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I was helping my sister learn percentages. We've got a simple exercise: product X costs $153.75$ including 23% tax. Calculate price without tax.

Now, the answer is of course to solve $1.23x = 153.75$. She however tried another way: $153.75 - 0.23 \cdot 153.75 = x$ with intuition to just subtract the tax from price. This is obviously wrong. I showed her the right way and we worked another example where the tax was 100% (and where the error is obvious).

She now knows that the second way is wrong, she sees the numbers that come out... but she told me that she still does not understand why exactly the second method does not work. And I couldn't find a way to explain this without using examples with specific numbers.

Question: How to explain this problem?

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Did you tell her that you need to take 23% on what you call $x$ and not $153.75$? From the way she subtracts, it seems that she thinks that the 23% is on $153.75$. – Rankeya Nov 5 '11 at 23:44
Also $0.76(1.23x)$ $\neq$ $x$. That would imply that 0.76 is the inverse of 1.23, which it isn't. – Rankeya Nov 5 '11 at 23:52
up vote 3 down vote accepted

As you said, it would be difficult to explain without specific numbers, but I'll give it a try. The original price gets a tax added, giving the total price. The tax added is a given percent of the original. When she subtracted the same percent from the total price, she was getting what the tax would be on the total price, a larger number than the tax on just the original price.

Let me try WITH numbers. :-) If the original price is $\$100$ (to make it easy), and the tax is 10% (again easy numbers), then the total price is $\$110$. The problem with taking the 'tax' back off of the total, trying to get back to the original, is that we would now subtract $\$11$ from the total, getting an 'original' price of only $\$99$. Tax on the original is not the same as tax on the larger total price.

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23% of $x$ is smaller than 23% of the larger quantity 153.75.

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Use distributivity (if she understands that)

Adding tax is $\text{price} + \text{tax}\cdot \text{price}$

In other words after applying distributivity: $\text{price}\cdot(1+\text{tax})$

Then you'll need to explain that $\dfrac{1}{1+\text{tax}} \neq(1-\text{tax})$

Or make the mistake really obvious:

You have $\text{taxedprice} = \text{price} + \text{tax}\cdot\text{price}$

She tried to solve price with $\text{price} = \text{taxedprice} - \text{tax}\cdot\text{taxedprice}$

Fill in the original $\text{taxedprice}$ in the formula and you'll get $\text{price} = (\text{price} + \text{tax}\cdot\text{price})- \text{tax}\cdot\text(\text{price} + \text{tax}\cdot\text{price})$ or $\text{price} = \text{price} + \text{tax}\cdot\text{price}- \text{tax}\cdot\text{price} -\text{tax}\cdot\text{tax}\cdot\text{price}$. Which would mean that $0 = -\text{tax}\cdot\text{tax}\cdot\text{price}$

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You can write $\text{tax}\cdot\text{price}$ or $\text{tax}\times\text{price}$. Using an asterisk for ordinary multiplication is a workaround for occasions when one is limited to the characters on the keyboard. Its use in MathJax or LaTeX or the like is vulgar. ${}\qquad{}$ – Michael Hardy Sep 25 '15 at 15:57

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