Question

Large reusable bottles cost $5 more than small ones. Ten large bottles cost $30 less than 15 small ones. How much does one small bottle cost?

Answer 1

The cost of one small bottle is $16.00

Step-by-step explanation

Problem Statement

The question tells us that Large bottles cost $5 more than smaller ones while 10 large bottles cost $30 less than 15 small ones. We are asked to find the cost of one small bottle.

Solution

To solve this question, we need to interpret the statements within the question and write them out as solvable equations. Thus, the steps we shall take are as follows:

Step 1: Analyze the statements and write them in form of equations.

Step 2: Solve the equations simultaneously

Step 1:

Statement 1:

"Large reusable bottles cost $5 more than small ones"

`\begin{gathered} \text{This implies that if the cost of a large bottle is L} \\ \text{And the cost of a small bottle is S},\text{ then,} \\ L\text{ is greater than S by \$5.00} \\ \\ \therefore L=S+5\text{ (Equation 1)} \end{gathered}`

Statement 2:

"Ten large bottles cost $30 less than 15 small ones"

`\begin{gathered} \text{This implies that:} \\ \text{If we have 10 large bottles each costing \$L, then the price of 10 large bottles is 10L} \\ \text{ If we have 15 small bottles each costing \$S, then the price of 15 small bottles is 15S} \\ \\ \text{ We are told that 10 large bottles cost less than 15 small bottles by \$30.00} \\ \text{Thus,} \\ 10L=15S-30\text{ (Equation 2)} \end{gathered}`

Now that we have our equations, we can proceed to solve for the cost of a small bottle S, by substitution.

Step 2:

`\begin{gathered} L=S+5\text{ (Equation 1)} \\ 10L=15S-30\text{ (Equation 2)} \\ \\ \text{From Equation 1, we can find an expression for L in which we can substitute into Equation 2.} \\ 10L=15S-30 \\ \text{But, L=S+5, from Equation 1.} \\ 10(S+5)=15S-30 \\ \text{Expand the bracket} \\ 10S+50=15S-30 \\ \text{Collect like terms} \\ 50+30=15S-10S \\ 80=5S \\ 5S=80 \\ Divide\text{ both sides by 5} \\ (5S)/(5S)=(80)/(5) \\ \\ \therefore S=16 \end{gathered}`

Final Answer

The cost of one small bottle is $16.00