# 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?

Sep 29, 2015

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

#### Explanation:

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

$5 + P + C = 50 \rightarrow P + C = 45$

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

$30 + 1.5 P + 4.5 C = 145 \rightarrow 1.5 P + 4.5 C = 115$

$\rightarrow 1.5 \left(P + C\right) + 3 C = 67.5 + 3 C = 115$

$\rightarrow 3 C = 47.5 \rightarrow C = \frac{47.5}{3}$

$\rightarrow P + \frac{47.5}{3} = 45$

$\rightarrow P = 45 - \frac{47.5}{3} = \frac{135 - 47.5}{3} = \frac{87.5}{3} \approx 29.2$