Understanding Maths Series:

*[ Percent (%) = per cent = in relation to (one) hundred]*

The following are a series of calculations on how to reach **∑ **(*the sum of*)

You should understand the relation of all figures in the equation. The *MAIN VARIABLE* will determine whether you use the multiply/divide by 0.*xx% or 1.xx%. *Generally, if the main variable is to increase, then we use 1.*xx% *but, if the main variable decreases then we should use 0.*xx% *. Therefore, identifying the main variable is imperative and find out whether we need to extract a larger or smaller figure from it.

Look out for this in the examples below:

**How do I find a percentage of a number?**

*Q. What is 12% of 48?*

The figure we are looking for is a smaller portion of 48, therefore 12% is represented as the decimal 0.12

*x*= ∑ 48 x 0.12

*x*= 5.76

In Q.1 the main variable is 48 . The extracted figure was **smaller** than the base, which is also the main variable. We multiplied it by **.12**

**How do I find out the result of***x*percent increase?

*Q. I have £48 and it increases by 12%*

In this calculation, the base figure (48) *increases* by 12% which means that *x* = 112% of the base.

112% is represented here by the decimal 1.12 (it is merely 112 represented as a rational number ÷100)

*x*= ∑ 48 x 1.12

*x*= 53.76

In Q.2 the main variable was still 48, this time the figure was to **INCREASE** so we multiply it by **1.12**

**NB**. the base is not increasing by 112% but is increasing only by 12%. To increase by 112% the calculation is more than 2 x base, the figure to multiply by would be 2.12

**How do I find out the result of***x*percent decrease?

*Q. I have £48 and it decreases by 12%*

In this calculation, the base figure (48) *decreases* by 12% which means that *x*= 88% of the base.

88% is represented here by the decimal 0.88 (it is merely 88 represented as a rational number ÷100)

*x*= ∑ 48 x 0.88

*x*= 42.24

Alternatively, if we use the 0.12 (12%) we will arrive at the sum 5.76 which you can subtract from the base, like so:

48 x 0.12 = 5.76 (this is 12% of the base)

48 – 5.76 = 42.24 (if you subtract the 12% from the base you arrive at the sum)

In Q.3 main variable is 48 yet again. The extracted figure was **smaller** than the base, which is also the main variable. We multiplied it by **.88**

**Percentages as a proportionate figure**

*Q. 48 is 12% of x, how do I find ‘x’?*

The result is 12% of *x *so we convert 12% into it’s decimal format: 0.12

*x (multiplied by) 0.12 = 48*

:. (therefore) 48 ÷ 0.12 = *x*

*x = 400*

Q.4 the main variable is ‘*x’*. The formula is actually *x (multiplied by) y% = z. *The figure represented by ‘*x*‘ is *larger *than the base figure(48), therefore to find 12% of *x * we would use the sum: ‘*x’ multiplied by* 0.12 but that is not the question.

What we are doing is finding a **smaller **figure from *x, the main variable*.

We know 12% of *x *is 48 so we divide 48 by 12% of it’s parent (*x*) using **0.12**

*‘x’ X 12% = 48 —-> ‘x’ X 0.12 = 48 —-> We now rearrange the formula:*

‘*x’ X 0.12 ÷ 0.12 = 48 ÷ 0.12 —-> *‘

*x’ X*~~0.12 ÷ 0.12~~ = 48

*÷*0.12**—->**‘*x’ = 48**÷*0.12*oh, alternatively! (48 ÷ 12)* X 100 🙂 but that’s too easy!

**Representative percentage increases**

*Q. I had £x and it increased by 12% to £48, what is x?*

I’m trying to find out what I used to have, so this becomes the main figure. The result is 112% of ‘*x’*

*x (multiplied by) 1.12 = 48*

:. 48 ÷ 1.12 = 42.86

Q.5 main variable = *x. The formula is actually x (multiplied by) y% = z. *

*‘x’ ***increased **in size therefore we use the figure: **1.12**

*x’ X 12% = 48 —-> ‘x’ X 1.12 = 48 —-> We now rearrange the formula:*

‘*x’ X 1.12 ÷ 1.12 = 48 ÷ 1.12 —-> *‘

*x’ X*~~1.12 ÷ 1.12~~ = 48

*÷ 1*.12**—->**‘*x’ = 48**÷ 1*.12**Representative percentage decreases**

*Q. I had £x and it decreased by 12% to £48, what is x?*

I’m trying to find out what I used to have, so this (*x*) becomes the main figure. The result decreased (by 12% which is equal to 88% value of ‘*x*‘) so *x * is obviously larger than 48.

We know that 48 is 88% (100 – 12) of ‘*x’*

*x (multiplied by) 0.88 = 48*

*:. 48 ÷ 0.88 = 54.54 (r)
*

*Q.6 the main variable is again x. This is because, like in question 4, the formula is actually x (multiplied by) y% = z. In this case we have x (multiplied by) 88% = 48 because we are trying to find 88% of ‘x‘ Once again, we are searching for a smaller figure of x *

Here is another example:

I have $48 to spend on CD’s. The shop has a discount of 20% on selected items, what is the highest pre-discount priced item I can purchase?

:. the formula is: *x **X y% = 48*

If we multiply *x by 1.2 it will = a mark the price upwards. This is not correct because* ‘*x’ *is greater than 48.

If we multiply *x by o.2 it will = a 20% reflection of the price not a 20% reduction. This is not correct.*

:. we need to represent *x* with a 20% decrease in value, this is done by multiplying by 0.8 (See Q.3)

:. *x X 0.8 = 48 —-> We now rearrange the formula:*

‘*x’ X 0.8 ÷ 0.8 = 48 ÷ 0.8 —-> *‘

*x’ X*~~0.8 ÷ 0.8~~ = 48

*÷ 0*.8**—->***x = 48**÷ 0*.8*x = 60*

**Percentage increase differences**

This is very simple.

*Q. I have 48 apples, yesterday I had 40, what is the % increase?*

the difference between the two figures is 48-40 = 8

8 ÷ previous amount —-> 8 ÷ 40 = 20

the increase is 20%

**Percentage decrease differences**

*Q. I have 48 apples, yesterday I had 60, what is the % increase?*

the difference between the two figures is 48-60 = 12

12 ÷ previous amount —-> 8 ÷ 60 = 20

###### the decrease is 20%

Hope that all makes sense!