. 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?
To solve this problem, we need to set up a system of equations based on the information provided. Let's break it down step by step:
Let's assume the merchant has x gold coins and he divides them into two unequal numbers. Let the smaller number be y and the larger number be z.
According to the problem, the difference between the two numbers (z - y) multiplied by 25 is equal to the difference between the squares of the two numbers (z^2 - y^2). Mathematically, this can be represented as:
25(z - y) = z^2 - y^2
Now, let's simplify the equation further.
Expanding the expression on the right side of the equation using the difference of squares formula, we have:
25(z - y) = (z + y)(z - y)
Simplifying it:
25(z - y) = (z + y)(z - y)
25(z - y) = z^2 - y^2
Since z - y is present on both sides of the equation, we can divide both sides by (z - y):
25 = z + y
We have now obtained our first equation.
Next, we need to consider the total number of gold coins. The merchant has x gold coins, which can be expressed as the sum of the two numbers:
x = y + z
We have now obtained our second equation.
To solve this system of equations, we need to find values for x, y, and z that satisfy both equations simultaneously.
By substituting the value of y + z from the first equation into the second equation, we get:
x = (25 - y) + y
x = 25
Now we have the value of x. The rich merchant has 25 gold coins.
To find the values of y and z, we substitute x = 25 into the first equation:
25 = z + y
This equation does not have a unique solution. There are multiple pairs of y and z that satisfy this equation. Therefore, we cannot determine the specific number of gold coins the merchant has.
In conclusion, based on the given information, we can only determine that the rich merchant has 25 gold coins, but we cannot determine the distribution between the two unequal numbers.