All About Percentages

The detail concepts of the Percentage is collected from various source. I guess it will useful for others. What is a Percentage? A fraction with denominator 100 is called percentage. To convert a fraction into percentage, multiply by 100 and put % sign.

Expressing one quantity as percentage of other:Ex: What percent is number 5 of number 25?

Out of 25-> 5,

Out of 1-> 5/25,

Out of 100-> (5/25)*100= 20%.

Percentage Change: General Formula: Percent change=\(((Change in Quantity)/(Original Quantity)*100)\)

Percentage increase=\((Increased value-original value)/(Original value)*100\)

Percentage decreased=\((Original value-Decreased value)/(Original value)*100\)

Percentage Change Advance: If a value R is increased by x%, then decrease the increased resultant value \(

(R+Rx/100)\) by \(

(x/(x+100)*100)%\) to get back the original value R.

If a value R is decreased by x%, then increase the decreased resultant value\(

(R-Rx/100)\) by\(

(x/(100-x)*100)%\) to get back the original value R.

Ex: 1000 is increased by 20% then resultant value is 1200.

20/120*100=16.67% (Rounded to nearest digit)

So, 1200 is reduced by 16.66% = 1200*16.67/100=200.04

1200- 200.04= 1000 – the approximation due to the round off- If we use 16.66666666666667 we will get the exactly 200.

Percentage Increased/ Reduced BY and Percentage Increased/ Reduced TO: If a quantity is reduced

by x% then result will (100-x)% of orginal, and if a quantity is reduced

to x%, then the new value is x% of original value.

By represent difference and to represents the final value.

Ex: if a rate of product is reduced by 30%, then the new rate will be 70% of original, and if the rate of the product is reduced to 30%, then new rate will be 30% of original.

Percentage in POPULATIONS:If the original population of region is A, and annual growth is x%. Then population after R years is \(A(1+x/100)^R\).

Increase in Population –\(

A[(1+x/100)^R-1]\)

If the original population of region is A, and decrease in population is x%. Then population after R years is \(A(1-x/100)^R\).

Decrease in Population- \(

A[1-(1+x/100)^R]\).

If a value of a number is first increased by R% and then decreased by R%, the net change is always a decrease or loss in original value.

Hence, % Loss or decrease = \((R/10)^2%.\)

Other Important Notes: If a value R is increased by x% to S, and S is again decreased to R by y%, Then x is always greater than y (Positive values only).

If a value R is decreased by x% to S, and S is again increased to R by y%, then x is always less than y.

If a value P is increased by x% then by y% and then by z%, the final value will be same even if we reverse the order of x,y,z. I.e., P is increased by z% first, then by y% and then by x%. (Same for the decrease)- Successive increase or decrease in value.

Similarly if value P is increased by X% then by Y% and then decreased by Z%. the final value will be same when P is decreased by Z% first, then increased by Y%, and then by X%.

If there is an increase of x/y in any value A, then the increased value will be \(A (1+x/y)\).

If there is a decrease of x/y in any value A, then the increased value will be \(A (1-x/y)\).