Question

The difference between $2$ numbers is $14$ and the difference between their squares is $448$. Find the numbers.

Answer 1

Hint: Here, we will make equations out of the data which is given to us in the question and then solve them to find the numbers. Finally we get the required answer.

Complete step-by-step solution:
Let the $2$ numbers be $a$ and $b$.
From the question we know that the difference between the $2$ numbers is $14$, mathematically it can be written as:
$ \Rightarrow a - b = 14 \to (1)$
And we also know that the difference between the squares of the numbers is $448$, mathematically is can be written as:
$ \Rightarrow {a^2} - {b^2} = 448$
Since the left-hand side of the equation is ${a^2} - {b^2}$ we can expand it. Therefore, the equation can be written as:
$ \Rightarrow (a + b)(a - b) = 448$
Now from equation $(1)$$a - b = 14$, on substituting we get:
$ \Rightarrow (a + b)14 = 448$
The bracket can be expanded and re-written as:
$ \Rightarrow 14a + 14b = 448 \to (1)$
Now to solve both $(1)$ and $(2)$ we multiply equation $(1)$ by $14$.
$ \Rightarrow 14a - 14b = 196$
$ \Rightarrow 14a + 14b = 448$
On adding both the equations we get:
$ \Rightarrow 28a = 644$
On sending across the $ = $ sign we get:
$ \Rightarrow a = \dfrac{{644}}{{28}}$
On dividing we get:
$ \Rightarrow a = 23$
On substituting the value of $a$ in equation $(1)$ we get:
$ \Rightarrow 23 - b = 14$
On transferring across the $ = $ sign we get:
$ \Rightarrow 23 - 14 = b$
On simplifying we get:
$ \Rightarrow b = 9$.

Therefore, the two numbers are $23$ and $9$ which is the required answer.

Note: To cross check whether the answer is correct, we substitute the values we have got:
Now we know the difference between the two numbers is supposed to be $14$
We can write it as,
$ \Rightarrow 23 - 9 = 14$ this is true.
Also, the difference of the squares of the numbers is supposed to be \[448\]:
So we can write it as,
$ \Rightarrow {(23)^2} - {(9)^2}$
The square of $23$ is $529$ and the square of $9$ is $81$ therefore,
It can be written as:
$ \Rightarrow 529 - 81$
On subtracting we get
$ \Rightarrow 448$, which is the supposed answer to the second equation, therefore both the numbers are correct and the answer is verified.