Percentage

The term percentage means per hundred. Percentages are fractions with a denominator as 100. Percentages are always written with symbol %. Hence, 7% means \(\frac{7}{100}\).

Percentages are ratios, that represent the part of a whole. Meaning, 12% is 12 parts of 100.

Basic Rules of Percents

Compute percentage, given part and the whole

This is done by dividing the part over whole and multiply by 100.

% = \(\frac{part}{whole}\)100

For example,

To find what percent of 120 is 24, we just divide 24 over 120 and multiply the value by 100. Hence, it is 20 %.

In the same way, to find 13 is 24% of what number, we divide 13 over 24%. Hence, the value is, 54.16.

Percentage change

When the value of a product changes from an old to the new value, its percentage change is calculated by, finding the difference between old and new value and dividing the value by old value and multiplying by 100.

percentage change = \(\frac{{old - new }}{old}\) 100

Keep in mind,

If the value of percentage change is negative, it means the value of the product has reduced.

Consecutive Percentage Changes

Till now we have seen how to calculate the percentage, now we will look into how to calculate the final value with two consecutive percentage changes.

In this case, we have a simple formula to calculate the net percentage change. If the successive percentage changes are x% and y% then, the net percentage change is \((x + y + \frac{xy}{100})\).

Keep in mind,

If at all there is a decrease in change in value, the formula is changed accordingly. For example, the value of a product is increased by a % and then decreased by b%, then the updated formula is, \((a - b - \frac{ab}{100})\).

Example, if the value of product is increased by 12% and further increased by 3%, then the net percentage change is, \((12 + 3 + \frac{36}{100})\) that is 15.36%.

In the same way, if the value of a product is increased by 14% and then given a discount of 10%, then the net percentage change is \((14 - 10 - \frac{140}{100})\) that is 2.6%.

Undoing Percentage Changes

The next step in percentage is, how to find the original value. Meaning, if a product is discounted and we know the final value and the percentage of discount, how to find its initial value.

If a is the percentage by which the value is decreased or increased and y is the final amount, then the initial value x is calculated using the following formula,

\(x = y (\frac{100}{{100 + a}})\)

Keep in mind,

If the value is discounted, then change the sign in the formula to negative.

For example,
If a product is being sold for 270 after giving a discount of 15%, then what is the original price of the product?

In order to answer this question, we use the above formula with negative sign as the value is being discounted. Hence, on applying the above formula we get,

\(x = 270 (\frac{100}{{100 - 15}})\)
On solving the above equation we obtain 317.64 as the value.

Population Formula

Example:
The population of a city was 2 years ago 24000, population decreases every year at the rate of 5 %. Find the present population of the city.
Solution :

population was decreases at a particular rate so

Present population of that city is
\(=24000 * ( 1 - 5 %)^2\)
\(=24000 * ( 1 - 5 / 100)^2\)
\(=24000 * ( 19 / 20) * (19 / 20 )\)
\(=21660.\)