## Ratio & Partnership

The concepts of ratios can be used in any math question. Furthermore, ratio based questions or cases appear in Data Interpretation frequently. Therefore, it is imperative for you to have a strong understanding of the concepts covered in this lesson. This lesson will cover the basics of ratios, it's properties and concepts, and Partnership.

### 1. Ratios

Ratio is the quantitative relationship or comparison of numbers. It indicates the relationship of one quantity with respect to another.

For instance, if there are $8$ chocolates and $12$ toffees in a bag, then the ratio of chocolates to toffees in the bag is $8 : 12$, i.e., $2 : 3$. The ratio indicates that for every $2$ chocolates in the bag, there are $3$ toffees.

A ratio can have 2 or more elements. For instance, if there are $20$ Sri Lankans, $30$ Indians and $40$ Americans attending a conference, then the ratio of number of Sri Lankans, Indians and Americans at the conference will be $2 : 3 : 4$. So, for every $2$ Sri Lankans at the conference, there are $3$ Indians and $4$ Americans.

A ratio of $2$ numbers can be expressed as a fraction in this way $\implies a : b = \dfrac{a}{b}$,

where a is called the antecedent and b is called the consequent

Like fractions, ratios can also be simplified and expressed.
Eg: Ratio of $8 : 32 = 1 : 4$

### Example 1

The ratio of girls to boys in a class was $2 : 1$. If there were $32$ boys, how many girls were there in the class?

### Solution

Let the number of girls in the class $= x$

∴ $\dfrac{\text{Girls}}{\text{Boys}} = \dfrac{2}{1} = \dfrac{x}{32}$

$\implies x = 64$

Answer: $64$

1.1 Applying ratios Ratios tell us the size of one item relative to the other. For instance, if the ratio of apples to mangoes in a bag is $1 : 4$, then the following can be concluded

$1$) For every apple in the bag, there are $4$ mangoes. Therefore, for every apple in the bag, there are a total of $5$ fruits in the bag.

$2$) Number of apples is $\dfrac{1}{4}$ or $25 \%$ of the number of mangoes.

$3$) Number of mangoes is $4$ times or $400 \%$ of the number of apples.

$4$) Number of apples is $\dfrac{1}{1 + 4} = \dfrac{1}{5}$ or $20 \%$ of the total number of fruits.

$5$) Number of mangoes is $\dfrac{4}{1 + 4} = \dfrac{4}{5}$ or $80 \%$ of the total number of fruits.

In most of the questions pertaining to ratios, the variables we assume for our calculations is what the question requires us to find.

### Example 2

In an orchard, if mangoes account for two-fifth of the fruits, and rest of the $240$ are apples, then how many mangoes are there?

### Solution

Let the total number of fruits be $x$.

Number of mangoes $= \dfrac{2x}{5}$

Number of apples $= x - \dfrac{2x}{5} = \dfrac{3x}{5} = 240$

$\implies x = 400$

Number of mangoes $= 400 – 240 = 160$

Answer: $160$

### Example 3

Rahul who is $1.6$ m tall and standing under the sun, casts a shadow that is $1.2$ m in length. At that moment, what is the length (in metres) of the shadow cast by a $120$ m pole?

### Solution

Let the length of the shadow of the pole be $x$.

At any time during the day, the ratio of the height of any standing object to the length of the shadow cast by the object will remain the same.

∴ $\dfrac{1.6}{1.2} = \dfrac{120}{x}$

$\implies x = 90$

Answer: $90$