**Problem**

Albert is as old as Bryan will be when Albert is twice as old as Bryan was when Albert's age was half the sum of their present ages. Bryan is as old as Albert was when Bryan was half the age he will be ten years from now. How old are Albert and Bryan?

**Answer Key**

Albert = 40 yrs old

Bryan = 30 yrs old

**Solution**

*A*= present age of Albert

*B*= present age of Bryan

Present | x-yrs hence |
y-yrs ago |
z-yrs ago |
10-yrs hence | |

Albert | A |
A + x |
A - y |
A - z |
A + 10 |

Bryan | B |
B + x |
B - y |
B - z |
B + 10 |

Albert is as old as Bryan will be...

$A = B + x$

$x = A - B$

... when Albert is twice as old as Bryan was...

$A + x = 2(B - y)$

$A + (A - B) = 2(B - y)$

$2A - B = 2B - 2y$

$2y = 3B - 2A$

... when Albert's age was half the sum of their present ages

$A - y = \frac{1}{2}(A + B)$

$2A - 2y = A + B$

$A - B = 2y$

$A - B = 3B - 2A$

$3A = 4B$

$B = \frac{3}{4}A$

Bryan is as old as Albert was...

$B = A - z$

$z = A - B$

... when Bryan was half the age he will be ten years from now

$B - z = \frac{1}{2}(B + 10)$

$B - (A - B) = \frac{1}{2}(B + 10)$

$2B - A = \frac{1}{2}(B + 10)$

$4B - 2A = B + 10$

$3B - 2A = 10$

$3(\frac{3}{4}A) - 2A = 10$

$\frac{1}{4}A = 10$

$A = 40 ~ \text{yrs old}$ ← present age of Albert (*answer*)

$B = \frac{3}{4}(40)$

$B = 30 ~ \text{yrs old}$ ← present age of Bryan (*answer*)