Diophantus of Alexandria, who is known as the father of algebra, lived in Greece about 250 A.D. We know how long he lived by this algebraic riddle that has been passed down over the centuries:

Question

of his life was spent as a boy.
of his life was spent as a youth.
of his life later he married.
5 years after marriage a son was born.
The son lived only 2 as long as his father did.
Diophantus died 4 years after his son's death.
Write an equation with one unknown and solve it to find how long Diophantus lived.

Answer 1

Let y = The number of years that Diophantus lived.
Equation: Solve for y: 1
Y 1 -1211 ± ± 5 + + 4
To add the fractions, express each with the lowest common denominator (LCD). LCD = 6 * 7 * 2 = 84

Note: The product of 2 and 6 is 12. Therefore, 12 is not included in the LCD. The common multiple 6 * 12 * 7 * 2 = 1008 can be used in place of the LCD, if you wish. The solution below uses the LCD of 84.
y = (14y / 84) +(7y / 84) + (12y / 84) + (5 * 84 / 84) + (42y / 84) + (4 * 84 / 84)
84y = (14 + 7 + 12 + 42)y + (9 * 84) 84y = 75y + (9 * 84)
84y — 75y = 75y — 75y + (9 * 84) 9y = 9 * 84
y= 84
Solution: Diophantus lived 84 years. Multiplied both sides by 84 • Added y coefficients on right Subtracted '75y from both sides Subtracted y terms
Divided both sides by 9

Check: of his life was spent as a boy.
u of his life was spent as a youth
of his life later he married
5 years after marriage a son was born
The son lived only z as long as his father did
Diophantus died 4 years after his son's death 1 * 84
6
1 * 84
12
* 84
7
1 * 84
2 =14
=7
= 12
=5
= 42
=4

Adding all the parts give: 14 + 7 + 12 + 5 + 42 + 4 = 84 The solution checks.