SOLUTION: Helen's age next year will be three times Dan's age last year. Their present ages total 32. How old is each now?

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Question 113120: Helen's age next year will be three times Dan's age last year. Their present ages total 32. How old is each now?
Answer by solver91311(24713) About Me  (Show Source):
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This year Helen is h years old, and Dan is d years old. Helen's age next year then must be h + 1. And Dan's age last year must have been d - 1. The problem tells us:
Helen's age next year (h + 1) will be (=) 3 times Dan's age last year 3(d - 1).

So we can write:
Eq. 1) h%2B1=3%28d-1%29

We also know that their present ages, h and d, total 32, so we can write:
Eq. 2)h%2Bd=32

Start by solving Eq. 2) for one of the variables (doesn't matter which, I'll choose to solve for h)
h=32-d, Adding -d to both sides.

Now we have an expression that represents h in terms of d that can be substituted into Eq. 1), thus:
%2832-d%29%2B1=3%28d-1%29

Now, simplify and solve for d:
33-d=3d-3 Distributive and Associative Properties
-d-3d=-3-33 Adding -3d and -33 to both sides
-4d=-36 Collecting terms
d=-36%2F-4=9 Divide both sides by -4, and now we know that this year, Dan is 9. Since the sum of their ages is 32, Helen must be 32 - 9 = 23.

Let's check the answer.
Helen's age next year will be 24. Dan's age last year was 8. 3 times 8 is 24. Furthermore, 9 + 23 = 32. So our numbers satisfy both of the conditions of the problem. Answer checks.

Hope that helps,
John