# 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?

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Copyright © 1996-2018 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

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

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

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

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