# Extract base value from a number assuming previous percentage markup?

I don't want to waste anyone's time with what is probably a really simple question, but I am having trouble extracting, and then correctly re-applying, some percentage markups to a budgeted value I am working with.

Obviously, under normal circumstances, I would start with a Cost and mark it up to get my final Sum Total, like so:

                   Labor         Material

Cost             10,000.00       10,000.00

Supervision 3%      300.00                    (Cost * Supervision)
Load 5%             515.00                    (Cost + Supervision) * Load
Waste 5%                            500.00    (Cost * Waste)
Tax 9.25%                           925.00    (Cost * Tax)
------------------------------------------
Sub Total        10,815.00       11,425.00

Overhead 10%      1,081.50        1,142.50    (Sub Total * Overhead)
Profit 10%        1,189.65        1.256.75    (Sub Total + Overhead) * Profit
------------------------------------------
Total            13.086.15       13,824.25

Sum Total        26,910.40


However, there are circumstances in which I will actually be working backwards from the Sum Total towards an initial Cost. The catch is I need to introduce two variables accounting for a Labor percent and a Material percent (something like 40% and 60%).

I've been able to track back correctly when one value is 100% and the other is 0%, but when I use any combination in the middle, I end up just a little off of my expected Sum Total.

I don't really know what information I need to supply to help find a solution (don't even know how to tag this question), but I'd really appreciate any help I can get on this scenario.

Thanks!

## 2 Answers

If you have a quantity $x$, and you want to add $y$% to it, you multiply by $\frac{100+y}{100}$.

Symmetrically, if you have a quantity $X$ which represents some $x$ plus $y$%, and you want to recover $x$, you multiply by $\frac{100}{100+y}$.

I'm assuming the following:

• You know the "Sum Total";
• You know what percentage of the "sum total" accounts for "Total Labor", and what percentage accounts for "Total Material."
• You know the percentage of Overhead (10%) and of Profit (10%).
• You know what percentages make up the "Sub Total"

You want to figure out "Cost" for each of Labor and Material.

You can find "Total Labor" (call it $\mathsf{TL}$) and "Total Material" (call it $\mathsf{TM}$) easily enough: if Labor is x% and Material is y% (with $x+y=100$), then $\mathsf{TL} =(\mathrm{Sum Total})(x/100)$ and $\mathsf{TM}=(\mathrm{Sum Total})(y/100)$.

The next two steps work the same for $\mathsf{TL}$ and for $\mathsf{TM}$, so we can just proceed with a single quantity $\mathsf{T}$ and work backwards.

1. $\frac{\mathsf{T}}{1.1}=\text{Subtotal}+\text{Overhead}$. (Takes off the 10% profit mark-up)
2. $\text{Subtotal} = \frac{1}{1.1}(\text{Subtotal}+\text{Overhead})$. (Takes off the 10% overhead mark-up).

Labor.

You have a subtotal $\mathrm{ST}$. Divide by $1.05$ to get the rid of the 5% Load. Then divide the result by $1.03$ to get rid of the 3% Supervision.

In summary: if Labor is x% of the total, \begin{align*} \text{Labor Cost} &= (\text{Total Cost})\left(\frac{x}{100}\right)\left(\frac{100}{110}\right)\left(\frac{100}{110}\right)\left(\frac{100}{105}\right)\left(\frac{100}{103}\right)\\ &= (\text{Total Cost})\frac{10000x}{(110)(110)(105)(103)}. \end{align*}

Material

Same idea: divide $\mathsf{ST}$ by $1.0925$ to strip away the tax, and then by 105 to strip away the waste.

In summary, if Material is y% of the total cost, \begin{align*} \text{Material Cost} &= \left(\text{Total Cost}\right)\left(\frac{y}{100}\right)\left(\frac{100}{110}\right)\left(\frac{100}{110}\right)\left(\frac{100}{109.25}\right)\left(\frac{100}{105}\right)\\ &= \left(\text{Total Cost}\right)\frac{10000y}{(110)(110)(109.25)(105)}. \end{align*}

It doesn't look like Labor and Material have anything to do with it. If you have a Sub Total, the overhead is 10% giving 110%, then profit is 10% of that=11% of Sub Total, giving Sum Total = 121% Sub Total. So Sub Total =$\frac{100}{121}$ Sum Total, Profit=$\frac{11}{121}$ Sum Total, Overhead=$\frac{10}{121}$ Sum Total