What are reverse percentages?

The percentage refers to the ratio of the fraction expressed in terms of $$100$$. To calculate the percentage of a number, divide it by whole and multiply the received number by $$100$$. Percentage means any number multiplied by $$100$$. It does not have dimensions because of which it is known as a dimensionless number. 

Where reverse percentage is used?

If the value of percentage increase or decrease is given along with the product's final value, and you need to find the initial value before the percentage increases or decreases at that time, the reverse percentage is used. Or it can be said that the value of some quantity after it has been either increased or decreased by a given percentage. Then, find the value of that quantity before it was increased or decreased, and it will be calculated with the help of reverse percentage.

For example, Radhika goes out for shopping shoes. She saw that a sale of $$25%$$ on shoes was there, whose sale price became $$40$$ dollars. From this, it is clear that $$40$$ dollars is the sale price after the discount of $$25%$$. Then, she started calculating the original price of the shoes before having a discount.

Another example of this is a car whose cost is increased by $$15%$$ to $$15630$$ dollars. Then, to calculate the original price before the increase, use reverse percentage to find them.

E1.12: Carry out calculations involving reverse percentages.

Reverse percentage

The reverse percentage formulation is nothing; however, the percentage system is used to discover the quantity or share of something in a hundred phrases. In its easiest form, the percentage potential is per hundred. It is described as a variety represented as a fraction of a hundred. It is denoted using the symbol $$%$$ and is mostly used to evaluate and discover ratios. Reverse percentages are a way of working a share hassle backwards to discover the authentic amount.

Calculation for reverse percentage

When there is an increase in the percentage, add the increased percentage in $$100$$, and find the original value. If there is a decrease in the percentage, then subtract the decreased percentage in $$100$$, and then find the original value. There are two types of problems in reverse percentages: a reverse percentage from $$1%$$ to $$100%$$, and a reverse percentage using decimals. 

For example, Utsav went to buy a t-shirt, and he found that there is a discount of $$10%$$ on the t-shirt, and after the discount, the cost of the t-shirt is $$30$$ dollars. Then, finding the original value before the discount is calculated by reverse percentage.

Worked examples

Example 1: Club membership was increased in price by $$20%$$ to $$30$$ dollars per month. Find the cost of the club before it increased in price.

Step 1: Initial the price increased

The total percentage of price increased is $$100%+20%=120%$$, equal to $$30$$ dollars.

Step 2: Calculate the value of $$1%$$ increase in price

$$120$$ will be equal to $$30$$.

So, $$1$$ will be equal to $$\frac{30}{120}=0.25$$.

Step 3: Multiply by $$100$$ for calculating the original price

By multiplying $$100$$ in $$0.25$$, we get $$25$$ dollars.

So, the original price is $$25$$ dollars per month.

Example 2: Sangeeta has a new record for $$150$$ metre sprint in $$12$$ seconds. This is $$6%$$ faster than her previous record. Find her previous record.

Step 1: Initial the time decreased

The total percentage of time decreased is $$100%-6%=94%$$, which is equal to $$12$$ seconds.

Step 2: Calculate the value of $$1%$$ decreased in time.

$$94$$ will be equal to $$12$$.

So, $$1$$ will be equal to $$\frac{12}{94}=0.127696$$.

Step 3: Multiply by $$100$$ for calculating the previous record.

By multiplying $$100$$ in $$0.127696$$, we get $$12.7696$$ seconds.

So, her previous record is $$12.7696$$ seconds.