Ratios as Fractions

Express the following ratios as fractions.

a. $1:2$

b. $5:6$

c. $3:2$

d. $13:12$

Solution

The antecedent of the ratio is the numerator of the fraction while the consequent of the ratio is the denominator of the fraction.

a. The ratio$1:2$becomes

$1:2=\frac{1}{2}$

b. The ratio $5:6$becomes

$5:6=\frac{5}{6}$

c. The ratio $3:2$ becomes

$3:2=\frac{3}{2}$

d. The ratio $13:12$ becomes

$13:12=\frac{13}{12}$

In the ratio 2:3, 2 is the antecedent while 3 is the consequent.

When 2:3 is converted to a fraction, it becomes $\frac{2}{3}$.

Note that the antecedent (2) is now the numerator of the fraction while the consequent (3) is now the denominator of the fraction.

10:5 cannot be converted to a fraction because 5 (consequent) is a factor of 10 (antecedent). Thus, when converted to a fraction 10:5 is simplified to a whole number which is 2.

For example, 6:10 being converted to a fraction becomes $\frac{6}{10}$ and has to be further simplified to be $\frac{3}{5}$.

given the ratio of two distances is 5cm is to 7cm. By conversion it becomes,

$5cm:7cm=\frac{5\cancel{cm}}{7\cancel{cm}}=\frac{5}{7}$

We consider the ratio of two distances is 50 cm: 1 m, hence before even putting this ratio into a fraction, we need to ensure that both antecedent and consequent have the same unit.

$1m=100cm50cm:1m=50cm:100cm=\frac{50\cancel{cm}}{100\cancel{cm}}=\frac{1}{2}$

The ratio $4.5:3.5$ would be converted by multiplying all through by 2 to become $9:7$which is $\frac{9}{7}$

Another way to convert the antecedent and the consequent into whole numbers is to multiply them by 10, to get

$4.5:3.5=45:35=\frac{45}{35}=\frac{9}{7}$

If biscuits are shared in the ratio 2:1:3 among three people, we cannot represent it as a fraction like

$\frac{2}{\frac{1}{3}}$.

However, we may be asked to express the value of the first as a ratio of the total and hence we have

$total=2+1+3=6first:total=2:6=\frac{2}{6}=\frac{1}{3}$

Simplify the following ratio

$18:24$

Solution

Step 1.

Express the ratio as a fraction,

$18:24=\frac{18}{24}$.

Step 2.

Find the HCF between the numerator and the denominator. The HCF between 18 and 24 is 6.

Step 3.

Divide the numerator and the denominator by the HCF,

$\frac{18÷6}{24÷6}=\frac{3}{4}$

So the answer is,

$\frac{3}{4}$.

Simplify the following

$18:24$

Solution

Step 1.

Express the ratio as a fraction. Thus,

$18:24=\frac{18}{24}$

Step 2.

Divide by using the lowest common factor. The lowest common factor between 18 and 24 is 2, Thus;

$\frac{\cancel{18}}{\cancel{24}}=\frac{9}{12}$

Between 9 and 12 what is the lowest common factor? The lowest common factor is 3. So divide through by 3,

$\frac{\cancel{9}}{\cancel{12}}=\frac{3}{4}$

Between 3 and 4 what is the lowest common factor? There is no common factor between 3 and for. Therefore our answer is;

$\frac{3}{4}$

Two teenagers Bill and Jill share a loaf of bread in the ratio 2:3. What is the fraction of the whole bread did Jill take?

Solution

Jill's share is 3. The total ratio is,

$2+3=5$

The fraction of the whole bread Jill take is,

$Jill\text{'}sshare:totalshare=3:5=\frac{3}{5}$

In a cinema, the ratio of horror to sci-fi to comedy movies is 2:3:7. Express horror movies as a fraction of all kinds of movies being viewed in the cinema.

Solution

We are told that the ratio of these kinds of movies is,

$2:3:7$

Find the total of the ratio,

$2:3:7=2+3+7=12$.

We are asked to express horror movies as a fraction of all the movies, and horror movies have 2 among the ratio. Thus divide the horror movies' quantity by the total ratio,

$\frac{horrormovies}{totalmovies}=\frac{2}{12}$

Simplify, by dividing by the HCF which is 2,

$\frac{2÷2}{12÷2}=\frac{1}{6}$.

One-fifth of Kohe's books are torn. What is the ratio of the books untorn to those torn?

Solution

Kohe's books comprise both those that are torn and untorn.

Whenever you are given a fraction to express proportion or ratio, note that the sum of all items is 1. In this case, we have

$torn+untorn=1\frac{1}{5}+untorn=1untorn=1-\frac{1}{5}untorn=\frac{4}{5}$

This implies that four-fifth of Kohe's books are untorn. But you are asked to find the ratio of untorn to torn. Thus,

$untorn:torn=\frac{4}{5}:\frac{1}{5}$

Recall that neither the antecedent nor the consequent of a ratio should be a fraction. Thus multiply through by 5;

$(\frac{4}{5}\times 5):(\frac{1}{5}\times 5)=4:1$

Thus, the ratio of untorn to torn is 4:1.

A bag contains billiard balls of three colors white, blue and amber in the ratio 4:5:6 respectively. That fraction of the ball is not amber?

Solution

We first find the total ratio,

$4:5:6=4+5+6=15$

We next find the portion of the ratio that is not amber. Since amber is 6 and the total is 15, the amount that is not amber is,

$Notamber=15-6=9$

Now you know the proportion that is not amber, you can now find the fraction that is not amber out of the total ratio to be;

$fractionnotamber=\frac{valuenotamber}{totalratiovalue}fractionnotamber=\frac{9}{15}$

Simplify by dividing through by 3,

$fractionnotamber=\frac{9÷3}{15÷3}fractionnotamber=\frac{3}{5}$