# Problem of the Week Problem E and Solution Mixture of Three

## Problem

Kanza is making and selling trail mix. She makes three different blends, each consisting of a mixture of cashews, dark chocolate, and almonds. All of these blends are sold at the same price of $$\18$$ per kg.

If she mixes cashews, dark chocolate, and almonds in the ratio of $$1:1:1$$, by mass, then she makes a profit of $$20\%$$.

If she mixes cashews, dark chocolate, and almonds in the ratio of $$3:2:1$$, by mass, then she makes a profit of $$8\%$$.

If she mixes cashews, dark chocolate, and almonds in the ratio of $$1:4:2$$, by mass, then she makes a profit of $$26\%$$.

1. What price, in dollars per kg, does Kanza pay for each of the cashews, dark chocolate, and almonds?

2. What percentage of a profit would she make if she mixes cashews, dark chocolate, and almonds in the ratio of $$2:3:4$$, by mass?

## Solution

Let $$c$$ be the price Kanza pays for cashews, in dollars per kg.
Let $$d$$ be the price Kanza pays for dark chocolate, in dollars per kg.
Let $$a$$ be the price Kanza pays for almonds, in dollars per kg.

Consider the blend where she mixes cashews, dark chocolate, and almonds in the ratio of $$1:1:1$$, by mass. In $$1$$ kg of this blend, $$\frac{1}{3}$$ kg is cashews, $$\frac{1}{3}$$ kg is dark chocolate, and $$\frac{1}{3}$$ kg is almonds. Also, $$1$$ kg of this blend will cost Kanza $$\frac{1}{3}c + \frac{1}{3}d + \frac{1}{3}a$$ and will be sold for $$\18$$. Since she makes a profit of $$20\%$$, we have $1.2\left( \frac{1}{3}c + \frac{1}{3}d + \frac{1}{3}a\right) = 18$ Multiplying by $$3$$, we obtain $1.2(c + d +a) = 54$ Dividing by $$1.2$$, we obtain $c + d + a = 45 \tag{1}$

Consider the blend where she mixes cashews, dark chocolate, and almonds in the ratio of $$3:2:1$$, by mass. In $$1$$ kg of this blend, $$\frac{1}{2}$$ kg is cashews, $$\frac{1}{3}$$ kg is dark chocolate, and $$\frac{1}{6}$$ kg is almonds. Also, $$1$$ kg of this blend will cost Kanza $$\frac{1}{2}c + \frac{1}{3}d + \frac{1}{6}a$$ and will be sold for $$\18$$. Since she makes a profit of $$8\%$$, we have $1.08\left( \frac{1}{2}c + \frac{1}{3}d + \frac{1}{6}a\right) = 18$ Multiplying by $$6$$, we obtain $1.08(3c + 2d + a) = 108$
Dividing by $$1.08$$, we obtain $3c + 2d + a= 100 \tag{2}$

Consider the blend where she mixes cashews, dark chocolate, and almonds in the ratio of $$1:4:2$$, by mass. In $$1$$ kg of this blend, $$\frac{1}{7}$$ kg is cashews, $$\frac{4}{7}$$ kg is dark chocolate, and $$\frac{2}{7}$$ kg is almonds. Also, $$1$$ kg of this blend will cost Kanza $$\frac{1}{7}c + \frac{4}{7}d + \frac{2}{7}a$$ and will be sold for $$\18$$. Since she makes a profit of $$26\%$$, we have $1.26\left( \frac{1}{7}c + \frac{4}{7}d + \frac{2}{7}a\right) = 18$

Multiplying by $$7$$, we obtain $1.26(c + 4d + 2a) = 126$
Dividing by $$1.26$$, we obtain $c + 4d + 2a= 100 \tag{3}$

We now need to solve the following system of equations. \begin{align*} c + d + a &= 45 \tag{1}\\ 3c + 2d + a&= 100 \tag{2}\\ c + 4d + 2a&= 100 \tag{3}\end{align*}

First, subtracting equation $$(1)$$ from equation $$(2)$$ gives $2c+d = 55 \tag{4}$ Second, doubling equation $$(1)$$ and then subtracting equation $$(3)$$ gives $c- 2d = -10 \tag{5}$

We will now use equations $$(4)$$ and $$(5)$$ to solve for $$c$$ and $$d$$. Doubling equation $$(4)$$ and then adding equation $$(5)$$ gives $5c= 100$ or $c = 20$

Substituting $$c=20$$ into equation $$(4)$$, we get $$2(20) + d = 55$$ or $$d=15$$.

Substituting $$c=20$$ and $$d=15$$ into equation $$(1)$$, we find $$a=10$$.

1. Therefore, Kanza pays $$\20$$ per kg for cashews, $$\15$$ per kg for dark chocolate, and $$\10$$ per kg for almonds.

2. Suppose she mixes cashews, dark chocolate, and almonds in the ratio of $$2:3:4$$, by mass. Then, in $$1$$ kg of the blend, $$\frac{2}{9}$$ kg is cashews, $$\frac{1}{3}$$ kg is dark chocolate, and $$\frac{4}{9}$$ kg is almonds.

Also, $$1$$ kg of this blend will cost Kanza $$\frac{2}{9}c + \frac{1}{3}d + \frac{4}{9}a = \frac{2}{9}(20) + \frac{1}{3}(15) + \frac{4}{9}(10) = \frac{125}{9}$$ dollars.

She sells $$1$$ kg of this blend for $$\18$$. Since $$18 \div \frac{125}{9} = 1.296$$, the percentage profit that Kanza makes on this mixture is $$29.6\%$$.