A rich merchant had collected many gold coins. He did not want anybody to know about them. One day, his

wife asked, "How many gold coins do you have?" After pausing a moment, he replied, "Well! If I divide the
coins into two unequal numbers, then 25 times the difference between the two numbers equals the difference
between the squares of the two numbers." How many gold coins does the rich merchant have?

25(x-y)=x^2-y^2

x^2-25x=y^2-25y

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To solve the equation, we can start by rearranging it:

x^2 - 25x = y^2 - 25y

Next, we can add 25x and subtract 25y from both sides of the equation:

x^2 - 25x + 25^2 = y^2 - 25y + 25^2

Simplifying further:

(x - 25)^2 = (y - 25)^2

Taking the square root:

x - 25 = y - 25

Since the difference between the two numbers is constant, we can simplify this further:

x = y

So, the two numbers are equal.

Let's assume the number of gold coins as x.

Therefore, x = y = the number of gold coins.

To determine the value of x, we can substitute it back into one of the original equations. For example:

25(x - y) = x^2 - y^2

25(0) = x^2 - x^2

0 = 0

Based on this, we can conclude that the equation holds true for any value of x (or y).

Therefore, there is no specific number of gold coins mentioned in the problem. The rich merchant could have any number of gold coins.

To solve this problem, we need to find the values of x and y that satisfy the equation:

x^2 - 25x = y^2 - 25y

Let's simplify the equation and rearrange it:

x^2 - y^2 = 25x - 25y

Now, we can factor the left side of the equation using the difference of squares formula:

(x + y)(x - y) = 25(x - y)

We notice that there is a common factor of (x - y) on both sides of the equation, so we can divide both sides by (x - y):

(x + y) = 25

Now we have a simple equation with only one variable. The sum of x and y is equal to 25.

Here's where we need to make an assumption based on the wording of the problem. The merchant wants to divide his coins into two unequal numbers. This means that x and y cannot be equal.

So, we need to find two numbers that add up to 25 but are not equal. After checking the possible options, we find that x = 24 and y = 1 satisfy the equation.

Therefore, the rich merchant has 24 gold coins.

Try this:

(x+y)(x-y) = 25(x-y)
x+y = 25