How do you create your equations when working on a mixture word problem?

May 27, 2015

Let's try to think about the general form of a word problem involving mixtures.

In general, we have the following scenario:

• a merchant sells two kinds of products (coffee, sweets, etc).
• we know the unit prices for both kinds of products and for the final mixture
${p}_{1}$ US dollars per pound for the first kind of product,
${p}_{2}$ US dollars per pound for the second kind of product
${p}_{m}$ US dollars per pound for the mixture

• we know the total quantity formed by the mixture of the two products ($q$ pounds)

• we have to find out the quantities of each product needed to form the mixture
(here we have the variables: $x$ denoting the quantity of the first kind of product and $y$ denoting the quantity of the second kind of product)

Now, we have sufficient information to work out the equations.

First, we know that the sum of the two quantities is $q$ pounds, which gives us the first equation:
$x + y = q$

Second, we know that the sale price is the product of quantity and unit price, which gives us the second equation:
${p}_{1} x + {p}_{2} y = {p}_{m} \cdot q$

Now, we have a system of two linear equations that can be easily solved by substitution.