# Equivalency of percentage formulas

I know 3 methods for calculating percentages,

one example, find 70% of 50:

1) 50/100 * 70

2) 70/(100/50)

3) 70/100 * 50

I do not undertand how this 3 methods can be equivalent, also conceptually i do not understand method number 3. Can someone explain or give an understandable proof?

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#1: divide 50 into 100 parts, and take 70 parts out of it. #3: 70/100 = 70%, because "%" means per cent (100). You can rearrange all the terms so that #3 <-> #1 <-> #2 <-> #3. Different expressions give different meanings, and I think the meaning of #2 is very vague. – Frenzy Li Aug 12 '12 at 12:24
1. and 3. are the same thing, by virtue of multiplication being commutative. – J. M. Aug 12 '12 at 12:47
As explained in Ed Gorcenski's answer, 3 is the most natural formulation, as it clearly shows the percentage. The others are algebraically equivalent. – Ross Millikan Aug 12 '12 at 14:35

The important thing to remember about percent is that it literally means "per hundred"; so 70% means 70 [things] per hundred [things]. This is equivalent to a ratio; so if you have 70 things out of 100 things, you have 70% = 70/100 = 0.7. Those are all equivalent ways of writing 70%. Notice that percent is a dimensionless quantity. The dimensionless nature comes from the fact that $$70\% = \frac{70 [\mathrm{things}]}{100 [\mathrm{things}]} = \frac{70}{100} = 0.7$$

Therefore, 70% of 100 [things] is exactly 70 [things], which is the expected answer when we multiply the dimensionless quantity 70% by the quantity of 100 [things]: $$70\% \cdot 100\ [\mathrm{things}] = \frac{70}{100}\cdot 100\ [\mathrm{things}] = 70\ [\mathrm{things}].$$

Now, what happens when you don't have an even hundred [things]? Say you only have 50 [things], but you still want to estimate what the percent is. Obviously, 70% of 50 [things] is less than 50; so you cannot just say 70% of 50 = 70/50. Instead, you have to scale it. Remember that percent is dimensionless, so we do exactly the same operation as before: $$70\% \cdot 50\ [\mathrm{things}] = \frac{70}{100}\cdot 50\ [\mathrm{things}] = 35\ [\mathrm{things}].$$

Now you should be able to see why your formulas (1) and (3) are equivalent. Remember that division a quantity is like multiplying by one over that quantity: $\frac{3}{x} = 3\cdot \frac{1}{x}$. Also remember that multiplication is commutative. So, we have the following result:

$$70\% \cdot 50\ [\mathrm{things}] = \frac{70}{100} \cdot 50\ [\mathrm{things}] = \frac{1}{100}\cdot 70 \cdot 50\ [\mathrm{things}] = \frac{1}{100}\cdot 50 \cdot 70\ [\mathrm{things}]$$

EDIT:

As far as your formula #2, this works through simple manipulation of order of operations:

$$70/(100/50) = \frac{70}{\frac{100}{50}} = \frac{70\cdot 50}{100} = 70\cdot \frac{50}{100}.$$

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