SOLUTION: A family has three daughters. The product of the daughters' ages is 200. The oldest daughter is twice the age of the middle daughter. What is the sum of their ages? (A) 18 (B) 2

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Question 244884: A family has three daughters. The product of the daughters' ages is 200. The oldest daughter is twice the age of the middle daughter. What is the sum of their ages?
(A) 18 (B) 20 (C) 22 (D) 27 (E) 31
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
We'll start be writing expressions for the three girls' ages.
Let x = youngest girl's age
Let y = middle girl's age
Since the oldest girl is twice the age of the middle girl:
2y = age of the oldest girl

Now we can write an equation:

or

Normally we need two equations to solve a problem with two variables. However, since ages are usually expressed as Natural Numbers (1, 2, 3, ...) we can use some logic and trial and error to find an answer.

We'll start by dividing both sides by 2x:

In this form we can see that 100/x (or 100 divided by the youngest girl's age) must be a perfect square. So we just have to try dividing 100 by some Natural Numbers to see if we can find one (or more) that result in a perfect square.

The first Natural Number is 1 and 100/1 is a perfect square! So if x = 1 then . This makes y = 10 and 2y = 20 and the sum 1 + 10 + 20 = 31. And we have already found the answer: E.

BTW, there is another possible set of ages. x = 2 and x = 3 do not work because 100/2 and 100/3 are not perfect squares. But x = 4 4 works because 100/4 = 25 which is a perfect square. This makes y = 5 and 2y = 10 and the sum 4 + 5 + 10 = 19. This sum is not one of the multiple choice answers but the ages do fit the problem.

P.S. Any other set of ages will not fit because the youngest child ends up older than the middle child.