# What is wrong with this simple equation?

Consider the following equation: $$\Bigl(\bigl((100 + 5 - 2) * 1.15 + 6\bigr) * (1.07) - 3\Bigr) = 130.1615.$$

I want to take the above and add 3% non-compounded, meaning I want to take 3% of 100 and add it to the value, but I don't want to do:

$$\Bigl(\bigl((100 + 5 - 2) * 1.15 + 6\bigr) * (1.07) - 3\Bigr) + (100 * 0.03).$$

Is there a way I can add 3 percent without using 100. I thought I could do:

$$\Bigl(\bigl((100 + 5 - 2) * 1.15 + 6\bigr) * (1.07) - 3\Bigr) * 1.03,$$ but this doesn't work, and $$\Bigl(\bigl((100 + 5 - 2) * 1.15 + 6\bigr) * (1.07) - 3\Bigr) * 0.03$$ doesn't work either

Here is basically what I have to do, I have to reverse out the percentages and flat amounts to get back to the original value. I do not know the original value. I am only given the final value and in what order the percentages and flat amounts were applied. My other posts were mainly just dealing with compound amounts only or non-compound amounts. I actually did ask about mixing compound, non-compound, and flat, but never got a straight answer. I have already written a program to reverse the final amount using the adjustments and when I do the following it works fine:

(((100 + 5 - 2) * 1.15 + 6) * (1.07))

As soon as I want to add 3% non-compounded to this, the reason I don't want to use 100 is because I actually won't know this. This is just test data I am setting up, so I am trying to figure out how to do it without using 100 * (0.03).

Essentially, I am given 133.1615 and the adjustments and I have to reverse them out, now if they are all compound, non-compound, or flat, I am fine, but if I mix them, my program doesn't do it correctly. Right now I have if I am given:

(((100 + 5 - 2) * 1.15 + 6) * (1.07)) + (100 * 0.03) = 136.1615, to reverse it I am doing:

(((136.1615 / 0.03 + 1.07) - 6) / 1.15) + 2 - 5 which I know is wrong, regarding the 136.1615 / 0.03 part.

In resposne to Arturo's answer:

My program loops through the adjustments and as long as it keeps seeing a flat value, it adds them together. When it sees a non-compounded or compounded adjustment, it adds the flat value to the amount. The same applies to compound and non-compound amounts. As long as the previous value was a compound, it just multiplies the current one to the previous one and if it is non-compounded, it adds it or subtracts it from the previous one, so given 130.1615, my program prints statements like this to show what it is doing:

130.1615 + 3
133.1615 / 1.07
124.45 - 6
118.45 / 1.1150 = 103
103 - 3 = 100

PURPOSE:

Given a dollar amount (value), remove adjustments (flat amounts, compound percentages, non-compound percentages) from the value to get back the initial value before the adjustments were applied. The adjustments don't necessarily have to just be added to the initial value, they also be removed, so in that case, you must add the adjustment to get back to the initial value.

INPUT:

Post-Adjusted Value
Set of Adjustments

DESIRED OUTPUT:

Pre-Adjusted Value

• Of course they don't work. This is essentially the same issue that you've been asking (and getting answers for) since you started. When you multiply the entire thing by 1.03, you're compounding. – Arturo Magidin Apr 22 '11 at 16:36
• @Xaisoft: My point is that you seem to not be learning much from previous answers, and continue to try multiplying or dividing by quantities without putting any thought into it, in the hopes that it will magically produce what you want. – Arturo Magidin Apr 22 '11 at 16:39
• @mixedmath: Look up the previous questions by the OP. They all have the same flavor/topic of trying to add basic percentages with or without compounding. – Arturo Magidin Apr 22 '11 at 16:40
• @Xaisoft: So I have to assume that you are not asking what you are really meaning to ask, because you will not always have "100" as the main quantity. What you really want is to have the total of the above expression for a given value of $X$, and you want to figure out how to manipulate that total to add 3% of the original $X$, knowing only the total above. Here;s a hint: to get a question answer, ask the question, not something else. And I have to say that contrary to your claim, you do not "understand form a formula mathematical standpoint", given your manipulations. – Arturo Magidin Apr 22 '11 at 16:49
• @Xaisoft: "I have no idea if they put +3 or -2 and +5. I have to assume they will put the worst case scenario." To me, this sentence is absolute nonsense insofar as computations go. Since your question is about computations, I still have no idea why you put "-2+5" instead of putting "+3", unless you really don't understand formulas. – Arturo Magidin Apr 22 '11 at 17:04

## 2 Answers

\begin{align*} \Bigl(\bigl( (X+3)*1.15 + 6\bigr)*1.07 - 3\Bigr) &= \Bigl(\bigl( 1.15X + 9.45\bigr)*1.07 - 3\Bigr)\\ &= 1.2305X + 7.1115 \end{align*} To get 3% of the original $X$, you want to take the total you have, $T$, subtract 7.1115, divide the result by 1.2305, and then multiply by $0.03$. So if $T$ is the total, you want $$\left(\frac{T - 7.1115}{1.2305}\right)0.03.$$

For $X=100$, $T$ is equal to $130.1615$. Subtracting $7.1115$ gives $123.05$. Dividing that by $1.2305$ gives $100$ (the original $X$). Multiplying by $0.03$ gives $3$.

To add 3% of the original quantity $X$ to the total $T$ you are given, simply take $$T + \left(\frac{T - 7.1115}{1.2305}\right)0.03.$$

If you are instead given $$1.2305X + 7.1115 + 0.03 X = 1.2605X + 7.1115$$ then subtract $7.1115$, and divide by $1.2605$ to get back the original quantity. If you were given 133.1615, subtracting 7.1115 gives you 126.05; dividing by 1.2605 gives you back the original quantity 100. (Your total of 136.1615 is incorrect; you had 130.1615 before adding 100*0.03, so you get 133.1615, not 136.1615).

• Arturo, using your first equation above, I will update my post and show you how my how my program is calculating it. – Xaisoft Apr 22 '11 at 17:22

Firstly, I suspect you don't just to add 3% of 100. 3% of 100 is 3, and so your full equation would simply have a " +3 " at the end.

Instead, I will assume that you really want to increase the overall result by 3%. What that means is that your previous answer, 130.1615, get's multiplied by 1.03 (representing 1 * 130.1615 + 0.03* 130.1615, a 3% increase). Writing it all together, one gets $[(( 100 + 5 - 2) * 1.15 + 6)(1.07) - 3](1.03)$.

But... this is compounding.