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?

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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}