Question

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 we have?" After pausing a moment he replied, "Well! If I divide the coins into two unequal numbers, then 48 times the difference between the two numbers equals the difference between the squares of the two numbers." The wife looked puzzled. Can you help the merchant's wife by finding out how many gold coins the merchant has?

• A. 96
• B. 53
• C. 43
• D. None of these

Answer 1

The Correct Option is D

To solve this problem, let's define the two unequal numbers that the merchant divides the coins into as x and y, with x > y.

According to the given condition:

"48 times the difference between the two numbers equals the difference between the squares of the two numbers."

This translates to the equation:

\(48(x-y) = x^2 - y^2\)

We know that the difference between the squares of two numbers can be expressed as:

\(x^2 - y^2 = (x-y)(x+y)\)

Substituting this into the original equation gives:

\(48(x-y) = (x-y)(x+y)\)

Assuming \(x \neq y\), we can factor out \(x-y\) from both sides:

\(48 = x+y\)

This implies that the sum of the two numbers is 48. The total number of coins is therefore:

\(x + y = 48\)

Given that the two sums should add up to the total number of coins,

Let's check the options provided to determine whether they can yield such a division:

• Option 1: 96 — Cannot divide into two numbers adding up to 48.
• Option 2: 53 — Cannot divide into two numbers adding up to 48.
• Option 3: 43 — Cannot divide into two numbers adding up to 48.

Since none of these options can produce a sum of 48 for x and y, none of the given options fit. Therefore, the correct answer is "None of these."