The ages of two brothers are in the ratio two to three, but in eight years, the ratio of their ages will be three to four. What is the age of the older brother?
18
16
24

-- Let's call the age of the older brother ' A '.

-- Their ages are in the ratio of 2 to 3, so the age of the younger guy is (2/3 A).

-- In 8 years, their ages will be  (A + 8)  and  (2/3 A + 8).

-- The ratio of their ages will be 3 to 4, so we can write

(2/3 A + 8) / (A+8)  =  3/4 .

Multiply each side by (A + 8):  

(2/3 A+8) = (3/4) (A + 8)

Eliminate the parentheses:

2/3 A + 8 = 3/4 A + 6

Subtract 2/3 A from each side:

8  =  1/12 A + 6

Subtract 6 from each side:

2 = 1/12 A

Multiply each side by 12:

24 = A

Check:

-- Their ages are in the ratio of 2:3 today ... they're 16 and 24

-- In 8 years, they'll be 24 and 32 .

-- The ratio of 24 to 32 is 3:4 .  yay !

by definition, an arc subtended by a central angle of some given measure has the same measure. so .