Question

The difference between the two numbers is 5 and the difference in their squares is 65. The larger number is:
A. 9
B. 10
C. 11
D. 12

Answer 1

Hint: According to the question we have to find the larger number when the difference between the two numbers is 5 and the difference in their squares is 65.
So, first of all we have to assume the two numbers which can be any integer like a and b or m and n and so on.
Now, we have to find the difference between that assumed number which is equal to 5 which is mentioned in the question.
Now, we have to find the difference in their squares of those assumed numbers which is equal to 65 which is mentioned in the question.
Now, we have to divide both equations obtained from the above passage and find the integer number which we have to assume.

Complete step-by-step answer:
Step 1: First of all we have to let the two different numbers that is $x$ and $y$ where $x > y$
Step 2: Now, we have to take the difference between the numbers which $x$ and $y$ equals to 5 which is mentioned in the question.
$ \Rightarrow $$5 = x - y...........................(1)$
Step 3: Now, we have to find the difference in their squares between numbers $x$ and $y$which is equal to 65 which is mentioned in the question.
$ \Rightarrow $$65 = {x^2} - {y^2}...........................(2)$
Step 4: Now, we have to divide the equation (2) by (1). We get,
$ \Rightarrow \dfrac{{{x^2} - {y^2}}}{{x - y}} = \dfrac{{65}}{5}$
As we know that ${a^2} - {b^2}$is $(a - b)(a + b)$
$
   \Rightarrow \dfrac{{(x - y)(x + y)}}{{x - y}} = 13 \\
   \Rightarrow x + y = 13........................(3) \\
 $
Step 5: Now, we have to solve both equations (1) and (3) by adding both.
$
   \Rightarrow x - y + x + y = 13 + 5 \\
   \Rightarrow 2x = 18 \\
   \Rightarrow x = 9 \\
 $
Now, we have to put the value of $x$ in equation (1) we get,
$
   \Rightarrow 9 - y = 5 \\
   \Rightarrow y = 9 - 5 \\
   \Rightarrow y = 4 \\
 $

So, from the solution above we can see that the larger number between 9 and 4 is 9.

Final solution: Hence, the larger number is 9. So, option (A) is correct.

Note:
It is necessary to assume the two numbers integers.
It is necessary to make two equations with the help of conditions given in the questions for finding larger numbers.