 fits.
When one amount is doubled, is the other one doubled too? When one is multiplied by 5, is the other multiplied by 5? If the answer is yes, you are keeping the ratio. If you find yourself adding the same number to both, stop and check.
Rates, and the price of one
So far, both amounts in each ratio were the same kind of thing: people and people, or cups and cans in the same pot. When a ratio compares two different kinds of amount, such as dollars and hours, miles and gallons, or dollars and pounds, it is called a rate.
Rates are usually said with the word per. Per means “for each one.” A job that pays $18 per hour pays $18 for each one hour. Apples at $2 per pound cost $2 for each one pound. Whenever you see the word per, you can read it as “for each one.”
The unit rate
A unit rate is a rate for exactly one of the second thing: the price of one pound, the pay for one hour, the miles for one gallon. (“Unit” here means “one.”) You find a unit rate by dividing, because dividing shares an amount out equally.
4 pounds of chicken cost $10. What is the price per pound?
Share the $10 out equally over the 4 pounds: 10 ÷ 4 = 2.50. The unit rate is $2.50 per pound.
Once you know the price of one, you can find the price of any amount by multiplying. 7 pounds cost 7 × $2.50 = $17.50.
Which number goes on top
Dividing the other way, 4 ÷ 10 = 0.4, is also a true rate: 0.4 pounds per dollar. It answers a different question: how much chicken one dollar buys. To get the rate you were asked for, look at the word after per. The thing after “per” is what you divide by. Dollars per pound means dollars ÷ pounds. Miles per gallon means miles ÷ gallons.
Which is the better buy?
Unit rates make it possible to compare packages of different sizes. Many supermarkets print a unit price in small type on the shelf tag, so shoppers can compare. Here is how that number is worked out.
A 16-ounce box of cereal costs $4.80. A 24-ounce box of the same cereal costs $6.48. Which is the better buy?
The prices cannot be compared as they stand, because the boxes are different sizes. So find the price of one ounce in each box.
Small box: 4.80 ÷ 16 = 0.30, which is 30 cents per ounce. Large box: 6.48 ÷ 24 = 0.27, which is 27 cents per ounce.
The large box is the better buy. Every ounce costs 3 cents less.
Practice: rates
Type a number. A dollar sign is fine but not needed.
You earn $136 for 8 hours of work. What is your pay per hour?
Pay per hour means dollars ÷ hours: 136 ÷ 8 = $17 per hour.
6 bagels cost $7.50. What is the price of one bagel?
7.50 ÷ 6 = $1.25 per bagel.
A car travels 150 miles on 5 gallons of gas. How many miles per gallon is that?
Miles per gallon means miles ÷ gallons: 150 ÷ 5 = 30 miles per gallon.
Which is the better buy: 3 cans for $4.50, or 5 cans for $7.00?
Price per can: 4.50 ÷ 3 = $1.50, and 7.00 ÷ 5 = $1.40. The 5-can pack costs 10 cents less per can.
At $2.50 per pound, what do 6 pounds cost?
Multiply the unit rate by the number of pounds: 6 × 2.50 = $15.
Proportions: two ratios that are equal
A proportion is a statement that two ratios are equal. It is usually written as two fractions with an equals sign between them:
32 = 64
This is read “3 is to 2 as 6 is to 4.” It says that the mix 3 : 2 and the mix 6 : 4 are the same mix, which Part 3 showed they are.
Telling whether two ratios are equal
There are two dependable ways to check.
Simplify both. 6 : 4 simplifies to 3 : 2, and 9 : 6 simplifies to 3 : 2. They simplify to the same ratio, so they are equal.
Compare the unit rates. This is usually quicker with prices.
Store A sells 4 pounds of rice for $10. Store B sells 6 pounds for $15. Store C sells 5 pounds for $13. Which of these prices are in proportion?
Price per pound at A: 10 ÷ 4 = $2.50. At B: 15 ÷ 6 = $2.50. At C: 13 ÷ 5 = $2.60.
A and B have the same unit rate, so their ratios are equal: 104 = 156 is a true proportion. C charges a little more per pound, so its ratio is not equal to the other two.
Setting up a proportion
Most proportion questions give you three numbers and ask for a fourth. Before you can find the missing number, you have to write the proportion down correctly, and that is where most mistakes happen. One rule prevents nearly all of them:
The same kind of amount goes in the same place on both sides. If dollars are on top on the left, dollars must be on top on the right. If pounds are underneath on the left, pounds must be underneath on the right. Write the labels first, then the numbers.
The missing number is written as a letter, usually x. A letter used this way is called an unknown: it stands for the number we are trying to find.
If 4 pounds cost $10, what do 7 pounds cost? Set up the proportion.
Write the labels first: dollars on top, pounds underneath, on both sides.
The left side is the price you know: $10 for 4 pounds. The right side is the price you want: x dollars for 7 pounds.
The proportion is 104 = x7. The next part shows how to find x.
Solving a proportion
There are three ways to find the missing number in a proportion. All three give the same answer. The first two are quick when the numbers are friendly, and the third works every time.
Way 1: find the scale factor
Solve 35 = x20.
Look at the numbers you know on the bottom: 5 on the left, 20 on the right. What is 5 multiplied by to make 20? 20 ÷ 5 = 4. The scale factor is 4.
The top must be multiplied by the same scale factor: x = 3 × 4 = 12.
Check: 12/20 simplifies to 3/5. It works.
Way 2: find the unit rate
If 4 pounds cost $10, what do 7 pounds cost?
One pound costs 10 ÷ 4 = $2.50. So 7 pounds cost 7 × 2.50 = $17.50.
This way takes an extra step, but every number along the way has a meaning you can check: $2.50 is a sensible price for one pound.
Way 3: cross-multiplying
When no scale factor is easy to see and the unit rate is an awkward number, there is a method that always works. It is called cross-multiplying, and it comes from a fact about proportions that is worth understanding rather than just remembering.
In the proportion 32 = 64, multiply along the two diagonals of the picture: 3 × 4 = 12, and 2 × 6 = 12. The two answers are equal.
That is true of every proportion: the two diagonal products are always equal. They are called the cross products.
Two fractions that are equal stay equal when both are multiplied by the same number. Multiply both sides of 32 = 64 by both bottom numbers, 2 and 4, which is multiplying by 8. On the left, 3/2 × 8 is 3 × 8 ÷ 2, which is 3 × 4 = 12. On the right, 6/4 × 8 is 6 × 8 ÷ 4, which is 6 × 2 = 12.
Multiplying by both bottom numbers gets rid of the fractions, and what is left on each side is one of the diagonal products: 3 × 4 on the left, and 6 × 2 on the right. Two equal amounts, multiplied by the same number, give equal answers. So the diagonal products are always equal.
Here is the method, using the rule.
Solve 32 = x7. (This is the rice and beans recipe: how much rice goes with 7 cans of beans?)
Multiply the diagonals and set them equal: 2 × x = 3 × 7, which is 2x = 21. (2x means 2 times x.)
Divide both sides by the number next to x. x = 21 ÷ 2 = 10.5. So 7 cans of beans need 10½ cups of rice.
Check that it makes sense: 7 cans is a little more than 3 batches of 2 cans, and 10.5 cups is a little more than 3 batches of 3 cups. It fits.
On a map, 2 inches stands for 3 miles. Two places are 5 inches apart on the map. How far apart are they really?
Labels first: inches on top, miles underneath. 23 = 5x.
Cross products: 2 × x = 3 × 5, so 2x = 15, and x = 15 ÷ 2 = 7.5 miles.
A car goes 140 miles on 4 gallons of gas. How many gallons does it need for 245 miles?
Labels first: miles on top, gallons underneath. 1404 = 245x.
Cross products: 140 × x = 4 × 245, so 140x = 980, and x = 980 ÷ 140 = 7 gallons.
Or by unit rate: 140 ÷ 4 = 35 miles per gallon, and 245 ÷ 35 = 7. The same answer, as it must be.
Put your answer back into the proportion and see whether the two ratios are equal. Then ask whether it makes sense: if the second amount is bigger than the first on one row, it should be bigger on the other row too.
Practice: solve for x
Use whichever of the three ways you like.
Solve x6 = 1015.
Cross products: 15 × x = 6 × 10 = 60, so x = 60 ÷ 15 = 4. Check: 4/6 and 10/15 both simplify to 2/3.
Solve 49 = 12x.
Scale factor: 4 × 3 = 12, so x = 9 × 3 = 27.
Solve 7x = 2130.
Scale factor: 7 × 3 = 21, so x × 3 = 30, and x = 10.
Solve 2.54 = x12.
Scale factor: 4 × 3 = 12, so x = 2.5 × 3 = 7.5.
5 notebooks cost $8.75. At the same price, what do 8 notebooks cost?
Unit rate: 8.75 ÷ 5 = $1.75 per notebook. 8 × 1.75 = $14. Or cross products: 5x = 8.75 × 8 = 70, so x = 14.
Proportional, or not?
Two amounts are proportional when they always keep the same ratio: when one is doubled, the other is doubled; when one is multiplied by 10, so is the other. Pay at a fixed hourly rate is proportional to the hours worked. The cost of gas at a fixed price per gallon is proportional to the number of gallons.
Many amounts go up together without being proportional, and a proportion gives the wrong answer for them. Two quick tests tell the difference.
- The doubling test. If you double the first amount, does the second amount double?
- The zero test. If the first amount is zero, is the second amount zero too? No hours worked, no pay. No gallons, no cost.
If either test fails, the amounts are not proportional.
A car service charges $6 to start a ride, and then $3 for each mile. Is the cost proportional to the miles ridden?
Zero test: a ride of 0 miles still costs the $6 starting charge. The cost is not zero. The test fails.
Doubling test: 1 mile costs 6 + 3 = $9. 2 miles cost 6 + 6 = $12, not $18. The test fails again.
So the cost is not proportional to the miles. A starting charge, a joining fee, or any amount you pay before anything else happens will always break a proportion.
When Tanya was 5, her brother was 10. Tanya is now 10. How old is her brother?
Tanya’s age doubled, from 5 to 10, and it is tempting to double her brother’s age too, to 20.
Zero test: when Tanya was born, age 0, her brother was not 0. He was 5. The ages are not proportional. What stays the same is the difference: he is always 5 years older. So he is 10 + 5 = 15.
What it looks like on a graph
If you plot pairs of amounts on a coordinate grid (the page The Coordinate Plane explains how), a proportional relationship always makes a straight line through the origin, the point (0, 0) where the two axes cross. The line for pay starts at zero pay for zero hours. The line for the car service is also straight, but it starts at $6, above the origin, so it is not proportional.
Practice: proportional or not?
The cost of gas, at a fixed price per gallon, and the number of gallons.
Zero gallons cost nothing, and twice the gallons cost twice as much. Both tests pass.
The total cost of a gym that charges a $50 joining fee and then $30 a month, and the number of months.
At 0 months you have still paid the $50 fee. The zero test fails.
The distance around a square (its perimeter) and the length of one side.
The perimeter is always 4 times the side. A side of 0 gives a perimeter of 0, and doubling the side doubles the perimeter.
A person’s height and their age.
A baby is not 0 inches tall at age 0, and a 20-year-old is not twice as tall as a 10-year-old.
The cost of pizza at $3.50 a slice, and the number of slices.
0 slices cost $0, and each extra slice adds the same $3.50. Both tests pass.
Percent means “out of 100”
The word percent comes from the Latin words per centum, which mean “for each hundred.” So 45 percent, written 45%, means 45 for each 100, or 45 out of 100.
That makes a percent a ratio in which the second number is always 100. 45% is the ratio 45 : 100, which is the fraction 45100, which is the decimal 0.45 (forty-five hundredths). These are four ways of writing one amount.
In the picture, a square is cut into 100 small squares, and 45 of them are shaded. The shaded part is 45% of the square.
Why percent is useful
A percent measures everything against the same whole, 100, so amounts with different wholes can be compared.
In one class, 18 of 24 students passed a test. In another class, 21 of 30 passed. Which class did better?
21 is more than 18, but the second class is bigger, so the counts alone do not settle it. Compare by dividing, as in Part 1.
First class: 18 ÷ 24 = 0.75, which is 75 out of 100, or 75%. Second class: 21 ÷ 30 = 0.70, which is 70%. The first class did better.
Changing a percent to a decimal, and back
Percent to decimal: divide by 100. Dividing by 100 moves the decimal point two places to the left. 45% becomes 0.45. 8% becomes 0.08: the zero after the decimal point keeps the 8 in the hundredths place. 125% becomes 1.25.
Decimal to percent: multiply by 100. Multiplying by 100 moves the decimal point two places to the right. 0.3 becomes 30%. 0.075 becomes 7.5%.
(If a whole number has no decimal point showing, it is at the end: 45 is 45.0, so 45% is 0.45.)
Changing a percent to a fraction, and back
Percent to fraction: write the percent over 100 and simplify. 45% = 45100, and dividing top and bottom by 5 gives 920.
Fraction to percent: divide the top by the bottom to get a decimal, then move the decimal point two places to the right. 3/8 = 3 ÷ 8 = 0.375, which is 37.5%.
Percents worth knowing by heart
| Percent | Decimal | Fraction |
|---|---|---|
| 100% | 1 | the whole |
| 75% | 0.75 | 3/4 |
| 50% | 0.5 | 1/2 |
| 25% | 0.25 | 1/4 |
| 20% | 0.2 | 1/5 |
| 10% | 0.1 | 1/10 |
| 5% | 0.05 | 1/20 |
| 1% | 0.01 | 1/100 |
More than 100%, and less than 1%
100% is the whole amount. A percent over 100 means more than the whole: 150% of $40 is one and a half times $40, which is $60.
A percent under 1 means less than one hundredth. 0.5% is half of 1%, which is 0.005 as a decimal, not 0.5. Read the percent sign carefully: 0.5 and 0.5% are a hundred times apart.
Try it: shade a percent
Tap a small square to shade every square up to it. The page shows the amount you shaded as a percent, a decimal, and a fraction.
Practice: percents, decimals, fractions
For a percent, type the number; the % sign is fine but not needed. Type a fraction with a slash, like 3/5.
Write 35% as a decimal.
Divide by 100: move the decimal point two places left. 35% = 0.35.
Write 0.08 as a percent.
Multiply by 100: move the decimal point two places right. 0.08 = 8%.
Write 3/4 as a percent.
3 ÷ 4 = 0.75, which is 75%.
Write 60% as a fraction in simplest form.
60/100. Divide top and bottom by 20: 3/5.
Write 1/8 as a percent.
1 ÷ 8 = 0.125, which is 12.5%.
Write 150% as a decimal.
150 ÷ 100 = 1.5. A percent over 100 gives a decimal over 1.
The three percent questions
Every percent question is about three numbers:
- the whole: the amount the percent is taken of, the amount everything is measured against;
- the part: the piece of the whole that the question is about;
- the percent.
The picture shows all three at once. The bar is the whole. It has two rulers: along the top, percents from 0% to 100%; along the bottom, amounts from 0 up to the whole. Both rulers measure the same bar, so a mark on one lines up with its match on the other.
Here the whole is 60, and 15% of the bar is shaded. The shaded part lines up with 9 on the bottom ruler. So 15% of 60 is 9.
One proportion for all three
The two rulers make a proportion. The part is to the whole as the percent is to 100:
partwhole = percent100
A percent question gives you two of the three numbers and asks for the third. So there are only three kinds of percent question, and all three are solved with this one proportion, by the cross-multiplying of Part 7.
What is 15% of 60?
The word of points to the whole: 15% of 60, so the whole is 60. The percent is 15. The part is unknown.
x60 = 15100. Cross products: 100 × x = 60 × 15 = 900. So x = 900 ÷ 100 = 9.
9 is 15% of what number?
Here “of what number” is the whole, and it is unknown. The part is 9 and the percent is 15.
9x = 15100. Cross products: 15 × x = 9 × 100 = 900. So x = 900 ÷ 15 = 60.
9 is what percent of 60?
The whole is 60 (it follows “of”). The part is 9. The percent is unknown.
960 = x100. Cross products: 60 × x = 9 × 100 = 900. So x = 900 ÷ 60 = 15, which means 15%.
Getting the whole right is the most important step. The word of usually points straight to it: in “15% of 60,” the whole is 60. When there is no “of,” ask what the percent is being measured against. It is usually the amount that came first: the price before the sale, the rent before the increase, the pay before the deductions.
A shorter way when the part is missing
When the part is missing, there is a one-step way: change the percent to a decimal and multiply it by the whole. 15% of 60 is 0.15 × 60 = 9. This is the same proportion with the steps combined, and it is how the site’s percent quizzes write it: the part is the percent times the whole. When the whole or the percent is missing, the proportion is the safer route.
Try it: the percent bar
Choose a whole, then slide to choose a percent. The bar shows the part that goes with it.
Practice: the three questions
First decide which of the three numbers is missing. Then set up the proportion.
What is 20% of 45?
Part missing. x/45 = 20/100, so 100x = 900, and x = 9. Or 0.20 × 45 = 9.
30 is 25% of what number?
Whole missing. 30/x = 25/100, so 25x = 3,000, and x = 120. Check: a quarter of 120 is 30.
12 is what percent of 48?
Percent missing. 12/48 = x/100, so 48x = 1,200, and x = 25. So 25%.
What is 150% of 40?
Part missing. 1.5 × 40 = 60. Over 100%, so the part is more than the whole.
7 is what percent of 20?
7/20 = x/100, so 20x = 700, and x = 35. So 35%.
18 is 40% of what number?
Whole missing. 18/x = 40/100, so 40x = 1,800, and x = 45.
Percents in everyday life
Most percent questions on the GED test are set in everyday situations: sales tax, tips, discounts, raises, and rent increases. They use the same three questions from Part 10, often with one extra step at the end.
Finding 10% and 1% in your head
10% of an amount is one tenth of it: move the decimal point one place to the left. 10% of $35 is $3.50. 1% is one hundredth: move the decimal point two places left. 1% of $35 is $0.35.
Other percents can be built from these. 20% is 10% twice: $7.00. 5% is half of 10%: $1.75. 15% is 10% plus 5%: $3.50 + $1.75 = $5.25. This is a quick way to work out a tip, and a good way to check a calculator answer.
Sales tax
New York City sales tax is 8.875%. What is the tax on a $40 purchase, and what is the total?
The whole is $40, since the tax is a percent of the price. Change the percent to a decimal: 8.875% = 0.08875.
Tax: 0.08875 × 40 = $3.55. Total: 40 + 3.55 = $43.55.
Check with 10%: 10% of $40 is $4.00, and the tax should be a little less than that. $3.55 is.
Discounts, and what you pay
Sneakers priced at $85 are 20% off. What do you pay?
The discount is 20% of $85: 0.20 × 85 = $17. You pay 85 − 17 = $68.
A shorter way: if 20% is taken off, you pay the other 80%, since the two pieces must make 100%. 0.80 × 85 = $68, in one step.
The same idea works for increases. Adding 8.875% tax means paying 108.875% of the price: 1.08875 × 40 = $43.55. A 5% raise means the new pay is 105% of the old pay.
Percent change
Percent change measures how much an amount went up or down, as a percent of where it started. The starting amount is called the original, and it is the whole.
percent change = the changethe original × 100
A rent goes from $1,600 a month to $1,680. What is the percent increase?
The change: 1,680 − 1,600 = $80. The original: $1,600.
80 ÷ 1,600 = 0.05, which is 5%.
A common mistake is to divide by the new rent instead: 80 ÷ 1,680 is about 4.8%. The arithmetic is right, but it measures the change against the wrong amount. The change is always measured against the original.
A price drops from $50 to $40. What is the percent decrease?
The change is $10. The original is $50. 10 ÷ 50 = 0.2, which is a 20% decrease.
Down and back up do not cancel
Suppose that $50 price, now $40, is later raised by 20%. Does it go back to $50?
No. The 20% raise is taken of $40, the new price: 0.20 × 40 = $8. The price becomes $48, not $50.
The two 20% changes were taken of two different wholes, $50 and then $40, so they were two different amounts of money: $10 off, then $8 back on. Whenever there are two percent changes in a row, the second one is a percent of the new amount.
Practice: percents in daily life
Round money to the nearest cent.
With New York City sales tax of 8.875%, what is the tax on a $25 purchase?
0.08875 × 25 = 2.21875, which rounds to $2.22.
You leave a 15% tip on a $60 meal. How much is the tip?
10% of $60 is $6. 5% is half of that, $3. 15% is $6 + $3 = $9.
A $48 shirt is 25% off. What do you pay?
You pay the other 75%: 0.75 × 48 = $36. Or: the discount is $12, and 48 − 12 = 36.
Your pay goes from $20 an hour to $21 an hour. What is the percent increase?
The change is $1. The original is $20. 1 ÷ 20 = 0.05, which is 5%.
A price drops from $80 to $60. What is the percent decrease?
The change is $20. The original is $80. 20 ÷ 80 = 0.25, which is 25%.
The words on this page, in one place
Difference: the answer to a subtraction; how much more one amount is than another.
Ratio: a comparison of two amounts by division, written 3 to 2, 3 : 2, or 3/2. The order matters.
Simplest form: a ratio or fraction whose two numbers cannot both be divided by any whole number bigger than 1.
Part to part, part to whole: a ratio that compares one part with another part; a ratio that compares one part with the whole. Add the parts to get the whole.
Equivalent ratios: ratios that describe the same mix, such as 3 : 2 and 6 : 4.
Scale factor: the number both amounts are multiplied by to make an equivalent ratio.
Ratio table: a table of equivalent ratios, side by side.
Rate: a ratio that compares two different kinds of amount, such as dollars and hours.
Per: for each one.
Unit rate: a rate for exactly one of the second thing, such as the price of one pound. Found by dividing.
Unit price: the price for one unit, such as one ounce, printed on many shelf tags.
Double number line: two number lines, one above the other, with matching amounts lined up.
Proportion: a statement that two ratios are equal, such as 3/2 = 6/4.
Unknown: a letter, usually x, standing for the number you are trying to find.
Cross products: in a proportion, the two diagonal products. In every proportion they are equal.
Proportional: two amounts that always keep the same ratio. They pass the doubling test and the zero test, and their graph is a straight line through the origin.
Percent (%): for each hundred; a ratio whose second number is 100.
Whole, part: in a percent question, the amount the percent is taken of, and the piece of it the question is about.
Percent change: the change divided by the original amount, written as a percent.
Original: the starting amount, before the change.
Fourteen questions on the whole page
Answer each one, then press Check. Each answer comes with its reasoning. Type a ratio with a colon, like 3 : 2, and a fraction with a slash, like 3/5. For a percent, type the number; the % sign is fine but not needed.
- 1.
A class has 14 women and 10 men. What is the ratio of women to men, in simplest form?
14 : 10. Both numbers divide by 2, giving 7 : 5. Women are named first, so 14 comes first.
- 2.
In the same class, what fraction of the students are men?
The whole class is 14 + 10 = 24. The men are 10 of the 24: 10/24, which simplifies to 5/12.
- 3.
Which ratio is equivalent to 4 : 6?
4 : 6 simplifies to 2 : 3, and so does 6 : 9 (divide both by 3). Adding the same number to both, as in 5 : 7 or 6 : 8, does not keep a ratio.
- 4.
A recipe uses 2 cups of rice for every 5 cups of water. How much water goes with 6 cups of rice?
6 cups of rice is 3 times 2 cups, so the scale factor is 3. Water: 5 × 3 = 15 cups.
- 5.
5 pounds of potatoes cost $4.25. What is the price per pound?
Dollars per pound means dollars ÷ pounds: 4.25 ÷ 5 = $0.85, which is 85 cents a pound.
- 6.
Which is the better buy: 6 rolls for $4.20, or 10 rolls for $6.50?
Price per roll: 4.20 ÷ 6 = $0.70, and 6.50 ÷ 10 = $0.65. The pack of 10 costs 5 cents less per roll.
- 7.
Solve 68 = 15x.
Cross products: 6 × x = 8 × 15 = 120, so x = 120 ÷ 6 = 20. Check: 15/20 and 6/8 both simplify to 3/4.
- 8.
On a map, 1 inch stands for 4 miles. Two towns are 3.5 inches apart on the map. How many miles apart are they?
Each inch is 4 miles, so 3.5 inches is 3.5 × 4 = 14 miles. As a proportion: 1/4 = 3.5/x, so x = 14.
- 9.
Which of these is proportional?
Pay at a fixed hourly rate passes both tests: no hours, no pay, and twice the hours, twice the pay. Each of the others has an amount that is there before anything else happens, so the zero test fails.
- 10.
Write 0.6 as a percent.
Multiply by 100, which moves the decimal point two places right: 0.6 = 0.60 = 60%.
- 11.
What is 35% of 80?
The part is missing. 0.35 × 80 = 28. Or: x/80 = 35/100, so 100x = 2,800, and x = 28.
- 12.
18 is what percent of 24?
The whole is 24, since it follows “of.” 18/24 = x/100, so 24x = 1,800, and x = 75. So 75%.
- 13.
30 is 60% of what number?
The whole is missing. 30/x = 60/100, so 60x = 3,000, and x = 50. Check: 60% of 50 is 30.
- 14.
A price rises from $40 to $50. What is the percent increase?
The change is $10. The original is $40. 10 ÷ 40 = 0.25, which is 25%. (Dividing by $50 instead would give 20%, which measures against the wrong amount.)
After this page
Six quizzes go further into each part of this page, with worked examples, ten questions each, and an answer key. They are listed here in the order of this page.