Question

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

Answer 1

Hint: In this particular question assume any two different variables be the numbers such that the first number is greater than second or vice versa, then construct the equations according to the given conditions, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
The difference between two numbers is 14 and the difference between their squares is 448.
Now we have to find out the numbers.
Let the first number be x and the second number be y.
Let, x > y.
So according to question
The difference between two numbers is 14.
Therefore, x – y =14................... (1)
Now it is also given that the difference between their squares is 448.
Therefore, ${x^2} - {y^2} = 448$
Now in the above equation use the property that $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$ so we have,
$ \Rightarrow {x^2} - {y^2} = \left( {x - y} \right)\left( {x + y} \right) = 448$
Now substitute the value of (x – y) from equation (1) in the above equation we have,
$ \Rightarrow \left( {14} \right)\left( {x + y} \right) = 448$
Now divide by 14 throughout we have,
$ \Rightarrow \left( {x + y} \right) = \dfrac{{448}}{{14}} = 32$............... (2)
Now add equation (1) and (2) we have,
$ \Rightarrow x - y + x + y = 14 + 32$
$ \Rightarrow 2x = 46$
$ \Rightarrow x = 23$
Now from equation (2) we have,
$ \Rightarrow \left( {23 + y} \right) = 32$
$ \Rightarrow y = 32 - 23 = 9$
So the numbers are 23 and 9.
So this is the required answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the property that $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$, so first construct the equations according to given information as above then apply this property as above and simplify we will get the required numbers.