Question

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?

Answer 1

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 :)