The difference between two numbers is a common mathematical concept that we encounter at various stages in our academic journey. Sometimes we are given information about the difference between two numbers, and sometimes we have to find it while solving a problem. One such problem is presented to us in the form of a statement that says, "The difference between two numbers is 3, and the difference between their squares is 63. Which is the larger number?" In this article, we will explore the methodology to solve this problem, step-by-step.

Let's assume that the two numbers we are talking about are x and y. The statement tells us that the difference between these two numbers is 3. We can write this mathematically as:

x - y = 3

Now, the second part of the statement tells us that the difference between the squares of the same two numbers is 63. We can write this as:

x² - y² = 63

The equation x² - y² = 63 can be simplified using the identity for the difference of squares, which is:

(a² - b²) = (a + b) (a - b)

Using this identity on our equation, we get:

(x + y) (x - y) = 63

We already know that x - y = 3 from the first equation, so we can substitute it in the above equation and simplify it further:

(x + y) (3) = 63

Dividing both sides by 3, we get:

x + y = 21

We now have two equations:

x - y = 3

x + y = 21

We can solve these equations simultaneously to get the values of x and y. Adding the two equations, we get:

2x = 24

Dividing both sides by 2, we get:

x = 12

Substituting the value of x in either of the two equations, we get:

y = 9

So, the two numbers that satisfy the given conditions are 12 and 9. But the question asks us to find which number is larger. To do that, we simply compare the values of x and y:

x > y

12 > 9

Therefore, we can conclude that the larger number is 12.

To summarize the above process:

1. Let x and y be the two numbers we want to find.

2. Use the given information to write two equations:

x - y = 3

x² - y² = 63

3. Simplify the second equation using the identity for the difference of squares:

(x + y) (x - y) = 63

4. Substitute x - y = 3 in the above equation and simplify:

(x + y) (3)= 63

x + y = 21

5. Solve the two equations simultaneously to get the values of x and y:

x = 12

y = 9

6. Compare the values of x and y to find the larger number:

x > y

12 > 9

7. Therefore, the larger number is 12.

In conclusion, the problem of finding the larger number when the difference between two numbers is given, along with the difference between their squares, can be solved by following the steps mentioned above. This is a good example of how we can use mathematical concepts and identities to simplify and solve complex problems.