Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.


share|cite|improve this question
up vote 1 down vote accepted

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).


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, $$\begin{align*} \text{Labor Cost} &= (\text{Total Cost})\left(\frac{x}{100}\right)\left(\frac{100}{110}\right)\left(\frac{100}{110}\right)\left(\frac{100}{105}\right)\left(\frac{100}{103}\right)\\ &= (\text{Total Cost})\frac{10000x}{(110)(110)(105)(103)}. \end{align*}$$


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, $$\begin{align*} \text{Material Cost} &= \left(\text{Total Cost}\right)\left(\frac{y}{100}\right)\left(\frac{100}{110}\right)\left(\frac{100}{110}\right)\left(\frac{100}{109.25}\right)\left(\frac{100}{105}\right)\\ &= \left(\text{Total Cost}\right)\frac{10000y}{(110)(110)(109.25)(105)}. \end{align*}$$

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.