1510.22 – Pebble's Long Life

Pebble lived one twelfth of his life as a small child and one sixth more as a young adult. He then married Bloomie and lived happily for one seventh of his life plus five years. They then had a son, Nelius, who, sadly, Pebble outlived by four years. Nelius lived to be half of Pebble’s final age. How old was Pebble when he died?

Solution

Suppose Pebble lived xx years. Then Nelius lived x2\dfrac{x}{2} years. To find Pebble’s final age, simply add the given bits together and perform some basic algebra.

x12+x6+x7+5+x2+4=x7x84+14x84+12x84+5+42x84+4=x75x84+9=x75x84=x−975x=84x−756756=84x−75x=9xx=84\begin{aligned}\dfrac{x}{12} + \dfrac{x}{6} +\dfrac{x}{7} + 5 +\dfrac{x}{2} +4 &= x \\[1em]\dfrac{7x}{84} + \dfrac{14x}{84} +\dfrac{12x}{84} + 5 +\dfrac{42x}{84} +4 &= x \\[1em]\dfrac{75x}{84} + 9 &= x \\[1em]\dfrac{75x}{84} &= x - 9 \\[1em]75 x &= 84x - 756 \\[1em]756 &= 84x - 75x = 9x \\[1em]x &= 84\end{aligned}