A gift puzzle

With Christmas around the corner, I thought this puzzle would be appropriate.

This is a nice problem that I found in a math puzzle book.

The other day Bob went to three stores and bought a small gift from each.

He then wanted to total up his receipts to know how much he spent. Bob entered the three numbers on his cell phone calculator and saw the result was 9.96. But Bob was clumsy and realized he made a mistake: he had pressed the multiplication symbol x instead of the addition symbol +.

Bob realized this and redid the calculation. To his surprise, the result was the same when he added the prices: 9.96! To be sure, Bob checked his calculations a couple times and verified this was indeed the correct total.

Bob was tickled that he found three numbers whose sum was the same as their product.

What were the prices of the gifts that Bob bought?

(And I know, 9.96 is really too little to spend on three gifts, but it’s a math problem…)

Can you solve it?

Give it a try. I have provided the answer below.

Reference: A mathematical jamboree

Answer below
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The answer

We know the sum of the gifts and the product are the same number:

x + y + z = xyz = 9.96

These equations imply

z = (x + y)/(xy – 1)

These facts lead us to investigate factors of 996 which are candidates for the gift prices. To convert back to the original problem where the terms add up to 9.96, we must divide one of the factors by 100.

A bit of trial and error and we can find the answer is 1.66, 7.50, and 0.80.