What a ratio is: language and notation
A band has 3 trumpets and 2 drums. What short phrase says how those two amounts go together, and still says it when the band gets bigger?
Where you are 3 in What a ratio is
Why this exists
Priya’s school band has 3 trumpets and 2 drums. There are two ways to compare those amounts. You can subtract: there is 1 more trumpet than drum. Or you can say how they go together: for every 3 trumpets there are 2 drums.
Now the school forms four sections, each set up like Priya’s: 12 trumpets and 8 drums. The first comparison has changed, since there are now 4 more trumpets than drums. The second has not: for every 3 trumpets there are still 2 drums, and the band sounds just as balanced. A recipe made bigger, a paint mixed in a bigger tin, a model built to scale: in each of them what stays the same is a “for every”, not a difference.
A ratio is the name and the notation for that kind of comparison. You already know how to compare amounts with >, < and =, which say only which is bigger, and how to say one amount is so many times as many as another. A ratio packs a comparison into two numbers written in a fixed order.
Definition
Day 1: saying and writing a ratio
A ratio compares two amounts by saying how many of one go with how many of the other. When there are a of one thing and b of another, the ratio of the first to the second can be said or written in three ways, and all three mean the same:
- in words with “to”: the ratio of trumpets to drums is 3 to 2;
- with a colon: the ratio of trumpets to drums is 3:2, read aloud “three to two”;
- with “for every”: for every 3 trumpets there are 2 drums.
The two numbers are the terms of the ratio. The order of the terms follows the order of the words: the thing named first gets the first number. The ratio of drums to trumpets in the same band is 2:3. One band, two ratios, depending on which amount you name first.
“For each” is “for every” with a 1 in it. “For each flute there are 4 drums” means for every 1 flute there are 4 drums: the ratio of flutes to drums is 1:4.
Day 2: comparing by multiplying
There are two ways to compare one amount with another.
An additive comparison asks how many more: it subtracts. 8 cats and 2 dogs: there are 6 more cats than dogs.
A multiplicative comparison asks how many times as many: it divides. 8 cats and 2 dogs: 8 = 4 \times 2, so there are 4 times as many cats as dogs, and for each dog there are 4 cats.
A ratio is a multiplicative comparison. “Four times as many cats as dogs” and “the ratio of cats to dogs is 4:1” say the same thing. An additive comparison like “6 more” is not a ratio at all.
Intuition
Picture each amount as counters you can stand in groups.
The ratio 3:2 is one group: 3 trumpets with 2 drums. However many groups the band has, every group looks the same, so the ratio describes all of them at once. That is why “for every 3 trumpets there are 2 drums” is true of a band of 12 and 8 just as of a band of 3 and 2. The ratio does not tell you how many groups there are. It tells you what one group looks like.
Comparing by subtracting and comparing by multiplying ask different things of the same counters.
On the left, each dog is matched with one cat and the leftovers are counted: 6 more cats. On the right, the cats are shared out so that each dog gets the same number: 4 cats for each dog. The first is a difference. The second is a ratio.
Results
Three ways to say one ratio. For a of one thing and b of another, these all say the same: the ratio of the first to the second is a to b; the ratio is a:b; for every a of the first there are b of the second.
Order matters. b:a is the ratio of the second thing to the first. It describes the same situation read the other way round. Unless a and b are equal, a:b and b:a are different ratios.
For each. “For each one of the second there are k of the first” is the ratio k:1, first to second, and 1:k, second to first.
Times as many. If there are k times as many of the first as of the second, the ratio of the first to the second is k:1. In the other direction, a ratio a:b with b not 0 says there are
A difference is not a ratio. Knowing that one amount is so many more than another does not tell you their ratio, and knowing the ratio does not tell you the difference.
Why it is true
Why “for every” survives growth. Say the counters stand in some number of groups, each with a of the first thing and b of the second, as in the band figure. With 4 groups of 3 trumpets and 2 drums there are 4 \times 3 = 12 trumpets and 4 \times 2 = 8 drums. Every group still holds 3 and 2, so “for every 3 trumpets there are 2 drums” stays true however many groups there are. The difference does not stay put: it is 3 - 2 = 1 in each group, so 1 in one group and 4 in four groups. That is why the last result says a difference cannot stand in for a ratio.
Why “times as many” is a ratio k:1. “There are k times as many cats as dogs” means the cats can be split into k rows, each with as many cats as there are dogs. Stand each dog under one column: each column then holds k cats, one from every row. So for each dog there are k cats, which is the ratio k:1.
Why a:b means \displaystyle \frac{a}{b} times as many. In one group there are a of the first and b of the second. By a fraction as division, \displaystyle \frac{a}{b} \times b = a, so inside the group the first amount is \displaystyle \frac{a}{b} times the second. With several groups, both amounts are multiplied by the number of groups, and \displaystyle \frac{a}{b} times the second total is still the first total. For 3:2 with 4 groups: \displaystyle \frac{3}{2} \times 8 = 12.
Worked examples
Examples 1 to 4 go with the first day, 5 to 7 with the second.
Writing a ratio three ways
Priya’s band has 3 trumpets and 2 drums. Write the ratio of trumpets to drums in three ways. Then write the ratio of drums to trumpets with a colon.
SolutionTrumpets are named first, so their count, 3, is the first term:
- 3 to 2;
- 3:2;
- for every 3 trumpets there are 2 drums.
For drums to trumpets the order of the words flips, so the order of the numbers flips: 2:3. Both ratios describe the same band; they read it from opposite ends.
Which way round
Anouk helps at an animal shelter that has 5 cats and 4 dogs. A sign reads “4 dogs for every 5 cats.” What ratio of dogs to cats does the sign give? What is the ratio of cats to dogs? Does the sentence “5 dogs for every 4 cats” describe the same shelter?
SolutionThe sign names dogs first, with 4, so the ratio of dogs to cats is 4:5. Naming cats first gives the ratio of cats to dogs, 5:4. Both match the shelter’s 5 cats and 4 dogs.
“5 dogs for every 4 cats” gives dogs to cats as 5:4, which would mean more dogs than cats. That is a different shelter. Swapping the two numbers while keeping the two words in place changes the ratio.
For each
In a marching band, for each flute there are 4 drums. There are 3 flutes. How many drums are there? Write the ratio of flutes to drums and of drums to flutes.
Solution“For each flute there are 4 drums” means every flute comes with its own group of 4 drums. Three flutes come with three such groups: 3 \times 4 = 12 drums.
“For each” means “for every 1”, so the ratio of flutes to drums is 1:4 and the ratio of drums to flutes is 4:1. The counts 3 and 12 fit it: 3 groups, each 1 flute with 4 drums.
Lengths in the same unit
On sports day Elsa ties a red ribbon 1 foot long and a blue ribbon 9 inches long to the finish post. Write the ratio of the red length to the blue length.
SolutionThe two numbers in a ratio of lengths must count the same unit. “1 to 9” would compare 1 foot with 9 inches as if they were the same size, and they are not. One foot is 12 inches (see Customary length units), so the red ribbon is 12 inches long.
The ratio of red to blue is 12 to 9, or 12:9: for every 12 inches of red there are 9 inches of blue.
More than, or times as many
The shelter now has 8 cats and 2 dogs. Compare them by subtracting and by multiplying. Then write the ratio of cats to dogs and of dogs to cats, each in a “for each” form.
SolutionSubtracting: 8 - 2 = 6, so there are 6 more cats than dogs.
Multiplying: 8 = 4 \times 2, so there are 4 times as many cats as dogs.
The ratio of cats to dogs is 8:2. Share the cats out so each dog has the same number, as in the second figure: 4 cats for each dog, the ratio 4:1. Read the other way, there is 1 dog for every 4 cats, so the ratio of dogs to cats is 1:4. The difference, 6, appears nowhere in these ratios.
The same difference, different ratios
Joaquin’s band has 3 flutes and 1 drum. Desmond’s band has 12 flutes and 10 drums. Show that each band has 2 more flutes than drums. Is each band’s ratio of flutes to drums 3:1?
SolutionThe differences: 3 - 1 = 2 and 12 - 10 = 2. Both bands have 2 more flutes than drums.
Joaquin’s band: 3 = 3 \times 1, so there are 3 times as many flutes as drums, and the ratio of flutes to drums is 3:1.
Desmond’s band: the ratio of flutes to drums is 12:10. If it were 3:1, each of the 10 drums would have 3 flutes, which needs 3 \times 10 = 30 flutes. There are only 12, not even 2 \times 10 = 20, so there are fewer than 2 flutes for each drum. The ratio is not 3:1.
The same “2 more” fits two bands that are put together very differently. That is the sense in which a difference is not a ratio.
A ratio that is not a whole number of times
At another shelter there are 3 cats for every 2 dogs. How many times as many cats as dogs is that? The shelter has 6 dogs. How many cats does it have, and how many more cats than dogs?
SolutionBy the last result, 3:2 means \displaystyle \frac{3}{2} times as many cats as dogs, and \displaystyle \frac{3}{2} = 1\frac{1}{2}. Picture it: one group of 2 dogs and 3 cats gives each dog 1 cat with 1 cat over, and that cat shared between the two dogs is half a cat each. So there are \displaystyle 1\frac{1}{2} cats for each dog.
With 6 dogs, the dogs make 3 groups of 2, and each group comes with 3 cats: 3 \times 3 = 9 cats. Check with “times as many”: \displaystyle 1\frac{1}{2} \times 6 = 6 + 3 = 9. It agrees.
There are 9 - 6 = 3 more cats than dogs.
Boundaries
Same unit when the amounts are measured. Two lengths, two weights or two times must be in the same unit before their numbers go into a ratio, as in Example 4. Two counts of different things, cats and dogs, are fine as they are, since each number counts whole things.
Order is part of the ratio. a:b and b:a are the same ratio only when a and b are equal. A ratio given without saying which amount comes first is not yet a ratio.
A ratio does not give the amounts. 3:2 fits 3 trumpets and 2 drums, and 12 and 8, and many more bands. Finding all the amounts that fit one ratio is scaling a ratio up or down.
Zero. “Times as many” needs the second amount not to be 0: with 5 cats and 0 dogs no number of times 0 makes 5, so there is no “for each dog”. 5 more cats than dogs is still a true difference.
Two amounts at a time. This lesson compares two amounts. A ratio of three, such as trumpets, drums and flutes, and the difference between comparing a part with a part and a part with the whole, are in part-to-part and part-to-whole ratios.
Where this goes next
Once a ratio can be said and written, the next questions are about families of ratios. Scaling a ratio finds every pair of amounts that keeps the same “for every”, and gives each ratio a simplest form. A ratio table and a double number line lay those pairs out so they can be read at a glance. Comparing two ratios decides which of two bands has more flutes for each drum, which Example 6 began. And the “for each” form, with a 1 in it, grows into rates: so many kilometers for each hour, so much cost for each kilogram.
Exercises
Day 1: saying and writing a ratio
- A band has 3 flutes and 7 trumpets. Write the ratio of flutes to trumpets in three ways.
- For the band in Exercise 1, write the ratio of trumpets to flutes with a colon.
- A shelter has 9 dogs and 4 cats. Write the ratio of cats to dogs and the ratio of dogs to cats.
- “For every 5 drums there are 2 flutes.” Write the ratio of flutes to drums and the ratio of drums to flutes.
- On sports day Hamid’s team won 7 races and lost 4. Write the ratio of races won to races lost, and of races lost to races won.
- At the shelter, for each cat there are 3 toys. Write the ratio of toys to cats and of cats to toys.
- The band stands in 3 rows. Each row has 2 drums and 5 trumpets. How many drums and how many trumpets are there? Write the ratio of drums to trumpets for the whole band, and a “for every” sentence that is true of every row.
- For each flute there are 4 drums. There are 5 flutes. How many drums are there? Write the ratio of drums to flutes in a “for each” form.
- A red ribbon is 1 foot long and a blue ribbon is 5 inches long. Write the ratio of red to blue and of blue to red.
- A shelter has cats to dogs in the ratio 5:2. Which of these describe the same shelter? (a) for every 5 cats there are 2 dogs; (b) for every 5 dogs there are 2 cats; (c) the ratio of dogs to cats is 2:5; (d) the ratio of dogs to cats is 5:2.
Day 2: times as many, or more than
- A shelter has 12 cats and 3 dogs. How many more cats than dogs are there? How many times as many cats as dogs? Write the ratio of cats to dogs, and say how many cats there are for each dog.
- There are 5 times as many flutes as trumpets, and there are 4 trumpets. How many flutes are there, and how many more flutes than trumpets? Write the ratio of flutes to trumpets with 1 as its second term.
- There are 4 times as many dogs as cats. Write the ratio of dogs to cats and of cats to dogs.
- On sports day there are 6 judges and 2 more helpers than judges. How many helpers are there? Are there twice as many helpers as judges? Write the ratio of helpers to judges.
- The ratio of flutes to drums is 1:4. How many times as many drums as flutes are there? If there are 7 flutes, how many drums are there?
- For every 2 dogs there are 5 cats. How many times as many cats as dogs is that? If there are 8 dogs, how many cats are there?
- The ratio of trumpets to drums is 5:4. How many times as many trumpets as drums is that? If there are 16 drums, how many trumpets are there, and how many more trumpets than drums?
- Shelter A has 10 cats and 5 dogs. Shelter B has 6 cats and 1 dog. Show that each has 5 more cats than dogs. How many times as many cats as dogs does each have? Write each shelter’s ratio of cats to dogs with 1 as its second term.
- On sports day Elsa runs 9 laps and Priya runs 3. Show that Elsa runs 3 times as many laps as Priya and that she runs 6 more. The next day each runs 2 more laps than before. Which of the two statements is still true?
Reasoning
- A band’s ratio of trumpets to drums is 3:2. Explain why a band whose ratio of drums to trumpets is 3:2 is a different band.
- Explain, with two bands of your own, why knowing that a band has 2 more trumpets than drums does not tell you the ratio of trumpets to drums.
- A shelter has 5 cats and 0 dogs. Can you say how many times as many cats as dogs there are? Can you say how many more? Explain.
- A band has 15 trumpets and 10 drums. Explain, using groups, why “for every 3 trumpets there are 2 drums” is true of it.
- When do a:b and b:a describe the same comparison? Explain, and give an example.
Answers
- 3 to 7; 3:7; for every 3 flutes there are 7 trumpets. Flutes are named first, so 3 comes first.
- 7:3. Trumpets are named first now, so their count comes first.
- Cats to dogs 4:9; dogs to cats 9:4.
- Flutes to drums 2:5; drums to flutes 5:2. The sentence gives 5 drums with 2 flutes, and each ratio puts the count of the thing named first in front.
- Won to lost 7:4; lost to won 4:7.
- Toys to cats 3:1; cats to toys 1:3. “For each cat” means for every 1 cat.
- 3 \times 2 = 6 drums and 3 \times 5 = 15 trumpets. The whole band’s ratio of drums to trumpets is 6:15, and in every row, for every 2 drums there are 5 trumpets: 2:5.
- 5 \times 4 = 20 drums. For each flute there are 4 drums, so drums to flutes is 4:1.
- One foot is 12 inches. Red to blue is 12:5; blue to red is 5:12.
- (a) and (c). (a) is the same ratio in words. (c) is the same shelter read from the other end. (b) and (d) both say there are more dogs than cats, but 5:2 says more cats.
- 12 - 3 = 9 more cats. 12 = 4 \times 3, so 4 times as many. Cats to dogs is 12:3, and there are 4 cats for each dog: 4:1.
- 5 \times 4 = 20 flutes, and 20 - 4 = 16 more flutes than trumpets. Flutes to trumpets is 5:1, since there are 5 flutes for each trumpet.
- Dogs to cats 4:1; cats to dogs 1:4.
- 6 + 2 = 8 helpers. Twice as many would be 2 \times 6 = 12, so no. Helpers to judges is 8:6.
- 4 times as many drums: each flute has 4 drums. With 7 flutes there are 7 \times 4 = 28 drums.
- \displaystyle \frac{5}{2} = 2\frac{1}{2} times as many cats. 8 dogs make 4 groups of 2, so there are 4 \times 5 = 20 cats. Check: \displaystyle 2\frac{1}{2} \times 8 = 20.
- \displaystyle \frac{5}{4} = 1\frac{1}{4} times as many trumpets. 16 drums make 4 groups of 4, so there are 4 \times 5 = 20 trumpets, and 20 - 16 = 4 more trumpets than drums.
- 10 - 5 = 5 and 6 - 1 = 5. Shelter A: 10 = 2 \times 5, so 2 times as many, the ratio 2:1. Shelter B: 6 = 6 \times 1, so 6 times as many, the ratio 6:1.
- 9 = 3 \times 3 and 9 - 3 = 6, so both are true. The next day Elsa runs 11 and Priya 5. 11 - 5 = 6, so “6 more” is still true. 3 \times 5 = 15, not 11, so “3 times as many” is no longer true.
- Drums to trumpets 3:2 means for every 3 drums there are 2 trumpets, so that band has more drums than trumpets. A band with trumpets to drums 3:2 has more trumpets than drums. The numbers are the same but the order of the things is swapped.
- One possible pair: 3 trumpets with 1 drum, and 4 trumpets with 2 drums. Both have 2 more trumpets than drums, but the first has 3 trumpets for each drum (3:1) and the second has 2 for each drum (2:1). The same difference, two different ratios.
- No number of times as many: any number times 0 is 0, never 5, so there is no “for each dog”. There are 5 more cats than dogs, since 5 - 0 = 5. So “how many more” still works, but “how many times as many” does not.
- The 15 trumpets make 5 groups of 3, and the 10 drums make 5 groups of 2. Pair each trumpet group with a drum group: every pair has 3 trumpets and 2 drums.
- When the two terms are equal. For example, 4 cats and 4 dogs: cats to dogs and dogs to cats are both 4:4, one cat for each dog either way. If the terms differ, one order puts the bigger number first and the other does not, so the two ratios describe different situations.