Basic accounting percentage question

Question

I'm looking at a question which I find a little bit confusing. The question is asking me to find the value of the sales tax (17.5%) from a gross figure (£7402).

I know that I can work out the correct answer by finding the difference between the net and the gross amounts:

Gross = 7402
Net = 7402 / 1.175 = 6299.57
Value of sales tax = 7402 - 6299.57 = 1102.43

However, in the book it tell me the answer should be derived from this equation:

7402 * 17.5 / 117.5 = 1102.43

This does give me the correct answer, but I don't understand why. Can someone please explain why this equation is giving me the correct answer?

Answer 1

Let's do it yet another way to start with:

7402 / 1.175 = 6299.57 is the net value, as you already calculated.
The sales tax is 17.5% of this net value, so you could calculate the sales tax as 6299.57 * 0.175 = 1102.43.

If you did these calculations all in one go you would have:
(7402 / 1.175) * 0.175 = 1102.43

Note that I can switch the order of the division and the multiplication and get the same result:
7402 * 0.175 / 1.175 = 1102.43

I can actually do the division first without changing the result: 7402 * (0.175 / 1.175) = 1102.43

And finally that fraction in the brackets will be the same if I multiply the top and the bottom by 100: 7402 * (17.5 / 117.5) = 1102.43

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An alternative way to think about it, is to say that the tax is 17.5% of the net price and the gross amount is 117.5% of the net price:

Tax = 17.5% of Net
Gross = 117.5% of Net

If you divide these the "% of Net" will cancel and you'll get:

Tax / Gross = 17.5 / 117.5
and so 
Tax = Gross * (17.5 / 117.5)

Answer 2

You can interprete like that: First you calculate 1% of the gross price. So you divide 7402 by 117.5, because the gross price is 117.5% (=100%+17.5%) of the net price (100%). Then you have to calculate the taxes from the net price, which is 17.5-times of 1% of the net-price. : $(7402 / 117.5)*17.5$

greetings,

calculus

Answer 3

Let $g=7402$ and $r=.175$. Then $$ g - \frac{g}{1 + r} = g \left (1 - \frac{1}{r+1} \right) = g \left( \frac{(r+1)-1}{r+1} \right) = g \left( \frac{r}{r+1} \right) = g \left( \frac{10 \cdot r}{10 \cdot( r+1)} \right). $$ The leftmost side was your computation, whereas the rightmost side was the one provided by the text.