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One idea in three forms: comparing quantities by multiplication, per one, and per hundred.
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In this lesson you bring together everything you have learned about ratios. You write part-to-part and part-to-whole ratios, simplify and scale them, share a total in a given ratio with a tape diagram, convert units with a rate, and turn a ratio into a percent so that groups of different sizes can be compared fairly. A ratio, a unit rate and a percent turn out to be the same comparison written three ways.
Earlier in this course you found unit rates, such as a price per pound, filled in ratio tables and plotted them as points on a line through the origin, and found a percent of a number. You can simplify a fraction by dividing the top and the bottom by the same number. This lesson pulls those pieces together. A ratio, a rate and a percent are the same idea seen three ways: two quantities compared by multiplication.
| Term | What it means |
|---|---|
| Ratio | A comparison of two quantities, written 3:2, 3 to 2, or 3 for every 2. |
| Part-to-part ratio | A ratio comparing one group with another group, such as boys to girls. |
| Part-to-whole ratio | A ratio comparing one group with the total, such as boys to all students. |
| Equivalent ratios | Ratios that make the same comparison, because both numbers of one are the other's numbers times the same factor, like 2:3 and 8:12. |
| Lowest terms | A ratio whose two numbers share no factor except 1. |
| Rate and unit rate | A ratio of two different units, such as dollars for yards; a unit rate is the amount for one unit, such as 4 dollars per yard. |
| Percent | A part-to-whole ratio written out of 100; 35% means 35 per hundred. |
| Tape diagram | A row of equal boxes, one for each part of a ratio, used to share a total. |
There are two ways to compare 6 and 9. By subtraction, 9 is 3 more than 6. By multiplication, 6 is two-thirds of 9, or there are 2 of the first for every 3 of the second. A ratio is the second kind of comparison, and it is the kind that survives when things grow. A fruit bowl with 6 apples and 9 oranges has the ratio 6:9. Double the bowl to 12 apples and 18 oranges and the difference has changed, from 3 to 6, but the ratio has not: it is still 2 apples for every 3 oranges.
Ratios come in two kinds. Part to part compares two groups: apples to oranges, 6:9. Part to whole compares one group with everything: apples to all the fruit, 6:15.
Every ratio has relatives. Divide both numbers by the same factor and you get an equivalent ratio in simpler form: 6:9 becomes 2:3. Multiply both by the same factor and you scale it up: 2:3 becomes 20:30.
Two special relatives have names. Scale the second number to 1 and you get a unit rate: 6 apples to 9 oranges is $\tfrac{2}{3}$ of an apple per orange. Scale a part-to-whole ratio so the whole is 100 and you get a percent: 6 out of 15 is the same as 40 out of 100, so 40% of the fruit is apples. A ratio, a unit rate and a percent are one comparison written three ways.
Another way: picture
Draw a tape diagram: two boxes for apples and three for oranges, all the same size. Whatever number goes in one box, the apples are two boxes and the oranges are three. If the bowl holds 15 pieces of fruit, each box is $15 \div 5 = 3$, so there are 6 apples and 9 oranges. The boxes are the ratio; the number inside each box is the scale.
Another way: story
A painter mixes 2 cans of red with 3 cans of white to make a pink. For a bigger wall she uses 4 red and 6 white, and the pink is exactly the same. If she had added one more can of each instead, 3 red and 4 white, the pink would be darker. Mixing keeps its color only when every part is multiplied by the same number.
Order matters in a ratio. '6 apples to 9 oranges' is 6:9, and '9 oranges to 6 apples' is 9:6, a different statement. Always write the numbers in the order the words give them.
To simplify a ratio, divide both numbers by their greatest common factor. For 24:36 the greatest common factor is 12, so $24:36 = 2:3$. If you do not spot the greatest one at once, divide by any common factor and repeat: $24:36 = 12:18 = 6:9 = 2:3$. You are finished when the two numbers share no factor but 1.
To check that two ratios are equivalent, simplify both and compare, or find the factor that turns one into the other. $15:25$ and $6:10$ both simplify to 3:5, so they are equivalent. $4:6$ and $6:8$ are not: they simplify to 2:3 and 3:4. A common trap is to think 4:6 and 6:8 match because both numbers went up by 2. Adding the same amount changes a ratio; only multiplying keeps it.
Many real problems give a ratio and a total and ask for the parts. A tape diagram makes the method clear.
The most common mistake is dividing the total by one number of the ratio, for example $72 \div 5$. The 5 and the 3 are both counts of parts; the total is shared among all 8.
A rate is a ratio of two different kinds of quantity, such as 12 dollars for 3 yards. Its unit rate is the amount for one unit: $12 \div 3 = 4$ dollars per yard. Unit conversions are rates too: 3 feet per yard, 12 inches per foot, 60 minutes per hour. To change 21 feet to yards, divide by the rate: $21 \div 3 = 7$ yards.
A percent is a part-to-whole ratio scaled so the whole is 100. To turn a part-to-part ratio into a percent, first make it part to whole by adding the parts. If the ratio of wins to losses is 3:2, the ratio of wins to games is 3:5, and
$$\frac{3}{5} = \frac{3 \times 20}{5 \times 20} = \frac{60}{100} = 60\%.$$
The percent does not depend on how many games were played. A team with 3 wins in 5 games and a team with 30 wins in 50 games have both won 60% of their games. That is the power of all three forms: they let you compare groups of different sizes fairly.
Which is better, 12 wins in 20 games or 14 wins in 25? The raw numbers mislead: 14 is more wins, but out of more games. To compare fairly, put the two ratios on the same footing. There are three good ways.
Choose whichever makes the arithmetic easiest. Totals that divide 100, like 20, 25 and 50, make percents quick. Totals like 7 or 12 are often easier as unit rates or with a common multiple. Whatever you choose, compare part to whole with part to whole, never part to part with part to whole.
To solve a ratio problem:
To check:
Builders often make concrete from cement, sand and gravel in the ratio 1:2:3 by volume, with water added to make it workable. That is $1 + 2 + 3 = 6$ parts. To fill a garden-path form that needs 18 cubic feet of dry mix, one part is $18 \div 6 = 3$ cubic feet, so the builder needs 3 cubic feet of cement, 6 of sand and 9 of gravel. Cement is one part out of six, so it is $\tfrac{1}{6}$ of the mix, about 17%. If the builder ran low on cement and just added extra gravel, the ratio would change and the concrete would be weaker. Scaling every part by the same factor is what keeps the path as strong as the recipe promises, whether the batch is one wheelbarrow or a whole truckload.
Food labels in the United States list a '% Daily Value' for nutrients. It is a percent: the amount in one serving as a part of the amount recommended for a whole day. For sodium, the Daily Value is 2,300 milligrams. A bowl of soup with 690 milligrams of sodium has
$$\frac{690}{2{,}300} = \frac{30}{100} = 30\%$$
of the Daily Value, because 690 and 2,300 both divide by 23, giving 30 and 100. The ratio 690:2,300 is the same comparison as 30:100. If you eat two bowls, both numbers of the comparison double to 1,380:2,300, which is 60%. The percent lets a shopper compare a can of soup with a bag of chips without doing any division in the store: 5% or less is low, and 20% or more is high.
Adding instead of multiplying. 2:3 scaled up is 4:6, not 3:4. Adding the same amount to both numbers changes the comparison.
Swapping the order. Apples to oranges, 2:3, is not oranges to apples, 3:2.
Using part to part for a percent. With 3 wins to 2 losses, the percent of wins is 3 out of 5, 60%, not 3 out of 2.
Dividing a total by one number of the ratio. To share in 5:3, divide by the 8 parts, not by 5.
Converting units the wrong way. Feet to yards makes the number smaller, so divide by 3; yards to feet makes it larger, so multiply.
A fruit bowl holds 6 apples and 9 oranges. Write the ratio of apples to oranges.
$6:9$
Part to part, in the order the words give.
Simplify the ratio by dividing both numbers by 3.
$6:9 = (6 \div 3):(9 \div 3) = 2:3$
3 is the greatest number dividing both, so 2 apples for every 3 oranges.
Find the whole.
$6 + 9 = 15 \text{ pieces of fruit}$
The whole is both parts together.
Write apples to all the fruit.
$6:15 = 2:5$
Part to whole; dividing by 3 again gives lowest terms.
Write it as a percent.
$\dfrac{2}{5} = \dfrac{2 \times 20}{5 \times 20} = \dfrac{40}{100} = 40\%$
A percent is a part-to-whole ratio scaled to 100.
Maya worked 5 hours and Leo 3 hours mowing lawns. They share the 72 dollars they earned in the ratio 5:3. Draw the tape diagram.
$\text{Maya: 5 boxes}, \qquad \text{Leo: 3 boxes}$
Every box holds the same amount of money.
Count the boxes.
$5 + 3 = 8 \text{ parts}$
The whole 72 dollars fills all 8 boxes.
Find one part.
$72 \div 8 = 9 \text{ dollars}$
Share the total equally among the boxes.
Find Maya's share.
$5 \times 9 = 45 \text{ dollars}$
Maya has 5 boxes.
Find Leo's share.
$3 \times 9 = 27 \text{ dollars}$
Leo has 3 boxes.
Check both ways.
$45 + 27 = 72, \qquad 45:27 = 5:3$
The shares make the total, and dividing both by 9 gives back the ratio.
Team A won 12 of its 20 games. Find its losses.
$20 - 12 = 8$
Every game was either a win or a loss.
Write Team A's wins to losses and simplify.
$12:8 = 3:2$
Dividing by 4: three wins for every two losses.
Write Team A's wins as a percent of its games.
$\dfrac{12}{20} = \dfrac{12 \times 5}{20 \times 5} = \dfrac{60}{100} = 60\%$
Wins to games is part to whole; times 5 turns 20 into 100.
Team B won 14 of its 25 games. Find its losses.
$25 - 14 = 11$
The rest of the games.
Write Team B's wins to losses.
$14:11$
14 and 11 share no factor but 1, so this is already in lowest terms, and it is hard to compare with 3:2 by eye.
Write Team B's wins as a percent.
$\dfrac{14}{25} = \dfrac{14 \times 4}{25 \times 4} = \dfrac{56}{100} = 56\%$
Times 4 turns 25 into 100.
Compare the percents.
$60\% > 56\%$
Team B won more games, but Team A won a larger share of the games it played, so Team A has the better record.
Write and simplify tomatoes to peppers.
$14:6 = 7:3$
Divide both by 2.
Find the whole garden.
$14 + 6 = 20 \text{ plants}$
Add the parts.
Write the tomatoes as a percent of the garden.
Three bands sold tickets in advance and at the door. Match each band's ratio of advance tickets to door tickets with the same ratio in lowest terms.
| 5:7 | 2:3 | 3:5 | |
|---|---|---|---|
| 20:28 | |||
| 12:18 | |||
| 18:30 |
Two friends run a lemonade stand. They agree to split the 80 dollars they earned in the ratio 7:3, because of the hours each worked. Complete the worked solution to find each share.
Count the equal parts.
$7 + 3 =$ p parts
Picture a tape diagram: 7 boxes for the first friend and 3 for the second, all the same size.
Find the size of one part.
$80 \div \text{parts} =$ k dollars
The money is split into that many equal boxes.
Find the first friend's share.
$7 \times \text{one part} =$ x dollars
The first friend gets 7 of the boxes.
Find the second friend's share.
$3 \times \text{one part} =$ y dollars
The second friend gets 3 of the boxes.
Check that the shares add up to the total.
$\text{first share} + \text{second share} = 80$
Together the two shares must be all the money.
An animal shelter has 14 cats and 19 dogs. Complete the ratio of cats to dogs and the ratio of cats to all the animals.
Cats to dogs is 14:pp, and cats to all the animals is 14:pw.
A trail mix uses peanuts and raisins in the ratio 8:3. A scout troop uses 32 cups of peanuts. Find the scale factor, the cups of raisins, and the total cups of trail mix.
The scale factor is f, the troop needs rai cups of raisins, and the mix is tot cups in all.
Three classes vote on a field trip. Class A votes 9 yes to 11 no, Class B votes 20 yes to 5 no, and Class C (two classes together) votes 26 yes to 24 no. Complete the table with each total and the percent that voted yes.
| Total votes | Percent yes | |
|---|---|---|
| Class A | ||
| Class B | ||
| Class C |
A craft store sells ribbon for 2 dollars per yard. Jada needs 24 feet of ribbon for a banner. How many yards is that, and what will it cost?
Jada needs yd yards of ribbon, which cost cost dollars.
In a parking lot the ratio of electric cars to gas cars is 3:7. There are 90 cars in the lot. How many are electric, and what percent of the cars is that?
n of the cars are electric, which is pct percent of the lot.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A juice blend is apple and grape in the ratio 9:11. A bottle holds 1100 milliliters. How many milliliters of apple juice and of grape juice are in it, and what percent of the bottle is apple?
The bottle has apple mL of apple juice and grape mL of grape juice, and it is pct percent apple.
You can use ratios, rates and percents together. Without looking: how do you share 60 dollars in the ratio 3:2, and what percent of a group is the first part of a 3:2 ratio?
16. Your turn: a garden has 14 tomato plants and 6 pepper plants, step 3
$\dfrac{14}{20} = \dfrac{70}{100} = 70\%$
Multiply top and bottom by 5.