1510.11 – Heap: An Egyptian Equation

What may be the earliest recorded equation is found in the ancient Egyptian records of Ahmes the Scribe in the following form:

“Heap: its two-thirds, its half, its seventh, its whole gives ninety-seven.”

If we interpret this as instructing us to add two thirds of the heap, half of the heap, one seventh of the heap, and one whole heap together, find the size of the heap.

Solution

Let $x$ be the size of the heap. Then,

\begin{aligned} \frac{2}{3}x + \frac{1}{2} x + \frac{1}{7} x + x = 97, \\ \frac{28}{42}x + \frac{21}{42} x + \frac{6}{42} x + \frac{42}{42} x = 97, \\ \frac{97}{42} x = 97, \\ x = 42. \end{aligned}

So the size of the heap is $42$ . Of course, we are not given any units so we are left to imagine the size of the heap. Nor do we know what it's a heap of. (Yes, this sentence ends in a preposition.)