Rhind Papyrus Problem 67

Problem

The following problem is from the Rhind Papyrus (1650 BC):

A shepherd had a flock of sheep. He had to give a portion of his flock to a lord as tribute. The shepherd was told to give two-thirds of one-third of his original flock. The shepherd gave 70 sheep. How many sheep did the shepherd original have?

Solution

Let $n$ be the size of the original flock. Two-thirds of one-third of the original flock translates to

$${\frac {2}{3}}\left({\frac {1}{3}}n\right).$$

Since 70 sheep were given as tribute, we get:

$${\frac {2}{3}}\left({\frac {1}{3}}n\right)=70$$

$${\frac {2}{9}}n=70$$

$$n={\frac {9(70)}{2}}$$

$$n=315.$$

The shepherd had 315 sheep in his original flock.