1964 AMC 12 Problem 23

Attempt Problem 23 of the 1964 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1964 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

23.

Two numbers are such that their difference, their sum, and their product are to one another as

1:7:24.

The product of the two numbers is:

6

12

24

48

96

Represent the difference, sum, and product by

k,

7k,

and

24k

The two numbers are

\frac{7k+k}{2}

and

\frac{7k-k}{2}

Let the difference be

k

and the sum be

7k.

The numbers are

4k

and

3k,

so their product is

12k^2.

But the ratio also says the product is

24k.

For distinct numbers

k\ne0,

k=2

and the product is

48.

Therefore, the correct answer is D.

Problem 23 in other years