Percentages

Understanding Percentages

  • A percentage is a measure of proportion or ratio. It is a way of expressing a number as a fraction of 100. The term ‘per cent’ actually means ‘per hundred’.
  • Percentages are used widely in everyday mathematics and are particularly relevant in topics such as interest rates, discounts, and statistics.
  • A percentage can be converted to a fraction by dividing it by 100 (e.g. 50% as a fraction is 50/100 or 1/2) or to a decimal by dividing by 100 (e.g. 50% as a decimal is 0.5).

Calculating Percentages of a Number

  • To find a percentage of a number, you can convert the percentage into a decimal or a fraction and then multiply it by the given number.
  • For example, to find 30% of 60: as a decimal calculation this would be 0.30 * 60 = 18, or as a fraction it would be 30/100 of 60 = 18.

Working with Percentage Increase and Decrease

  • Percentage increase and percentage decrease are calculations often used to compare changes in quantities, especially in business and financial contexts.
  • To calculate a percentage increase: find the difference between the new and original quantities, divide that by the original quantity, and then multiply by 100 to convert it to a percentage.
  • To calculate a percentage decrease: find the difference between the original and new quantities, divide that by the original quantity, and then multiply by 100 to convert it to a percentage.

Reverse Percentage Problems

  • In reverse percentage problems, you’re often given the final amount (after an increase or decrease) and the percentage of the change, and you’re asked to find the original quantity.
  • To solve this, you can consider that the final quantity represents 100% plus or minus the percentage of the change.

Practice Problems

  • Problem: Calculate 45% of 220.
    • Solution: 99 (0.45 * 220 = 99)
  • Problem: If a dress originally cost £80 and is then reduced in a sale by 25%, what is the new price?
    • Solution: £60 (80 - 80*25/100 = 60)
  • Problem: A population of rabbits increases by 20% to 720. What was the original population?
    • Solution: 600 (720 / 120 * 100 = 600)

Key Takeaways

  • Percentages are a hugely useful tool for comparing proportions and ratios, and you can convert them into decimals or fractions to make calculations easier.
  • You’ll often find percentages in everyday use, particularly in financial contexts such as interest rates and discounts.
  • Percentage increase and percentage decrease help us measure changes in quantity, and reverse percentages allow us to find original quantities from the final amount and rate of change.
  • Practice is crucial, so keep working with percentages in different forms and contexts to become more comfortable and proficient with them.