Let
A = present age of Albert
B = present age of Bryan

Present 
xyrs hence 
yyrs ago 
zyrs ago 
10yrs 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)
Comments
Question:
Question:
Why is the mathematical expression for ...when Albert is twice as old as Bryan was...
A + x = 2(B  y)
And not
A = 2(B  y) since "when Albert is" refers to the present?
"When Albert is" refers to
"When Albert is" refers to the time when "Albert is as old as Bryan will be" (the previous statement) which is x yrs in the future. Similar interpretation for "Bryan is as old as Albert was when Bryan was half the age he will be ten years from now"