Mixture Problems

This chapter discusses a common type of word problem that can be solved by linear equations: mixture problems. Read the chapter, watch the videos, and work through examples. Complete the review exercise at the end of the chapter.

Mixture Problem: Coins

Janine empties her purse and finds that it contains only nickels (worth 5 cents each) and dimes (worth 10 cents each). If she has a total of 7 coins and they have a combined value of 55 cents, how many of each coin does she have?

Since we have 2 types of coins, let's call the number of nickels $x$ and the number of dimes $y$ . We are given two key pieces of information to make our equations: the number of coins and their value.

# of coins equation: $x+y = 7$ (number of nickels) + (number of dimes)

value equation: $5x + 10y = 55$ (since nickels are worth $5c$ and dimes $10$ .

We can quickly rearrange the first equation to isolate $x$ .

$x=7-y$	now substitute into equation 2:
$5(7-y)+10 y=55$	distribute the 5:
$35-5 y+10 y=55$	collect like terms
$35+5 y=55$	subtract 35 from both sides:
$5 y=20$	divide by 5:
$\underline {y=4}$	substitute back into equation 1:
$x+4=7$	subtract 4 from both sides:
$\underline {x=3}$

Janine has 3 nickels and 4 dimes.

Sometimes a question asks you to determine (from concentrations) how much of a particular substance to use. The substance in question could be something like coins as above, or it could be a chemical in solution, or even heat. In such a case, you need to know the amount of whatever substance is in each part. There are several common situations where to get one equation you simply add two given quantities, but to get the second equation you need to use a product. Three examples are below.

Type of mixture	First equation	Second equation
Coins (items with $ value)	total number of items $\left(n_{1}+n_{2}\right)$	total *value* (item value x no. of items)
Chemical solutions	total solution volume $\left(V_{1}+V_{2}\right)$	amount of *solute* (vol x concentration)
Density of two substances	total amount or volume of mix	total *mass* (volume x density)

For example, when considering mixing chemical solutions, we will most likely need to consider the total amount of solute in the individual parts and in the final mixture. (A solute is the chemical that is dissolved in a solution. An example of a solute is salt when added to water to make a brine.) To find the total amount, simply multiply the amount of the mixture by the fractional concentration. To illustrate, let's look at an example where you are given amounts relative to the whole.