Percentages - Formulas & Tricks

Percentage is a crucial concept in quantitative aptitude frequently tested in competitive exams. It represents a part of a whole expressed in hundredths and is commonly used to calculate discounts, interest rates, and comparisons between values. Understanding percentages is essential for solving various mathematical problems efficiently.

Formulas and Quick Tricks for Percentage Questions

1. Percentage means per 100, i.e., p% means p / 100.
2. To convert a fraction to a percentage, we multiply by 100 and add the “%” sign. For example, to express 1 / 5 in percentage, we simply multiply by 100, (1 / 5) × 100 = 20 %.
3. To convert a percentage to a fraction, we simply divide by 100. For example, 25 % = 25 / 100 = 1 / 4.
4. Expenditure = Price × Consumption.
5. If the price of an article increases by P %, the necessary reduction in consumption to avoid an increase in expenditure = [( P / (100 + P) ) × 100] %.
6. If the price of an article decreases by P %, the necessary increase in consumption to keep the same expenditure = [( P / (100 – P) ) × 100] %.
7. Population: If the population of a group/community/country/place(etc.) is currently P and if it increases by R % every year, then
   1. Population after ‘n’ years = P × [1 + (R / 100)]ⁿ
   2. Population before ‘n’ years = P / [1 + (R / 100)]ⁿan
8. Depreciation: If the price (or value) of an article is currently P and if it depreciates by R % every year, then:
   1. Price (or value) after ‘n’ years = P × [1 – (R / 100)]ⁿ.
   2. Price (or value) before ‘n’ years = P / [1 – (R / 100)]ⁿ.
9. x % of y and y % of x is the same. For example, 10 % of 100 and 100 % of 10 are the same.
10. A successive increase of a% and b% is equivalent to a net increase of a + b + ((a × b) / 100) %.
11. A successive decrease of a% and b% is equivalent to a net decrease of a + b – ((a × b) / 100) %.
12. A successive increase of a% and decrease of b% is equivalent to a net change of a – b + ((a × (-b) / 100) % = a – b – ((a × b) / 100) %.
13. A successive decrease of a% and increase of b% is equivalent to a net change of b – a + (((-a) × b) / 100) % = b – a – ((a × b) / 100) %.
14. An increase by n % and a successive decrease by n % are equal to an equivalent decrease of (n/10)² %. For example, if the price of an article is increased by 10 %, and is then successively decreased by 10 %, then this is equal to a decrease of (10/10)² = 1 %.

Note – If there is a % decrease instead of a % increase, then we take the (-) negative sign.

Examples - Percentages

Example 1:

Problem Statement: A quality control machine rejects 0.1% of all tennis rackets. If the machine rejected 50 rackets on a certain day, how many rackets were manufactured that day?

Solution:

Let the total number of bottles produced be n.
=> 0.1% of n = 50
=> (0.1/100) × n = 50
=> n = 50 × (100/0.1)
=> n = 50,000

Therefore, the total number of bottles manufactured on that day = 50,000.

Example 2:

Problem Statement: The population of the city is 200,000 this year. If it grows at a rate of 4% per annum, what will the population be in 5 years?

Solution:

We know that if the population is currently P and it increases by R% every year, then the population after n years is given by:

Population after n years = P × [1 + (R/100)]ⁿ

• Population after 5 years = 200,000 × [1 + (4/100)]⁵
  • = 200,000 × (1.04)⁵
  • = 200,000 × 1.216652902 ≈ 243,330

Therefore, the population in 5 years will be approximately 243,330.

Related Articles: 

• Quiz on Percentage
• Problems on Percentage | Set-2.

Summary

Percentages are a fundamental concept in mathematics and are widely used in various fields, including finance, statistics, and everyday calculations. A percentage is a way of expressing a number as a fraction of 100. It’s often used to describe how large one quantity is about another. In aptitude tests, percentage questions typically involve calculating percentage increases or decreases, finding original values from percentages, comparing percentages, and solving real-world problems involving percentages.