Finding total after percentage has been used?

Tried my best with the title. Ok, earlier while I was on break from work (I have low level math, and want to be more fond of mathematics)

I work retail, and 20% of taxes are taken out, and I am wanting to find out how much I made before 20% is taken out.

So I did some scribbling, and this is without googling so please be gentle. but lets say I make 2000 dollars per paycheck, so to see how much money I get after taxes I do 2000 * .8 which gives me 1600. I can also do 2000 * .2 which gives me 400 and then do 2000 – 400 (imo I don’t like this way since I have a extra unnecessary step)

Anyways, I was pondering how to reverse that to have the answer be 2000, and what I did was 1600 * 1.20 (120%) which gave me 1920, and I thought that is odd so I did 1600 * 1.25 and answer was 2000 exact.

My question is why did I have to add extra 5% (25%) to get my answer? I am sure I did something wrong, and I fairly confident the formula I am using is a big no no.

edit; wow thank you all for your detailed answers. I am starting to like mathematics more and more.

Solutions Collecting From Web of "Finding total after percentage has been used?"

If $20\%$ of your pay is removed for taxes, then you retain $80\%$ of it. Thus, the amount you receive after taxes is

$$\text{net} = 80\%~\text{of gross} = \frac{80}{100}~\text{gross} = \frac{4}{5}~\text{gross}$$

To determine your gross pay from your net pay, you must multiply your net pay by $5/4$ since

$$\text{gross} = \frac{5}{4} \cdot \frac{4}{5}~\text{gross} = \frac{5}{4}~\text{net}$$

Since $$\frac{5}{4} = \dfrac{125}{100}$$ your gross pay is actually $125\%$ of your net pay.

When $20\%$ is taken out, you have $80\%$ or $0.80$ of the original amount. If the amount was $x$ you now have $0.80x$. If you want to recover $x$, you need to divide by $0.80$. As $\frac 1{0.8}=1.25$, you can multiply by $1.25$ When the percentage $p$ is small, multiplying by $1-p$ and then multiplying by $1+p$ gets you back close as the product is $1-p^2$. As the percentage gets larger, to undo multiplying by $1-p$ you need to divide by $1-p$

Thanks for your well-asked question!

If your gross pay is $\$2,000$ and $20\%$ is taken out, you’re left with $\$1,600$.

So, $\$1,600$ is $(100-20)\%$ of $\$2,000$.

Now you want to know what percent of $\$1,600$ is $\$2,000$.

In other words, $1600x = 2000$.

Solving for $x$ we get $x = 2000/1600 = 1.25 = 125\%.$

In other words, you add $25\%$ of $\$1,600$ to $100\%$ of $\$1,600$ to get $\$2,000$.

Or, $\$2,000$ is $(100+25)\%$ of $\$1,600$.

Hope this helps!

Lets say $x$ is how much you earned before taxing and $y$ is the amount of money after taxing. As you calculated yourself in an instance, this follows the formula

$0.8 \cdot x = y$. To get from $y$ to $x$, you can simply divide by $0.8$

$\Rightarrow x = \frac{y}{0.8}$.

Also we see $0.8 = \frac{4}{5} \Rightarrow \frac{1}{0.8} = \frac{5}{4} = 1.25$.

$80\% = 80/100$. Since we’ve multiplied this into the gross pay to get the net pay, we have to do the opposite, which is to say divide by this, to go from net pay to gross pay. Dividing by $a/b$ is the same as multiplying by $b/a$, so you have to multiply your net pay by $100/80=5/4=125/100$ to get your gross pay back.

In puzzling situations like this it’s often instructive to imagine an extreme example. Suppose 50% of your salary went for taxes (yes that’s a high tax rate, but good for this problem). Then you take home half your gross pay. If you know your take home pay you should double it to find your gross. In the equations in the other answers that corresponds to
$\frac{1}{1-0.5} = 2$.

For an even more extreme example, think about a 100% tax rate. Then there’s no way to recover your gross from a takehome pay of 0. If you try the algebra, $\frac{1}{1-1} = \frac{1}{0}$ makes no sense.

I see from your profile that you’re a programmer. Extreme cases like these are analogous to testing boundary conditions in your code (things like loops that execute 0 times).

Let x to be your gross salary and y = 1600 to be your net salary. Because tax is 20% of gross salary then we have

x = y + tax = 1600 + 0.2x
=> 0.8x = 1600
=> x = 2000 USD