Question

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

Answer 1

Hint:We use the concepts of solving linear equations in two variables, to solve this problem. A linear polynomial is a polynomial of degree 1. We will see the solving of the linear equations, by removing one variable and simplifying. This is also called the elimination method.

Complete step by step answer:
There are two items in this problem. A soap and a sponge.
Let the cost of a single bar of soap be \[x{\text{ cents}}\].
And let the cost of a single sponge be \[{\text{y cents}}\].
It is given that, cost of three bars of soap and five sponges is \[{\text{\$ 2}}{\text{.31}}\] which is equal to \[{\text{231 cents}}\].
So, we can write this situation in mathematical form as,
\[3({\text{cost of single soap) + 5(cost of single sponge) = 231 cents}}\]
\[ \Rightarrow 3x{\text{ + 5y = 231}}\] -----(1)

And in the second case, the cost of five soaps and three sponges is \[\$ 3.05\] which is equal to \[305{\text{ cents}}\].
We can write this as,
\[5({\text{cost of single soap) + 3(cost of single sponge) = 305 cents}}\]
\[ \Rightarrow 5x{\text{ + 3y = 305}}\] ------(2)
Now, consider these two equations.
\[3x{\text{ + 5y = 231 and 5}}x + 3y = 305{\text{ }}\]
Multiply the first equation by 5 and the second equation by 3, to make the coefficients of \[x\] in both equations equal.
\[ \Rightarrow 15x{\text{ + 25y = 1155 and 15}}x + 9y = 915{\text{ }}\]

Now, subtract the second equation from the first on both sides.
\[ \Rightarrow \left( {15x + 25y} \right) - \left( {15x + 9y} \right){\text{ = 1155}} - 915{\text{ }}\]
\[ \Rightarrow 15x + 25y - 15x - 9y = 240\]
So, finally, we get it as,
\[ \Rightarrow 16y = 240\]
\[ \Rightarrow y = \dfrac{{240}}{{16}} = 15\]
So, value of single sponge is \[15{\text{ cents}}\]
Now, substitute this value either in (1) or (2)
\[ \Rightarrow 3x + 5(15) = 231\]
\[ \Rightarrow 3x = 156\]
\[ \therefore x = \dfrac{{156}}{3} = 52\]

So, the value of a single bar of soap is \[52{\text{ cents}}\].

Note:Make a note that, \[1\$ = 100{\text{ cents}}\] and make conversions as per this. After getting the values of variables, substitute back in the situations and verify your answer. By making coefficients of a variable equal, we can eliminate that variable and our problem becomes easier. In this problem, we made the coefficients of \[x\] equal. But instead, we can also make the coefficients of \[y\] equal.