Formula to get price to sell at depending on margin

Question

$sp$ = sell price

$cp$ = cost price

$p$ = profit

$m$ = margin

So to get $p,$ we do:

$$sp - {sp\over6} - 0.08sp - 0.024sp - cp$$

or, with figures:

$$12.99 - {12.99 \over6} - 12.99 \times0.08 - 12.99 \times0.024 - 8.63$$

This gives $0.84$ profit.

To get margin, we do:

$0.84 / (12.99 / 1.2)$

This gives: $7.8\%$ margin

What would the formula be if we wanted to get the sell price / price to sell at, with a margin of $5\%$?

Apologies for the formatting, I'm new here.

Answer 1

You say profit

$$p = s_p - \frac{s_p}{6} - 0.08s_p - 0.024s_p - c_p$$

and margin is

$$m = \frac{p}{s_p / 1.2} $$

We can simplify these two expressions to give

$$ p = \frac{547}{750}s_p - c_p \quad \text{ and } \quad m = \frac{6p}{5s_p} $$

We then substitute:

$$ m = \frac{\frac{547}{125}s_p - 6c_p}{5s_p} = \frac{547}{625} - \frac{6c_p}{5s_p} $$

Rearranging:

$$ s_p = \frac{6c_p}{\frac{547}{125} - 5m} = \frac{750c_p}{547 - 625m} $$

If margin is $5\%$ and $c_p = 8.63$ then $s_p = 12.55$.

Answer 2

You want:

$$\frac{p}{sp/1.2} = 0.05 \quad (1)$$

$$p= sp - {sp\over6} - 0.08sp - 0.024sp - cp \quad (2)$$

$$cp = 8.63$$

Substitute $p$ as in $(2)$ to $(1)$ and solve for $sp$