Claire bought three bars of soap and five sponges for $2.31. Steve bought five bars of soap and three sponges for $3.05. How do you find the cost of each item?

1 Answer
Jan 12, 2017

A bar of soap costs #52¢#, or #$0.52#
A sponge costs #15¢#, or #$0.15#

Explanation:

This question involves setting up a system of equations. If you are not familiar with systems of equations, I'd suggest you watch this video before proceeding.

Now, to set up a system, we need variables. Let's call the price of one soap bar #x#, and the price of one sponge #y#. With this information, we can construct the following:

Claire:
3 soaps and 5 sponges for $2.31 #=> 3x + 5y = 231#

Steve:
5 soaps and 3 sponges for $3.05 #=> 5x + 3y = 305#

Note: I have converted dollars to cents, simply to keep everything in whole numbers. We'll convert back to dollars in the end.

Now we have a system. Notice, however, that nothing cancels by simply adding or subtracting equations. The common step is to manipulate one equation so that things will cancel, but doing this here will lead to a lot of messy fractions. Hence, we will manipulate both equations. We will multiply Claire's equation by 5, and Steve's equation by 3. This gives us:

#15x + 25y = 1155#
#15x + 9y = 915#

Now, notice that we have a #15x# in both equations. This means that if we subtract one equation from another, the #x#'s will cancel each other out completely, leaving us with one variable - #y# - to solve for. As shown:

#(15x + 25y = 1155) - (15x + 9y = 915)#

#=> 16y = 240#

Now, to solve for #y#. Dividing both sides by 16 give us:

#y = 240/16 = 15#

Now that we know what #y# is, we can plug it into any of our two initial equations, and solve for #x#. I will chose Claire's equation (the first one):

#3x + 5(15) = 231#
#=> 3x = 156#
#=> x = 52#

Now we have #x# and #y#, let's go back to what they actually mean in the context of this problem:

A bar of soap (#x#) costs #52¢#, or #$0.52#
A sponge (#y#) costs #15¢#, or #$0.15#

Hope that helped :)