Question

The difference of two numbers is $3$ their sum is $13.$ How do you find the number?

Answer 1

Hint:
Let the two such numbers are $''x''$ and $''y''$ from the assumed value of numbers prepare two equations and then by solving the equations determine the value of unknown quantity.
Like, $\left( x-y \right)=3$and $\left( x+y \right)=13.$
Hence, by solving the above equation, determine the value of both the numbers.

Complete step by step solution:
As we have to find such two numbers whose difference is equal to $3$ and sum of equal to $13$
As, for determining the value of number we need to assume the value of numbers,
Let the first number is $''x''$
And value of the second number is equal to $''y''$
So, as per information given in the question,
Considering the first part of question,
As it is mentioned that,
Difference of both numbers is $3$
So, we can say that,
$\left( x-y \right)=3...(1)$
Now, from second part of question,
As the difference between both the number is $13$
So, we can convert mathematically by,
$\left( x+y \right)=13...(2)$
Now,
As from above,
We have two different equation, and we have to determine the value of two unknown quantities,
For that,
We can conclude that,
From both of above given equation,
If we add both the equation, then the positive value of $''y''$ will get cancelled by negative value of $''y''$ and thus by using calculation technic we can determine the value of $''x''$
Like,
Adding $\left( 1 \right)$ and $\left( 2 \right)$
As when we add the both equations,
Then the left-hand side of first equation will be add – up with left hand side of second equation, and similarly right-hand side of first equation will be add-up with right hand side of second equation.
So,
$\Rightarrow \left( x-y \right)+\left( x+y \right)=3+13$
$\Rightarrow \left( x-y+x+y \right)=16$
$\Rightarrow 2x=16$
As here, value of $''2x''$ is equal to $16,$
So, for determining the value of $''x''$ we need to shift $2$ from left side to right side,
So, we can say that,
Value of $''x''$ will be equal to $\dfrac{16}{2}=8$
Now, for determining the value of second number,
We can put the value of $''x''$ in any of equation,
Let’s putting the value of $''x''$ in equation $\left( 1 \right),$
We will get,
$\left( x-y \right)=3$
$\Rightarrow x-y=3$
$\Rightarrow y=8-3=5$
So, we can say that,
Value of second number will be $5$
Hence,

We have two such numbers $8$ and $5$.

Additional Information:
We can also check that, whether the obtained number are correct or not,
As per solution, we have two numbers as $8$ and $5$
So, the difference between the value of numbers will be $=\left( 8-5 \right)=3$
And, sum of the value of numbers will be $=\left( 8+3 \right)=13$
Hence, we can say that,
The obtained numbers satisfied both the conditions given in the question.

Note:
We can also, consider numbers as $''x''$ and $\left( 13-x \right)$ as it is given in the question as the sum of both number is $13$
So, from here,
As the difference of the number is $3$
So, $\left( 13-x \right)-x=3$
$\Rightarrow 13-x-x=3$
$\Rightarrow 2x=13x-3$
$\Rightarrow 2x=10$
$x=\dfrac{10}{2}=5$
Hence, from here, second number will be $=\left( 13-5 \right)=8$