# 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...

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

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Let the original quantity of barley be ( x ) units.

Given that ( \frac{2}{3} ) of ( \frac{2}{3} ) of ( x ) is taken, the quantity remaining is ( \frac{1}{3} ) of ( \frac{2}{3} ) of ( x ), which is ( \frac{2}{9}x ).

Then, when 100 units of barley are added, the total quantity becomes ( \frac{2}{9}x + 100 ).

To recover the original quantity, the total amount must be doubled, as half was taken initially.

So, we have the equation:

[ 2(\frac{2}{9}x + 100) = x ]

Solving for ( x ):

[ 2 \times \frac{2}{9}x + 2 \times 100 = x ] [ \frac{4}{9}x + 200 = x ] [ 200 = x - \frac{4}{9}x ] [ 200 = \frac{5}{9}x ] [ x = \frac{200 \times 9}{5} ] [ x = 360 ]

Therefore, the original quantity of barley was 360 units.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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