# 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.

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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