Ratios and Fractions

How a ratio gives fractions and a fraction gives a ratio: a fraction of the other quantity or a fraction of the whole, always saying of what, shown on bar models.

Before you start

A ratio $a : b$ compares two quantities of the same kind and is read " $a$ to $b$ ". The numbers $a$ and $b$ are its terms, and they follow the order of the words: if a class has 19 boys and 21 girls, the ratio of the number of boys to the number of girls is $19 : 21$ , and the ratio of the number of girls to the number of boys is $21 : 19$ . A ratio has no unit of measurement.

A ratio can compare a part with a part, or a part with the whole. In a Robotics Club, 11 members come to school by bus, 7 by MRT and 5 on foot, so the club has $11 + 7 + 5 = 23$ members. The ratio of the number who come by MRT to the number who come on foot is $7 : 5$ (a part with a part), and the ratio of the number who come by bus to the total number of members is $11 : 23$ (a part with the whole).

In a bar model each quantity is a bar made of equal units. Here a unit is one equal part of a bar, not a unit of measurement. A unit is the same amount in every bar, and the model shows the ratio, not the actual amounts.

A fraction is always a fraction of something. $\frac{3}{4}$ of 20 is 15: cut 20 into 4 equal parts of 5 and take 3 of them. A fraction greater than 1 can be written as an improper fraction or as a mixed number: $\frac{3}{2} = 1\frac{1}{2}$ , and $\frac{3}{2}$ of 20 is 30.

From a Ratio to Fractions

In a class, the ratio of the number of boys to the number of girls is $2 : 3$ . On a bar model the boys are 2 units and the girls are 3 units. Fractions can be read from the same model, but each fraction must say what it is a fraction of.

Compare the boys with the girls. The girls are 3 units, so 1 unit is $\frac{1}{3}$ of the number of girls. The boys are 2 units, so the number of boys is $\frac{2}{3}$ of the number of girls. In the same way, 1 unit is $\frac{1}{2}$ of the number of boys, so the number of girls is $\frac{3}{2}$ of the number of boys.

Compare the boys with the whole class. The boys and the girls together make up the class, so the class is $2 + 3 = 5$ units and 1 unit is $\frac{1}{5}$ of the class. The boys are $\frac{2}{5}$ of the class and the girls are $\frac{3}{5}$ of the class.

So $\frac{2}{3}$ and $\frac{2}{5}$ are both true of the boys, because they are fractions of different things: $\frac{2}{3}$ of the number of girls, and $\frac{2}{5}$ of the class. "The boys are $\frac{2}{3}$ " says nothing until we know of what.

Check with numbers. Suppose 1 unit stands for 6 students. Then there are $2 \times 6 = 12$ boys and $3 \times 6 = 18$ girls, and the class has $12 + 18 = 30$ students. The number of boys is $\frac{12}{18} = \frac{2}{3}$ of the number of girls, and the boys are $\frac{12}{30} = \frac{2}{5}$ of the class.

Rule: a ratio and its fractions

Suppose the ratio of a quantity A to a quantity B is $a : b$ .

A is $\frac{a}{b}$ of B, and B is $\frac{b}{a}$ of A. The term of the quantity being described is the numerator. The term of the quantity it is a fraction of is the denominator.
If A and B together make up a whole, the whole is $a + b$ units, and A is $a$ of those units. So as a fraction of the whole, A is its term over the sum of the terms. With $2 : 3$ , A is $\frac{2}{2 + 3} = \frac{2}{5}$ of the whole and B is $\frac{3}{5}$ of the whole.
If A and B do not make up a whole between them, $a + b$ units is only A and B together, not the whole. Do not take the sum of the terms as the whole unless the quantities make up the whole.

Example 1: From a ratio to fractions

A hawker stall sells only chicken rice and duck rice. On Monday, the ratio of the number of plates of chicken rice sold to the number of plates of duck rice sold was $7 : 4$ .
(a) Express the number of plates of chicken rice as a fraction of the number of plates of duck rice.
(b) What fraction of all the plates sold were plates of chicken rice?
(c) What fraction of all the plates sold were plates of duck rice?

Solution

Draw a bar model: 7 units for the chicken rice and 4 units for the duck rice.

(a) The duck rice is 4 units, so 1 unit is $\frac{1}{4}$ of the number of plates of duck rice. The chicken rice is 7 units, which is $\frac{7}{4}$ of the number of plates of duck rice. The fraction is greater than 1 because more chicken rice than duck rice was sold: $\frac{7}{4} = 1\frac{3}{4}$ .

(b) The stall sells only these two dishes, so all the plates are $7 + 4 = 11$ units. The chicken rice is 7 of the 11 units, which is $\frac{7}{11}$ of all the plates.

Check: the two fractions of the whole add up to the whole. $\frac{7}{11} + \frac{4}{11} = \frac{11}{11} = 1$

(a) The number of plates of chicken rice was $\frac{7}{4}$ of the number of plates of duck rice.
(b) $\frac{7}{11}$ of all the plates sold were plates of chicken rice.
(c) $\frac{4}{11}$ of all the plates sold were plates of duck rice.

Which Whole?

Before you add the terms, ask what their sum stands for. In the Robotics Club, the ratio of the number who come by MRT to the number who come on foot is $7 : 5$ . Here $7 + 5 = 12$ is only the members who come by MRT and on foot together. It is not the club, because the members who come by bus are in neither term. So the members who come by MRT are $\frac{7}{12}$ of the members who come by MRT or on foot, but only $\frac{7}{23}$ of the club.

A ratio of a part to the whole gives a fraction of the whole at once. In the Robotics Club, the ratio of the number of members who come by bus to the total number of members is $11 : 23$ , so $\frac{11}{23}$ of the members come by bus. The second term is already the whole, so nothing is added to it. The members who do not come by bus are the rest of the club, $23 - 11 = 12$ members, so the ratio of the number who come by bus to the number who do not is $11 : 12$ .

From a Fraction to a Ratio

A fraction can be turned back into a ratio. The first question to ask is always the same: the fraction is a fraction of what?

Rule: from a fraction to a ratio

If A is $\frac{a}{b}$ of B, then B is $b$ units and A is $a$ units, so A : B $= a : b$ .
If A is $\frac{a}{b}$ of the whole, then the whole is $b$ units and A is $a$ of them. The rest of the whole is $b - a$ units, so A : the rest $= a : (b - a)$ , and A : the whole $= a : b$ .

Example 2: From a fraction to a ratio

(a) Ravi's height is $\frac{4}{5}$ of his father's height. Find the ratio of Ravi's height to his father's height.
(b) $\frac{3}{7}$ of the members of a Chess Club are girls and the rest are boys. Find the ratio of the number of girls to the number of boys.

Solution

(a) Ask first: $\frac{4}{5}$ of what? Of his father's height. So cut his father's height into 5 equal units. Ravi's height is 4 of those units.

Ravi's height : his father's height $= 4 : 5$

(b) $\frac{3}{7}$ of what? Of all the members, which is the whole club. So the whole club is 7 units and the girls are 3 of them. The boys are the rest: $7 - 3 = 4$ units.

Number of girls : number of boys $= 3 : 4$

In (a) the fraction compared one quantity with the other, so its numerator and its denominator were the two terms. In (b) the fraction was a fraction of the whole, so the other part had to be found first. $3 : 7$ is the ratio of the number of girls to the number of members.

(a) The ratio of Ravi's height to his father's height is $4 : 5$ .
(b) The ratio of the number of girls to the number of boys is $3 : 4$ .

Example 3: A fraction greater than 1

A Secondary 1 class has only boys and girls. The number of girls is $\frac{5}{4}$ of the number of boys.
(a) Find the ratio of the number of girls to the number of boys.
(b) Express the number of boys as a fraction of the number of girls.
(c) What fraction of the class are boys?
(d) Find the ratio of the number of boys to the number of students in the class.

Solution

(a) $\frac{5}{4}$ of what? Of the number of boys. So cut the number of boys into 4 equal units. The girls are 5 of those units, more than the boys, because the fraction is greater than 1.

Number of girls : number of boys $= 5 : 4$

(b) The girls are 5 units, so 1 unit is $\frac{1}{5}$ of the number of girls. The boys are 4 units: $\frac{4}{5}$ of the number of girls.

(c) The class has only boys and girls, so they make up the class, and the class is $5 + 4 = 9$ units. The boys are 4 of the 9 units: $\frac{4}{9}$ of the class.

(d) The boys are 4 units and the class is 9 units: $4 : 9$ .

Check with numbers. If 1 unit is 3 students, there are $5 \times 3 = 15$ girls and $4 \times 3 = 12$ boys, and $15 + 12 = 27$ students in the class. $\frac{15}{12} = \frac{5}{4}$ , $\frac{12}{15} = \frac{4}{5}$ and $\frac{12}{27} = \frac{4}{9}$ .

(a) The ratio of the number of girls to the number of boys is $5 : 4$ .
(b) The number of boys is $\frac{4}{5}$ of the number of girls.
(c) $\frac{4}{9}$ of the class are boys.
(d) The ratio of the number of boys to the number of students in the class is $4 : 9$ .

Common Mistakes

Giving a fraction without saying of what. If the ratio of the number of boys to the number of girls is $2 : 3$ , the boys are $\frac{2}{3}$ of the number of girls and $\frac{2}{5}$ of the class. They are not $\frac{2}{3}$ of the class.
Taking the sum of the terms as the whole when the two quantities do not make up the whole. In the Robotics Club, $7 + 5 = 12$ is only the members who come by MRT and on foot together; the club has 23 members. In Example 2, Ravi's height and his father's height are not two parts of one whole: Ravi's height is $\frac{4}{5}$ of his father's height, and $4 + 5 = 9$ units is only the two heights added together.
Adding the terms of a ratio that already compares a part with the whole. The ratio of the number of members who come by bus to the total number of members is $11 : 23$ . The club is 23 units, not $11 + 23 = 34$ units.
Writing a fraction of the whole as a ratio of the two parts. If $\frac{3}{7}$ of a Chess Club are girls and the rest are boys, the ratio of the number of girls to the number of boys is $3 : 4$ , not $3 : 7$ . $3 : 7$ compares the girls with the whole club.
Reading the terms as the actual numbers. If the ratio of the number of boys to the number of girls in a class is $2 : 3$ , the class is 5 units, not 5 students. 1 unit may stand for any number of students: if it is 6, the class has 30.

Practice

Work each question on paper before you open its solution.

Question 1

One plot of a school garden has only chilli plants and tomato plants. The ratio of the number of chilli plants to the number of tomato plants is $5 : 8$ .
(a) Express the number of chilli plants as a fraction of the number of tomato plants.
(b) What fraction of the plants in the plot are chilli plants?
(c) What fraction of the plants in the plot are tomato plants?
(d) Express the number of tomato plants as a fraction of the number of chilli plants.

ShowHide solution

Draw a bar model: 5 units for the chilli plants and 8 units for the tomato plants.

(a) The tomato plants are 8 units, so 1 unit is $\frac{1}{8}$ of the number of tomato plants. The chilli plants are 5 units: $\frac{5}{8}$ of the number of tomato plants.

(b) The plot has only these two kinds of plant, so all the plants are $5 + 8 = 13$ units. The chilli plants are 5 of the 13 units: $\frac{5}{13}$ of the plants in the plot.

(d) The chilli plants are 5 units, so 1 unit is $\frac{1}{5}$ of the number of chilli plants. The tomato plants are 8 units: $\frac{8}{5} = 1\frac{3}{5}$ of the number of chilli plants.

(a) The number of chilli plants is $\frac{5}{8}$ of the number of tomato plants.
(b) $\frac{5}{13}$ of the plants in the plot are chilli plants.
(c) $\frac{8}{13}$ of the plants in the plot are tomato plants.
(d) The number of tomato plants is $\frac{8}{5}$ of the number of chilli plants.

Question 2

The bar model shows the number of red apples and the number of green apples in a box. There is no other fruit in the box.

(a) Find the ratio of the number of green apples to the number of red apples.
(b) Express the number of green apples as a fraction of the number of red apples.
(c) What fraction of the apples in the box are red?
(d) Find the ratio of the number of red apples to the number of apples in the box.

ShowHide solution

(a) The green apples are 8 units and the red apples are 3 units. The green apples are named first: $8 : 3$

(b) The red apples are 3 units, so 1 unit is $\frac{1}{3}$ of the number of red apples. The green apples are 8 units: $\frac{8}{3} = 2\frac{2}{3}$ of the number of red apples.

(c) The box has only red and green apples, so all the apples are $3 + 8 = 11$ units. The red apples are 3 of the 11 units: $\frac{3}{11}$ of the apples.

(d) The red apples are 3 units and all the apples are 11 units: $3 : 11$

(a) $8 : 3$
(b) The number of green apples is $\frac{8}{3}$ of the number of red apples.
(c) $\frac{3}{11}$ of the apples in the box are red.
(d) $3 : 11$

Question 3

(a) The mass of a papaya is $\frac{3}{10}$ of the mass of a watermelon. Find the ratio of the mass of the papaya to the mass of the watermelon.
(b) $\frac{3}{10}$ of the fruits in a basket are papayas and the rest are mangoes. Find the ratio of the number of papayas to the number of mangoes.

ShowHide solution

(a) $\frac{3}{10}$ of what? Of the mass of the watermelon. So the mass of the watermelon is 10 units and the mass of the papaya is 3 of those units. The ratio of the mass of the papaya to the mass of the watermelon is $3 : 10$ .

(b) $\frac{3}{10}$ of what? Of all the fruits in the basket, which is the whole. So the whole basket is 10 units and the papayas are 3 of them. The mangoes are the rest: $10 - 3 = 7$ units.

Number of papayas : number of mangoes $= 3 : 7$

The same fraction gives two different ratios, because in (a) it is a fraction of the other quantity and in (b) it is a fraction of the whole.

(a) The ratio of the mass of the papaya to the mass of the watermelon is $3 : 10$ .
(b) The ratio of the number of papayas to the number of mangoes is $3 : 7$ .

Question 4

Every stall at a hawker centre sells either food or drinks, and $\frac{2}{11}$ of the stalls sell drinks.
(a) What fraction of the stalls sell food?
(b) Find the ratio of the number of drinks stalls to the number of food stalls.
(c) Find the ratio of the number of food stalls to the number of stalls at the hawker centre.
(d) Express the number of food stalls as a fraction of the number of drinks stalls.

ShowHide solution

$\frac{2}{11}$ of what? Of all the stalls, which is the whole. So all the stalls are 11 units and the drinks stalls are 2 of them. The food stalls are the rest: $11 - 2 = 9$ units.

(a) The food stalls are 9 of the 11 units: $\frac{9}{11}$ of the stalls.

(b) Drinks stalls : food stalls $= 2 : 9$

(d) The drinks stalls are 2 units, so 1 unit is $\frac{1}{2}$ of the number of drinks stalls. The food stalls are 9 units: $\frac{9}{2} = 4\frac{1}{2}$ of the number of drinks stalls.

(a) $\frac{9}{11}$ of the stalls sell food.
(b) $2 : 9$
(c) $9 : 11$
(d) The number of food stalls is $\frac{9}{2}$ of the number of drinks stalls.

Question 5

The mass of Ali's school bag is $\frac{7}{5}$ of the mass of Ravi's school bag.
(a) Find the ratio of the mass of Ali's bag to the mass of Ravi's bag.
(b) Express the mass of Ravi's bag as a fraction of the mass of Ali's bag.
(c) Ali says, "The mass of my bag is $\frac{7}{12}$ of the mass of the two bags together." Is he right?

ShowHide solution

(a) $\frac{7}{5}$ of what? Of the mass of Ravi's bag. So the mass of Ravi's bag is 5 units and the mass of Ali's bag is 7 of those units: $7 : 5$

(b) Ali's bag is 7 units, so 1 unit is $\frac{1}{7}$ of the mass of Ali's bag. Ravi's bag is 5 units: $\frac{5}{7}$ of the mass of Ali's bag.

(c) Ali compares his bag with the two bags together, and the two bags together are exactly his bag and Ravi's bag. So here the sum of the terms is the whole he names: the two bags together are $7 + 5 = 12$ units, and Ali's bag is 7 of them, which is $\frac{7}{12}$ of the mass of the two bags together.

(a) $7 : 5$
(b) The mass of Ravi's bag is $\frac{5}{7}$ of the mass of Ali's bag.
(c) Yes. The two bags together are 12 units and his bag is 7 of them.

Question 6

In a choir of boys and girls, the ratio of the number of boys to the number of girls is $5 : 6$ . Say whether each statement is true or false, and correct the false ones.
(a) $\frac{5}{6}$ of the choir are boys.
(b) The number of boys is $\frac{5}{6}$ of the number of girls.
(c) $\frac{6}{11}$ of the choir are girls.
(d) The number of girls is $\frac{6}{5}$ of the number of boys.
(e) There must be exactly 11 students in the choir.

ShowHide solution

The boys are 5 units and the girls are 6 units. Together they make up the choir, so the choir is $5 + 6 = 11$ units.

(a) False. The boys are 5 of the 11 units, so $\frac{5}{11}$ of the choir are boys. $\frac{5}{6}$ is a fraction of the number of girls, not of the choir.

(b) True. The girls are 6 units, so 1 unit is $\frac{1}{6}$ of the number of girls, and the boys are 5 units: $\frac{5}{6}$ of the number of girls.

(d) True. The boys are 5 units, so 1 unit is $\frac{1}{5}$ of the number of boys, and the girls are 6 units: $\frac{6}{5}$ of the number of boys.

(e) False. 11 is the number of units, and 1 unit may stand for more than 1 student. If 1 unit is 3 students, there are $5 \times 3 = 15$ boys and $6 \times 3 = 18$ girls, which is 33 students.

(a) False: $\frac{5}{11}$ of the choir are boys.
(b) True
(c) True
(d) True
(e) False: the ratio does not give the number of students.

Question 7

Hui Min has 30 stickers. 13 of them are animal stickers and the rest are flower stickers.
(a) Find the ratio of the number of animal stickers to the number of flower stickers.
(b) Find the ratio of the number of flower stickers to the number of all her stickers.
(c) What fraction of her stickers are animal stickers?
(d) Express the number of animal stickers as a fraction of the number of flower stickers.

ShowHide solution

Find the other part first. Number of flower stickers $= 30 - 13 = 17$

(a) 13 animal stickers and 17 flower stickers: $13 : 17$

(b) 17 flower stickers and 30 stickers altogether: $17 : 30$

(d) There are 13 animal stickers and 17 flower stickers, so the number of animal stickers is $\frac{13}{17}$ of the number of flower stickers.

(a) $13 : 17$
(b) $17 : 30$
(c) $\frac{13}{30}$ of her stickers are animal stickers.
(d) The number of animal stickers is $\frac{13}{17}$ of the number of flower stickers.

Question 8

In a Secondary 1 class, the ratio of the number of students who take the MRT to school to the total number of students in the class is $5 : 12$ .
(a) What fraction of the class take the MRT to school?
(b) Find the ratio of the number of students who take the MRT to the number of students who do not.
(c) Express the number of students who do not take the MRT as a fraction of the number who do.

ShowHide solution

This ratio compares a part with the whole. The class is 12 units and the students who take the MRT are 5 of them. 12 is already the whole, so do not add 5 and 12.

(a) 5 of the 12 units take the MRT: $\frac{5}{12}$ of the class.

(b) The students who do not take the MRT are the rest of the class: $12 - 5 = 7$ units.

Number who take the MRT : number who do not $= 5 : 7$

(c) The students who take the MRT are 5 units, so 1 unit is $\frac{1}{5}$ of their number. The students who do not are 7 units: $\frac{7}{5} = 1\frac{2}{5}$ of the number who do.

(a) $\frac{5}{12}$ of the class take the MRT to school.
(b) The ratio of the number who take the MRT to the number who do not is $5 : 7$ .
(c) The number who do not take the MRT is $\frac{7}{5}$ of the number who do.

Question 9

A bowl holds apples, oranges and pears. The ratio of the number of apples to the number of oranges is $3 : 4$ .
(a) Express the number of apples as a fraction of the number of oranges.
(b) What fraction of the apples and oranges together are oranges?
(c) Siti says, " $\frac{3}{7}$ of the fruits in the bowl are apples." Explain why she is wrong.

ShowHide solution

The apples are 3 units and the oranges are 4 units.

(a) The oranges are 4 units, so 1 unit is $\frac{1}{4}$ of the number of oranges. The apples are 3 units: $\frac{3}{4}$ of the number of oranges.

(b) The apples and oranges together are $3 + 4 = 7$ units, and the oranges are 4 of them: $\frac{4}{7}$ of the apples and oranges together.

(c) $3 + 4 = 7$ units is only the apples and the oranges together. The bowl holds pears as well, and the pears are in neither term, so all the fruits in the bowl are more than 7 units. The apples are $\frac{3}{7}$ of the apples and oranges together, but less than $\frac{3}{7}$ of all the fruits in the bowl.

(a) The number of apples is $\frac{3}{4}$ of the number of oranges.
(b) $\frac{4}{7}$ of the apples and oranges together are oranges.
(c) The apples and oranges do not make up the whole bowl, because there are pears too. The apples are $\frac{3}{7}$ of the apples and oranges together, and less than $\frac{3}{7}$ of all the fruits.

Question 10

Mrs Goh makes a drink by mixing 2 cups of syrup with 9 cups of water.
(a) Find the ratio of the volume of syrup to the volume of water.
(b) Find the ratio of the volume of syrup to the volume of the drink.
(c) What fraction of the drink is syrup?
(d) Mr Tan mixes 9 cups of syrup with 2 cups of water. He says that his drink is the same as Mrs Goh's, because it is also made with the numbers 2 and 9. Is he right?

ShowHide solution

(a) Both are volumes in cups: $2 : 9$

(b) The drink is the syrup and the water together, so it is the whole: $2 + 9 = 11$ cups. The ratio of the volume of syrup to the volume of the drink is $2 : 11$ .

(d) For Mr Tan's drink, the ratio of the volume of syrup to the volume of water is $9 : 2$ , and 9 of its $9 + 2 = 11$ cups are syrup: $\frac{9}{11}$ of the drink. $9 : 2$ and $2 : 9$ are different ratios, and his drink is far sweeter.

(a) $2 : 9$
(b) $2 : 11$
(c) $\frac{2}{11}$ of the drink is syrup.
(d) No. His ratio of syrup to water is $9 : 2$ , not $2 : 9$ : the order of the terms matters.

Key Points

Every fraction says what it is a fraction of. When the ratio of boys to girls is $2 : 3$ , the boys are $\frac{2}{3}$ of the number of girls and $\frac{2}{5}$ of the class.
If the ratio of A to B is $a : b$ , then A is $\frac{a}{b}$ of B and B is $\frac{b}{a}$ of A.
If A and B together make up a whole, the whole is $a + b$ units: A is $\frac{a}{a + b}$ of the whole and B is $\frac{b}{a + b}$ of the whole.
If A and B do not make up the whole, $a + b$ units is only A and B together, not the whole. Do not take the sum of the terms as the whole unless the quantities make up the whole.
A ratio of a part to the whole, such as $11 : 23$ , gives the fraction of the whole at once, $\frac{11}{23}$ . Nothing is added.
To turn a fraction into a ratio, ask "of what?". A fraction of the other quantity gives the two terms at once. A fraction of the whole gives the part and the whole; find the rest by subtracting before you write the ratio of the two parts.