Question

# The sum of two numbers is equal to $20$ and their difference is $2.5$. Find the ratio of the numbers.A) $9:7$B) $7:9$C) $3:5$D) $2:7$

Hint:This question can be solved using simultaneous linear equations. In this method, the value of the two numbers will be replaced by particular variables. Further, the statements and conditions can be converted into mathematical forms and equations. These equations can be solved to attain the required value of the numbers. Then, their ratio can also be calculated.

It can be observed that the whole question revolves around two numbers, whose ratio is to be found.
We can replace their values with variables so that we will be able to apply mathematical approach to solve this problem.
Let the first number$= x$
Let second number$= y$
Now it is given that the sum of the two numbers is $20$. This can be mathematically written as,
Sum of $x$ and $y = 20$
$x + y = 20………...(1)$
Further, it is written that their difference is $2.5$. The fact that the numbers have a difference means the numbers are not equal. We can safely assume one of them to be larger than the other. So, let $x$ be greater than $y$.
So, now,
Difference of $x$ and $y = 2.5$
$x - y = 2.5………...(2)$
We can achieve the original values of $x$ and $y$ by simply solving the simultaneous equations (1) and (2)
$x + y = 20$ and $x - y = 2.5$
We will get
$\left( {x + y} \right) + \left( {x - y} \right) = 20 + 2.5 \\ x + y + x - y = 22.5 \\ 2x = 22.5 \\ x = \dfrac{{22.5}}{2} \\ x = 11.25 \\$
Also, $x + y = 20$
$11.25 + y = 20 \\ y = 20 - 11.25 = 8.75 \\$
The required numbers are $11.25$ and $8.75$
Their ratio$= \dfrac{x}{y} = \dfrac{{11.25}}{{8.75}} = \dfrac{9}{7}$
Hence the ratio of the numbers is $9:7$.

So, the correct answer is “Option A”.

Note:The student might get confused while dealing with simultaneous linear equations. The only correct way to avoid errors is reading the statements given in the question word by word.Further, the student will get confused between the options $9:7$ and $7:9$. To understand this, remember that their difference is given. So the first number has to be the larger one (in the ratio).