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Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion


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March 23rd, 2017, 02:38 AM   #1
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Percentage problem - which is probably really simple!!

Hi,

I just need to say in advance that I know my problem is probably quite simple, but for some reason I have completely confused myself with this and now I need someone to explain the logic behind it as it's driving me insane and I can't continue on with my work until I have the answer:

I'm trying to calculate the balance on an account - I work in accounts - and I am inputting the VAT value - which is 20% of the gross figure.

Gross total: £110.35
VAT: 18.35
Net: 91.96 (round to 2.dp)

The percentage problem:

To calculate the net figure, I need to remove 20% of VAT from the gross total.

Usually, I would divide by 1.2:

110.35 / 1.2 = 91.958333 which rounded up = 91.96

The confusion:

As I understand it, when I divide the gross figure by whole number of 1 with decimal of 0.20 (1.2), this removes 20% from the figure of 110.35

What I don’t understand is when I try to remove 20% by multiplying 110.35 by 0.2 and then subtracting this from the gross figure, I get a completely different Net value.

i.e.

I Calculate 20% of 110.35

20/100 x 110.35 = 22.07


Then subtract this from 110:35

110.35 - 22.07 = 88.28

Why does dividing by 1.2 give a completely different answer to simply working out the percentage of the gross number and subtracting that from it?

From my understanding on both occasions I am removing 20% of the number.

But the methods give completely different answers.

Sorry if this is long-winded, but any help would be much appreciated.

Thanks.

Fi

Last edited by skipjack; March 23rd, 2017 at 03:56 AM.
Tubbyfej is offline  
 
March 23rd, 2017, 03:06 AM   #2
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If £n is the net price and £g is the price including VAT,
g = n + 20% of n = n(1 + 20%) = n × 1.2, and so n = g/1.2.

Equivalently, n = g - g/6, because 1 - 1/6 = 5/6 = 1/1.2.
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March 23rd, 2017, 03:28 AM   #3
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Quote:
Originally Posted by Tubbyfej View Post
Hi,

I'm trying to calculate the balance on an account - I work in accounts - and I am inputting the VAT value - which is 20% of the gross figure.

Gross total: £110.35
VAT: 18.35
Net: 91.96 (round to 2.dp)

The percentage problem:

To calculate the net figure, I need to remove 20% of VAT from the gross total.

Usually, I would divide by 1.2:

110.35 / 1.2 = 91.958333 which rounded up = 91.96
This operation is not "reducing the gross total by 20%". To do that, you perform

$\displaystyle 110.35 \times 0.8 = 88.28$

But I don't think that's correct in the context of your work.

---

Firstly, let's set VAT to 20%.

Secondly, let's remember that percentages are applied to particular values. In our case, VAT is a percentage tax applied to the net value. Repeat... it is applied to the net value. Therefore, if you start with a net value of 91.96, the gross value (net + tax) can be calculated using

$\displaystyle 91.96 \times 1.2 = 110.352.$

However, what if you started with the gross value and wanted the net value? You do the inverse operation to get what you started:

$\displaystyle 110.352 / 1.2 = 91.96$

And this is why you divided by 1.2.

---

Extra info:

If you want to derive the percentage of gross value that is tax, you'll get a different value from your VAT, and rightly so because that percentage would apply to the gross value, not the net value. Different value; different percentage. The percentage is obtained using

percentage = $\displaystyle \left(1 - \frac{91.96}{110.352}\right) \times 100 \% = 16.\dot{6}\%$

You would then specify the relationships between net and gross values as follows:

$\displaystyle 91.96 / 0.8\dot{3} = 110.352$

and

$\displaystyle 110.352 \times 0.8\dot{3} = 91.96$

Notice the divide and multiplication signs have swapped from your original problem? It's because this percentage is applied to the gross value, not the net value.
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Last edited by skipjack; March 23rd, 2017 at 03:57 AM.
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March 23rd, 2017, 07:49 AM   #4
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Thank you so much for this detailed explanation!!!!

It really makes sense now....you've made my day

Fi
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