Question

It cost George $\$678$ to make wreaths. How many wreaths must he sell at $\$$15 apiece to make a profit?

Answer 1

Hint: We need to find the number of wreaths to be sold such that George makes profit. We will first assume the number of wreaths he sold as k. Then, we will find the revenue that he gets, which is the amount he gets by selling k wreaths at the price of \[\$15\] per piece. We have a cost price as \[\$678\]. To make a profit, the amount of revenue must be more than the cost price of wreaths.

Complete step by step solution:
We are asked to find the minimum number of wreaths George should sell to make a profit. We are given that the cost of making wreaths is \[\$678\], and that he will sell each wreath at the price of \[\$15\] per piece.
Let us assume that the number of wreaths he should sell for a profit gain is $k$.
We know that the selling price of one piece is $\$15$, so we get the total revenue he gets by selling $k$number of wreaths by multiplying $\$15$ and $k$ as \[\$15k\].
We know that to make profit the revenue must be greater than the cost price. Thus, we can write an inequality as
\[\begin{align}
  & \Rightarrow 678<15k \\
 & \Rightarrow 15k>678 \\
\end{align}\]
Dividing both sides by 15, we will get
\[\Rightarrow \dfrac{15k}{15}>\dfrac{678}{15}\]
\[\Rightarrow k>45.2\]
Since k is the number of wreaths, it must be a positive integer. By taking the nearest integer we get that, George must sell at least 46 wreaths for $\$15$ to make profit.

Note: Here we take the inequality sign as greater than because George wants to make a profit. If we were asked to find quantities such that he makes no loss, we would have used ‘greater than and equal to \[\left( \ge \right)\] in the inequality. Calculation mistakes should be avoided.