Computing the rate of commission

Question

Benyamin receives 45% of the commission on every painting he sells. If he recently sold a painting of \$256,000 and received a commission of \$7,488, what was the total rate of commission?

I am not getting what exactly is being asked since we do not know the total number of painting?

My Try:

Let $x$ be the total commission. then $(0.45)x=\$7488$ and so $x=\$16640$. Now we have got total commission; how to proceed?

Answer 1

Like you say, the total commission for the painting is $\$16\,400$, so the commission rate is $$\frac{\$16\,400}{\$256\,000} = 0.065 = 6.5\%.$$

Here's another way to organize a solution:

If the sell price of the painting is $P$, and the rate of commission is $\color{red}{r}$, then the total commission is $Pr$; Benyamin receives a fraction $f = 45\% = 0.45$ of this, so his commission is $$B = P\color{red}{r}f. \qquad(\ast)$$

For this painting, we have $P = \$256\, 000$ and Benyamin's commission is $B = \$7\,488$, so substituting these known values into $(\ast)$ gives $$(\$7\,488) = \$256\,000 \times \color{red}{r} \times 0.45.$$ Solving for the rate of commission $\color{red}{r}$ by dividing gives $$\color{red}{r} = \frac{\$7\,488}{0.45 \times \$256\,000} = 0.065 = 6.5\%.$$

Answer 2

Let $A$ denote the sales price, $B$ denote the commission, and $C$ denote Benyamin's share of that commission.

We have $C=0.45B$ and $B=xA$, where $x$ is what we want to learn. We also know $A=256000$ and $C=7488$.

Can you finish from here?