A merchant has 5 pounds of mixed nuts that cost $30. He wants to add peanuts that cost $1.50 per pound and cashews that cost $4.50 per pound to obtain 50 pounds of a mixture that costs $2.90 per pound. How many pounds of peanuts are needed?

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Answer 1 · 2015-09-29T16:34:29

The merchant needs $29.2$ $29.2$ pounds of peanuts to make his mixture.

Be $P$ $P$ the quantity of peanuts added to the mixture and $C$ $C$ the quantity of cashews added to the mixture. We have:

$5 + P + C = 50 \to P + C = 45$ $5+P+C=50 rarr P+C=45$

If the mixture is to cost $$ 2.90$ $$2.90$ per pound, then 50 pounds will cost $$ 145$ $$145$ . Therefore we have:

$30 + 1.5 P + 4.5 C = 145 \to 1.5 P + 4.5 C = 115$ $30+1.5P+4.5C=145 rarr 1.5P+4.5C=115$

$\to 1.5 (P + C) + 3 C = 67.5 + 3 C = 115$ $rarr 1.5(P+C)+3C=67.5+3C=115$

$\to 3 C = 47.5 \to C = \frac{47.5}{3}$ $rarr 3C=47.5 rarr C=47.5/3$

$\to P + \frac{47.5}{3} = 45$ $rarr P+47.5/3=45$

$\to P = 45 - \frac{47.5}{3} = \frac{135 - 47.5}{3} = \frac{87.5}{3} \approx 29.2$ $rarr P=45-47.5/3=(135-47.5)/3=87.5/3~~29.2$