What Is a Ratio?

What a ratio says about two quantities, how to write it and read it in the right order, when two quantities can be compared by a ratio, and how to draw a ratio as a bar model.

Before you start

Quantities are measured in units of measurement, such as centimetres (cm), metres (m), kilograms (kg), millilitres (ml) and dollars. 1 m = 100 cm.

Two numbers can be compared by subtracting: 5 is $5 - 3 = 2$ more than 3. This lesson shows a second way of comparing them.

Comparing Two Quantities

Siti makes a bracelet with 3 red beads and 5 blue beads. One way to compare the two numbers is to subtract: there are 2 more blue beads than red beads. A second way is to set the two numbers side by side: 3 red beads to 5 blue beads. This second way of comparing is called a ratio.

We say that the ratio of the number of red beads to the number of blue beads is $3 : 5$ , and we read it as "3 to 5". The numbers 3 and 5 are the terms of the ratio.

The drawing below shows this ratio. Each bead is one box, and the boxes of each colour are joined to make a bar. A drawing like this is called a bar model. This lesson comes back to it.

Definition: ratio

A ratio is a comparison of quantities of the same kind. It shows how much there is of one quantity against how much there is of another.

The ratio of $a$ to $b$ is written $a : b$ and read " $a$ to $b$ ". Here $a$ and $b$ stand for numbers greater than 0. They are the terms of the ratio: $a$ is the first term and $b$ is the second term.

A ratio has no unit of measurement.

Two ribbons are 7 cm and 9 cm long. The ratio of the length of the first ribbon to the length of the second ribbon is $7 : 9$ . We do not write "7 cm : 9 cm". Both lengths are measured in centimetres and the ratio only sets the two numbers side by side, so the unit of measurement is left out. In your working, write the quantities with their units of measurement first, as "7 cm and 9 cm" or "7 cm to 9 cm", and then the ratio without them, $7 : 9$ .

A ratio can compare three quantities as well. In this lesson every ratio compares two.

The Order of the Terms

The terms of a ratio are written in the same order as the quantities are named. For Siti's bracelet, the ratio of the number of red beads to the number of blue beads is $3 : 5$ , and the ratio of the number of blue beads to the number of red beads is $5 : 3$ . If Siti wrote that the ratio of red beads to blue beads is $5 : 3$ , she would be describing a different bracelet, one with more red beads than blue beads.

Rule: the order of the terms

Write the terms in the order in which the quantities are named in words: the quantity named first gives the first term. Unless $a$ and $b$ are equal, $a : b$ and $b : a$ are different ratios.

Example 1: A ratio from a sentence

Class 1B has 21 girls and 19 boys.
(a) Find the ratio of the number of boys to the number of girls.
(b) Find the ratio of the number of girls to the number of boys.
(c) Are the two ratios the same?

Solution

(a) The question names the boys first, so the number of boys is the first term, even though the sentence about the class gives the girls first.

Number of boys : number of girls $= 19 : 21$

(b) Now the girls are named first, so the number of girls is the first term.

Number of girls : number of boys $= 21 : 19$

(c) $19 : 21$ has the smaller number first and $21 : 19$ has the larger number first. They are different ratios, and each is right only with its own words.

(a) The ratio of the number of boys to the number of girls is $19 : 21$ .
(b) The ratio of the number of girls to the number of boys is $21 : 19$ .
(c) No. The two ratios are different.

The Same Kind and the Same Unit of Measurement

Two quantities are of the same kind when they can be measured in the same unit of measurement, or counted as things of the same sort: two lengths, two masses, two amounts of money, or two numbers of things that are counted, such as students, beads or animals. Only quantities of the same kind can be compared by a ratio.

A coach travels 150 km in 2 hours. A distance and a time are quantities of different kinds: no unit of measurement measures both, and it makes no sense to ask what fraction of 150 km the 2 hours are. So " $150 : 2$ " is not a ratio. Comparing quantities of different kinds gives a rate, and a rate keeps its units of measurement: 150 km in 2 hours. Rates are taught in the chapter on rate and speed.

Quantities of the same kind must also be in the same unit of measurement before the ratio is written. Ravi's pencil is 13 cm long and a metre rule is 1 m long. Both are lengths, but the ratio of the length of the pencil to the length of the rule is not $13 : 1$ : that ratio would make the pencil the longer of the two. In centimetres the lengths are 13 cm and 100 cm, so the ratio is $13 : 100$ . Ratios of quantities given in different units of measurement have a lesson of their own later in this chapter.

Before you write a ratio

Check that the two quantities are of the same kind. If they are not, there is no ratio.
Check that they are in the same unit of measurement. If they are not, change one of them so that both are in the same unit of measurement (for example, 1 m = 100 cm).
Write the two numbers in the order asked, without the unit of measurement.

Example 2: Can it be a ratio?

Can the two quantities be compared by a ratio? If they can, write the ratio of the first quantity to the second.
(a) 7 kg of rice and 4 kg of flour
(b) 300 ml of milk and 200 g of sugar
(c) $9 and $20
(d) $12 and 3 kg of prawns

Solution

(a) Both are masses, and both are in kilograms. The ratio of the mass of rice to the mass of flour is $7 : 4$ .

(b) The milk is measured by its volume (in ml) and the sugar by its mass (in g). A volume and a mass are different kinds of quantity, so there is no ratio.

(d) An amount of money and a mass are different kinds of quantity, so there is no ratio. Together they give a rate: the price of the prawns for each kilogram.

(a) Yes. The ratio is $7 : 4$ .
(b) No. A volume and a mass cannot be compared by a ratio.
(c) Yes. The ratio is $9 : 20$ .
(d) No. Money and mass cannot be compared by a ratio.

A Part and a Part, a Part and the Whole

Look again at Class 1B, with its 19 boys and 21 girls. The boys and the girls are two parts of the class, and the class is the whole: $19 + 21 = 40$ students. A ratio can compare a part with another part, or a part with the whole.

A part with a part: the ratio of the number of boys to the number of girls is $19 : 21$ .
A part with the whole: the ratio of the number of boys to the number of students in the class is $19 : 40$ .
The whole with a part: the ratio of the number of students in the class to the number of girls is $40 : 21$ .

Read the words to see which is wanted. "To the number of students in the class", "to the total number" and "to all the students" each mean the whole. If the whole is not given, add all the parts that make it up. If the whole is made of two parts, and the whole and one of them are given, the other part is the whole minus the part that is given.

Example 3: Ratios from a table

The table shows how the members of a Robotics Club come to school.

How they come to school	Number of members
By bus	11
By MRT	7
On foot	5

(a) Find the ratio of the number of members who come by MRT to the number who come on foot.
(b) Find the ratio of the number of members who come by bus to the total number of members.
(c) Find the ratio of the total number of members to the number who come on foot.

Solution

(a) Both are parts of the club. From the table, 7 members come by MRT and 5 come on foot. The ratio is $7 : 5$ .

(b) The total is not in the table, so add all the parts: $11 + 7 + 5 = 23$ members.

11 members come by bus, so the ratio of the number who come by bus to the total number of members is $11 : 23$ .

Notice that in (a) the terms 7 and 5 add up to 12, which is not the number of members: the members who come by bus are in neither term. The sum of the terms stands for the whole only when the two quantities compared make up the whole between them.

(a) The ratio of the number who come by MRT to the number who come on foot is $7 : 5$ .
(b) The ratio of the number who come by bus to the total number of members is $11 : 23$ .
(c) The ratio of the total number of members to the number who come on foot is $23 : 5$ .

Ratios on a Bar Model

A bar model is a drawing of a ratio, like the drawing of Siti's beads. Its equal boxes are called units. It is drawn by three rules.

How a bar model is drawn

Draw one bar for each quantity, one under the other, all starting at the same line on the left. Write beside each bar what it stands for.
Cut every bar into equal units. A unit is the same size in every bar of the model.
Give each bar as many units as its term of the ratio. (In this lesson every term is a whole number.)

The ratio of Ali's savings to Mei Ling's savings is $4 : 7$ . In the bar model below, Ali's bar has 4 units and Mei Ling's bar has 7 units. The bracket on the right joins the two bars: together they have $4 + 7 = 11$ units.

In a bar model the word unit means one of the equal parts of a bar. It is not a unit of measurement such as cm or kg. The model does not say how much money 1 unit is: it may be $1, $5 or any other amount. What the model does say is that 1 unit is the same amount in both bars. If Ali's savings are cut into 4 equal parts, Mei Ling's savings are 7 parts of that size. So a ratio tells us the sizes of two quantities against each other. It does not tell us the quantities themselves.

To read a ratio from a bar model, count the units in each bar.

Example 4: Ratios from a bar model

The bar model shows the number of storybooks and the number of comics on Hui Min's shelf. There are no other books on the shelf.

(a) Find the ratio of the number of storybooks to the number of comics.
(b) Find the ratio of the number of comics to the number of storybooks.
(c) Find the ratio of the number of storybooks to the total number of books on the shelf.
(d) Can you tell from the model how many storybooks there are?

Solution

(a) Count the units. The bar for the storybooks has 5 units and the bar for the comics has 3 units, and every unit is the same size. The ratio is $5 : 3$ .

(b) The comics are named first, so their 3 units give the first term: $3 : 5$ .

(c) The shelf has only storybooks and comics, so all the books are $5 + 3 = 8$ units. The ratio of the number of storybooks to the total number of books is $5 : 8$ .

(d) No. The model shows 5 units of storybooks, but not how many books 1 unit stands for. If 1 unit is 1 book, there are 5 storybooks. If 1 unit is 4 books, there are $5 \times 4 = 20$ storybooks.

(a) The ratio of the number of storybooks to the number of comics is $5 : 3$ .
(b) The ratio of the number of comics to the number of storybooks is $3 : 5$ .
(c) The ratio of the number of storybooks to the total number of books is $5 : 8$ .
(d) No. The model gives the ratio, not the number of books.

Common Mistakes

Writing the terms in the order of the numbers, not of the words. "A class has 21 girls and 19 boys. Find the ratio of the number of boys to the number of girls" has the answer $19 : 21$ , not $21 : 19$ .
Taking the sum of the terms as the whole when the two quantities do not make up the whole. In Example 3 the ratio of the number who come by MRT to the number who come on foot is $7 : 5$ . The $7 + 5 = 12$ members are only those two parts together. The club has 23 members, not 12: the members who come by bus are in neither term. To find the whole, add all the parts.
Writing a ratio for quantities of different kinds. 150 km and 2 hours give a rate, not a ratio.
Leaving a unit of measurement in the ratio, or not making the units of measurement the same. 7 cm and 9 cm are in the ratio $7 : 9$ , not "7 cm : 9 cm". 13 cm and 1 m are in the ratio $13 : 100$ , not $13 : 1$ .
Reading the terms as the actual numbers. A ratio of boys to girls of $2 : 3$ does not say that there are 2 boys and 3 girls. It says that the boys are 2 units and the girls are 3 units of the same size.

Practice

Work each question on paper before you open its solution.

Question 1

Mr Tan's fruit stall has 15 pineapples and 8 watermelons.
(a) Write the ratio of the number of watermelons to the number of pineapples.
(b) How is the ratio in (a) read?
(c) Write the ratio of the number of pineapples to the number of watermelons.

ShowHide solution

(a) The watermelons are named first, so their number is the first term: $8 : 15$

(b) It is read "8 to 15".

(a) The ratio of the number of watermelons to the number of pineapples is $8 : 15$ .
(b) "8 to 15"
(c) The ratio of the number of pineapples to the number of watermelons is $15 : 8$ .

Question 2

At an animal shelter, the ratio of the number of cats to the number of dogs is $9 : 4$ .
(a) Which term goes with the cats?
(b) Are there more cats or more dogs at the shelter?
(c) Siti says, "The ratio of the number of dogs to the number of cats is $9 : 4$ ." Is she right?

ShowHide solution

(a) The cats are named first, so the first term, 9, goes with the cats. The second term, 4, goes with the dogs.

(b) The cats are 9 units and the dogs are 4 units of the same size. 9 is greater than 4, so there are more cats.

(c) In Siti's sentence the dogs are named first, so the term for the dogs must come first. The ratio of the number of dogs to the number of cats is $4 : 9$ .

(a) The first term, 9, goes with the cats.
(b) There are more cats than dogs.
(c) No. The ratio of the number of dogs to the number of cats is $4 : 9$ .

Question 3

Can the two quantities be compared by a ratio? If they can, write the ratio of the first quantity to the second.
(a) 5 litres of paint and 12 litres of water
(b) 80 km and 3 hours
(c) 11 girls and 14 boys
(d) 25 g of salt and 2 litres of water
(e) a string 47 cm long and a string 1 m long

ShowHide solution

(a) Both are volumes in litres. The ratio is $5 : 12$ .

(b) A distance and a time are different kinds of quantity, so there is no ratio. Together they give a rate.

(d) A mass and a volume are different kinds of quantity, so there is no ratio.

(e) Both are lengths, so they can be compared by a ratio, but they are in different units of measurement. 1 m is 100 cm, so the lengths are 47 cm and 100 cm. The ratio is $47 : 100$ , not $47 : 1$ .

(a) Yes. The ratio is $5 : 12$ .
(b) No.
(c) Yes. The ratio is $11 : 14$ .
(d) No.
(e) Yes. The ratio is $47 : 100$ .

Question 4

The table shows the medals that a school won at a swimming meet.

Medal	Number won
Gold	3
Silver	5
Bronze	11

(a) Find the ratio of the number of gold medals to the number of bronze medals.
(b) Find the ratio of the number of silver medals to the number of gold medals.
(c) Find the ratio of the number of bronze medals to the total number of medals.
(d) Find the ratio of the total number of medals to the number of silver medals.
(e) Which of your ratios compare a part with a part?

ShowHide solution

(a) 3 gold medals and 11 bronze medals: $3 : 11$

(b) 5 silver medals and 3 gold medals: $5 : 3$

(c) Add all the parts to find the whole: $3 + 5 + 11 = 19$ medals. The ratio of the number of bronze medals to the total number of medals is $11 : 19$ .

(d) The total is named first: $19 : 5$

(e) The ratios in (a) and (b) compare a part with a part. The ratio in (c) compares a part with the whole, and the ratio in (d) compares the whole with a part.

(a) $3 : 11$
(b) $5 : 3$
(c) $11 : 19$
(d) $19 : 5$
(e) The ratios in (a) and (b).

Question 5

A bus is carrying 43 passengers. 19 of them are sitting and the rest are standing.
(a) Find the ratio of the number of passengers sitting to the number standing.
(b) Find the ratio of the number of passengers standing to the number of passengers on the bus.
(c) Find the ratio of the number of passengers on the bus to the number sitting.

ShowHide solution

The whole (43 passengers) and one part (19 sitting) are given, so the other part is the whole minus that part. Number standing $= 43 - 19 = 24$

(a) 19 sitting and 24 standing: $19 : 24$

(b) 24 standing and 43 passengers in all. This compares a part with the whole: $24 : 43$

Check: $19 + 24 = 43$ , the number of passengers on the bus.

(a) The ratio of the number sitting to the number standing is $19 : 24$ .
(b) The ratio of the number standing to the number of passengers on the bus is $24 : 43$ .
(c) The ratio of the number of passengers on the bus to the number sitting is $43 : 19$ .

Question 6

The bar model shows the lengths of two ropes, P and Q.

(a) Find the ratio of the length of rope P to the length of rope Q.
(b) Find the ratio of the length of rope Q to the length of rope P.
(c) Find the ratio of the length of rope Q to the total length of the two ropes.
(d) Rope P is cut into 3 equal pieces. How many pieces of that length can be cut from rope Q?

ShowHide solution

(a) Rope P is 3 units and rope Q is 8 units: $3 : 8$

(b) Rope Q is named first: $8 : 3$

(d) Rope P is 3 units, so each of its 3 equal pieces is 1 unit long. Rope Q is 8 units, so 8 pieces of that length can be cut from it.

(a) $3 : 8$
(b) $8 : 3$
(c) $8 : 11$
(d) 8 pieces can be cut from rope Q.

Question 7

Ravi wrote four sentences, and each of them has a mistake. Find the mistake in each and correct it.
(a) "The ratio of the mass of a 3 kg bag of flour to the mass of a 5 kg bag of rice is 3 kg : 5 kg."
(b) "A car uses 7 litres of petrol to travel 100 km, so the ratio of the petrol to the distance is $7 : 100$ ."
(c) "A stick 23 cm long and a stick 1 m long are in the ratio $23 : 1$ ."
(d) "Class 1E has 17 boys and 20 girls, so the ratio of the number of girls to the number of boys is $17 : 20$ ."

ShowHide solution

(a) Wrong. Both are masses in kilograms, so they can be compared by a ratio, but a ratio has no unit of measurement. The masses are 3 kg and 5 kg, and the ratio is $3 : 5$ .

(b) Wrong. A volume of petrol and a distance are different kinds of quantity, so there is no ratio. 7 litres for every 100 km is a rate.

(c) Wrong. The lengths are in different units of measurement. 1 m is 100 cm, so the lengths are 23 cm and 100 cm, and the ratio is $23 : 100$ . The ratio $23 : 1$ would make the shorter stick the longer one.

(d) Wrong. The girls are named first, so their number is the first term: the ratio is $20 : 17$ .

(a) The ratio is $3 : 5$ , with no unit of measurement.
(b) There is no ratio: it is a rate.
(c) The ratio is $23 : 100$ .
(d) The ratio of the number of girls to the number of boys is $20 : 17$ .

Question 8

The ratio of the mass of flour to the mass of sugar in a cake recipe is $5 : 2$ .
(a) Draw a bar model to show this ratio.
(b) How many units stand for the flour and the sugar together?
(c) Write the ratio of the mass of sugar to the total mass of flour and sugar.

ShowHide solution

(a) Draw one bar for the flour and one bar for the sugar, starting at the same line on the left. Cut the bar for the flour into 5 equal units. Give the bar for the sugar 2 units of the same size.

(b) $5 + 2 = 7$ units. These 7 units stand for the flour and the sugar together and nothing else. They are not the whole cake mixture: a cake has other things in it as well, such as eggs and butter, and they are in neither term.

(a) A bar of 5 units for the flour and a bar of 2 units of the same size for the sugar.
(b) 7 units stand for the flour and the sugar together.
(c) The ratio of the mass of sugar to the total mass of flour and sugar is $2 : 7$ .

Question 9

On a bracelet, the ratio of the number of red beads to the number of blue beads is $4 : 9$ .
(a) Draw a bar model to show this ratio.
(b) Ali says, "The bracelet has 4 red beads and 9 blue beads." Can he be sure?
(c) Each unit of the bar model stands for 2 beads. How many red beads and how many blue beads are on the bracelet?

ShowHide solution

(a) Draw a bar of 4 equal units for the red beads and a bar of 9 units of the same size for the blue beads.

(b) No. The ratio says that the red beads are 4 units and the blue beads are 9 units of the same size, but not how many beads 1 unit stands for. If 1 unit is 1 bead, there are 4 red beads and 9 blue beads. If 1 unit is 3 beads, there are $4 \times 3 = 12$ red beads and $9 \times 3 = 27$ blue beads.

(c) 1 unit is 2 beads, and a unit is the same in both bars. The red beads are 4 units: $4 \times 2 = 8$ beads. The blue beads are 9 units: $9 \times 2 = 18$ beads.

(a) A bar of 4 units for the red beads and a bar of 9 units of the same size for the blue beads.
(b) No. The ratio does not say how many beads 1 unit stands for, so the bracelet need not have 4 red beads and 9 blue beads.
(c) There are 8 red beads and 18 blue beads.

Question 10

Each week Ravi spends $13 of his pocket money on food and $6 on transport, and he saves the other $10. He has no other spending.
(a) Find the ratio of the amount he spends on food to the amount he saves.
(b) Find the ratio of the amount he saves to his pocket money for the week.
(c) Find the ratio of his pocket money for the week to the amount he spends on transport.
(d) Mei Ling says, "From (a), the ratio is $13 : 10$ , so his pocket money is $23." What has she done wrong?

ShowHide solution

(a) $13 on food and $10 saved. Both are amounts of money in dollars: $13 : 10$

(b) His pocket money is the whole, and it is not given. Add all the parts that make it up: $13 + 6 + 10 = 29$ , so his pocket money is $29 a week. $10 saved and $29 in all: $10 : 29$

(d) She added only the two parts named in (a): $13 + 10 = 23$ , so the food and the savings together are $23. The money for transport is in neither term. The whole is $13 + 6 + 10 = 29$ , so his pocket money is $29.

(a) $13 : 10$
(b) $10 : 29$
(c) $29 : 6$
(d) She left out the $6 for transport. His pocket money is $29 a week, not $23.

Key Points

A ratio compares quantities of the same kind. $a : b$ is read " $a$ to $b$ ", and $a$ and $b$ are its terms.
The terms follow the order of the words. Unless $a$ and $b$ are equal, $a : b$ and $b : a$ are different ratios.
The quantities must be of the same kind and in the same unit of measurement. The ratio itself has no unit of measurement: 7 cm and 9 cm are in the ratio $7 : 9$ .
Quantities of different kinds, such as a distance in kilometres and a time in hours, give a rate, not a ratio.
A ratio can compare a part with a part, a part with the whole, or the whole with a part. If the whole is not given, add all the parts that make it up.
A bar model has one bar for each quantity, made of equal units. A unit is the same amount in every bar. The model shows the ratio, not the actual amounts.