Billy is twice as old as Sally

Outline Mathematics
Word Problems

Here's a problem to tackle:

Billy is twice as old as Sally was when Billy was as old as Sally is now. And the sum of their ages is 28. How old are they now?

Solution


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Copyright © 1996-2017 Alexander Bogomolny

Solution

Billy is twice as old as Sally was when Billy was as old as Sally is now. And the sum of their ages is 28. How old are they now?

Assume that right now Bill is b years old and Sally is s. The problem sounds like Bill is older than,older than,younger than,the same age as Sally. This means that b > s. Let d be the difference in their ages: d = b - s,d = b - s,d = s - b,d = 10.

d years ago, when Bill was b - d = s, Sally was s - d,s + d,s - d,5,10,twice as old, or 2s - b,s - 5,b - 5,s + b,2s - b, because s - d = s - (b - s) = 2s - b. This number multiplied by 2,4,1/2,2,s is the present Bill's age. We thus get an equation b = 2(2s - b),b = 2(2s - b),b = s + 2,b = s - 2,b = 2s,b = 2(s + b). Another equation is derived directly from the formulation: b + s = 28,b - s = 2,b + s = 2b,b + s = 28. We arrive at the system of two equations we'll have to solve:

b = 2(2s - b)
b + s = 28.

The first equation can be rearranged:

3b = 4s
b + s = 28.

There are many ways to solve a system of two linear equations. Sometimes a way to handle them is suggested by the manner in which they have been written. Just as a possibility, let's multiply the second equation by, say, 3:

3b + 3s = 84.

Now recollect that 3b = 4s, so that a substitution of 4s instead of 3b into the equation leads to a single equation:

4s + 3s = 84.

It follows that s = 12,b,10,11,12,13. From 3b = 4s we obtain that b = 16,16,17,19,21.

Do not forget to check your solution.

References

  1. J. A. H. Hunter, Mathematical Brain-Teasers, Dover Publications, 1976

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