The difference between two numbers is 5 and the difference between their squares is 65. The larger number is
a. 9
b. 10
c. 11
d. 12

Hint: We start solving the problem by assigning variables for the larger and smaller numbers. We then use the condition of the difference of numbers to get a relation between them. We then use the condition difference between the squares and we substitute the relation obtained before and make subsequent calculations to get the value of the larger number.

Complete step by step answer:
According to the problem, we have given that the difference between two numbers is 5 and the difference between their squares is 65. We need to find the value of the greater number.
Let us assume the greatest number be y and smallest number be x.
We have the difference between numbers as 5. So, we get $y-x=5$.
$x=y-5$ ---(1).
According to the problem, the difference between the squares of these numbers is 65.
${{y}^{2}}-{{x}^{2}}=65$.
From equation (1), we get
${{y}^{2}}-{{\left( y-5 \right)}^{2}}=65$.
${{y}^{2}}-\left( {{y}^{2}}-10y+25 \right)=65$.
${{y}^{2}}-{{y}^{2}}+10y-25=65$.
$10y=65+25$.
$10y=90$.
$y=\dfrac{90}{10}$.
$y=9$.
We have found the value of the greatest number as 9.
∴ The value of the greatest number is 9.

So, the correct answer is “Option a”.

Note: We can also solve the problem by finding the value of the smaller number first and then using it to find the value of the larger number from the first relation obtained. Whenever we get this type of problem, it is better to start solving by assigning the variables and getting the relations between them to find the required values.