# Suppose, 2/3 of 2/3 of a certain quantity of barley is taken, 100 units of barley are added and the original quantity recovered. find the quantity of barley? This is a real question from the Babylonian, posited 4 millenia ago...

Sep 12, 2016

$x = 180$

#### Explanation:

Let the quantity of barley be $x$. As $\frac{2}{3}$ of $\frac{2}{3}$ of this is taken and $100$ units added to it, it is equivalent to

$\frac{2}{3} \times \frac{2}{3} \times x + 100$

It is mentioned that this is equal to the original quantity, hence

$\frac{2}{3} \times \frac{2}{3} \times x + 100 = x$ or

$\frac{4}{9} x + 100 = x$ or

$\frac{4}{9} x - \frac{4}{9} x + 100 = x - \frac{4}{9} x$ or

$\cancel{\frac{4}{9} x} - \cancel{\frac{4}{9} x} + 100 = x - \frac{4}{9} x = \frac{9}{9} x - \frac{4}{9} x = \frac{9 - 4}{9} x = \frac{5}{9} x$ or

$\frac{5}{9} x = 100$ or

$\frac{9}{5} \times \frac{5}{9} x = \frac{9}{5} \times 100$ or

$\frac{\cancel{9}}{\cancel{5}} \times \frac{\cancel{5}}{\cancel{9}} x = \frac{9}{5} \times 100 = \frac{9}{\cancel{5}} \times 20 \cancel{100} = 180$ i.e.

$x = 180$