# Profit or loss? Prem buys two kinds of lemons at the rate of 6 and 9 for a rupee. He mixes them and sells the whole lot at 8 for a rupee. Find his gain or loss percentage.

Let the number of the first kind of lemon Prem buys be x, and the number of the second kind of lemon be y. Prem's cost is then:

C = 1/6 x + 1/9 y = 0.167x + 0.111y

Prem mixes them together to get x + y lemons, and sells them for 1/8 rupee each. His revenue is:

R = 0.125(x + y)

His profit is:

P = R - C = 0.125(x + y) - 0.167x - 0.111y = -0.042x + 0.014y

The gain, or ratio of profit to cost, is P/C:

P/C = (-0.042x + 0.014y) / (0.167x + 0.111y)

Obviously, if Prem buys too many of the more expensive lemons, he won't make a profit. So the trick to this question is to examine the number of expensive lemons to the number of inexpensive lemons, the ratio r = x/y. So, multiply the profit ratio by 1/y / 1/y:

P/C = P/C * (1/y / 1/y) = (-0.042x/y + 0.014) / (0.167x/y + 0.111)

P/C = (-0.042r + 0.014) / (0.167r + 0.111)

So, what ratio r is necessary to break even, or when P = 0?

P = -0.042r + 0.014 = 0  --> r = 0.014/0.042 = 4/7

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