Problem E and Solution

Mixture of Three

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 \(26\%\).

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

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?

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 \(a=10\).

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

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 \(29.6\%\).