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Ratios as Fractions

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So many times we are given a **ratio** with which items are to be shared, like a **ratio** between boys and girls in your class.

Do you know that this could be converted to fractions and can be used in many ways?

In this article, you would be learning how ratios can be represented as fractions.

Ratio as a fraction occurs when rations are written in the form of fractions.

The ratio $X:Y$ is expressed as a fraction in the form$\frac{X}{Y}$.

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

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

Below are several properties of ratios as fractions and some direct examples of each property.

a. The antecedent of the ratio is the numerator of the fraction while the consequent of the ratio is the denominator of the ratio. This applies only when the fraction is not in the simplified form.

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.

b. A ratio can only be converted to a fraction when the consequent is not a factor of the antecedent, otherwise a whole number is formed.

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.

c. When a ratio is converted to a fraction, the fraction must be reduced to its simplified form.

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

d. Ratios as fractions have no unit because both the antecedent and consequent are of same units.

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

$5cm:7cm=\frac{5\overline{)cm}}{7\overline{)cm}}=\frac{5}{7}$

e. Since ratios have no units, it means that if the antecedent has a unit, the consequent must have the same unit so that the ratio is unit free.

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.

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

f. The antecedent and the consequent of a ratio should be expressed as whole numbers.

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

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

g. Ratios containing more than one quantity cannot be expressed as a fraction. The ratio a:b:c should not express as a fraction unless each quantity is expressed as a fraction of the total quantities.

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

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

There are several methods used to write ratios as fractions in the simplest form.

When simplifying ratios in the form of fractions, we divide through by the highest common factor of the numerator and the denominator. Note that this is a once-and-for-all process because the highest common factor divides the numerator and denominator once to arrive at the final answer.

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,

$\frac{18\xf76}{24\xf76}=\frac{3}{4}$

So the answer is,

$\frac{3}{4}$.

When you are to simplify ratios that are in the form of fractions, in this case, you need to find the lowest common factor between the numerator and denominator. Thereafter, you should divide the fraction by using the lowest common factor continuously until there is no common factor between them. This is a more rigorous method when dealing with ratios of large numbers.

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;

$\frac{\overline{)18}}{\overline{)24}}=\frac{9}{12}$

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

$\frac{\overline{)9}}{\overline{)12}}=\frac{3}{4}$

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

$\frac{3}{4}$

Knowing these properties would enable us to understand problems regarding ratios as fractions much better.

We may encounter problems where ratios are not expressed as fractions. All we have to do is to carefully convert details to the right expression as a ratio.

Afterward, we convert such a ratio to its corresponding fraction, then we make sure to simplify the fraction to find the final answer depending on the requirement of the question.

Till now, we have only dealt with ratios of parts, now we expand more on ratios out of a whole.

Sometimes we are required to find the ratio between a part of the ratio and the total ratio. In order to do so, we find the sum of all the values which make up the ratio before expressing the value as a ratio of the total. We will see this in the following example.

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,

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

The best way to understand how ratio as fractions is calculated is through examples. You shall also be able to see word problems involving ratios as fractions hereafter.

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,

$\frac{2\xf72}{12\xf72}=\frac{1}{6}$.

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,

$fractionnotamber=\frac{9\xf73}{15\xf73}fractionnotamber=\frac{3}{5}$

- Ratio as fraction deals with the expression of ratios in the form of fractions.
- There are several properties of ratios as fractions that ease the calculation of ratios.
- To simplify ratios as a fraction, you could either use the HCF method or the LCF method.
- When calculating ratios as fractions ensure that fractions are simplified.
- When solving word problems involving ratios as fractions make sure details of the question are well interpreted as expressions.

Yes! Ratios can be written as fractions.

An example of representing a ratio to fraction would be converting 2:3 to 2/3.

More about Ratios as Fractions

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