If the difference of two numbers is 3 and the difference of their squares is 39 ; then the larger number is : (a) 9 (b) 12 (c) 13 (d) 8

Given that the difference between two numbers is 3 and the difference between their squares is 39, we can set up the following equations:

1. \(x – y = 3\)
2. \(x^2 – y^2 = 39\)

We know that \(x^2 – y^2\) can be factored into \((x – y)(x + y)\). Substituting the first equation into the second gives:

\[
(x – y)(x + y) = 39
\]

Since \(x – y = 3\), we substitute that into the equation:

\[
3(x + y) = 39
\]

Simplifying:

\[
x + y = 13
\]

We now have two equations:

1. \(x – y = 3\)
2. \(x + y = 13\)

Adding these two equations together to eliminate \(y\), we get:

\[
2x = 16
\]

Solving for \(x\), we find:

\[
x = 8
\]

To find \(y\), we substitute \(x = 8\) back into one of the original equations:

\[
8 – y = 3
\]

Solving for \(y\), we get:

\[
y = 5
\]

Therefore, the larger number is \(8\).

The correct option is (d) 8.