Tom buys $18$ lobsters at the same cost. He sells $16$ lobsters at a $60\%$ profit. Each of the remaining lobsters are sold at the same loss. It is given that the overall percentage profit is $50\%$.
A) Find the percentage of loss of selling the remaining lobsters.
B) Suppose that the total selling price of the lobster $\$1755$, find the loss for each remaining lobster.
This is what I tried: let $x$ be the cost of $1$ lobster. Therefore, the profit will be $9.6x$ for selling $16$ lobsters. Also, the selling price for $16$ lobsters would then be $25.6x$ That’s all I was able to get so far!
Best Answer
Let $c$ be each lobster's cost price. Then the total revenue (= total/net selling price) from selling the $18$ lobsters is $1.5$ times their total cost price, which is $1.5(\$18c) = \$27c.$ This total revenue is made up of two parts: (i) The total revenue from selling $16$ lobsters each at a selling price of $\$1.6c,$ which is $16(\$1.6c) = \$25.6c.$ (ii) The total revenue from selling $2$ lobsters each at a selling price of $2\left(1 - \frac{r}{100}\right)(\$c),$ where $r\%$ is the percentage loss for selling each of these $2$ lobsters. Therefore, we have
$$ \$27c \;\; = \;\; \$25.6c \; + \; \$2\left(1 - \frac{r}{100}\right)c $$
Dividing both sides by $c$ and rearranging a bit (and working with pure numbers) gives
$$ 27 - 25.6 \; = \; 2 - \frac{r}{50} $$
$$ -0.6 \; = \; -\frac{r}{50} $$
$$ r \; = \; 30 $$
Therefore, the answer to (A) is $30\%.$
For (B), the total selling price is equal to $\$1755.$ Therefore, $\$27c = \$1755,$ or $c = \$65,$ and hence the loss amount for each of the $2$ remaining lobsters is $30\%$ of $\$65,$ which equals $(0.3)(\$65) = \$19.50.$