Interest & Depreciation (AQA GCSE Maths)

What is simple interest?

• Interest is extra money added every year (or month) to an original amount of money
• Simple interest is interest that is the same amount each time
  • It can be good: for example, putting £100 into a bank account and the bank rewarding you with simple interest of 10% every year
    • After one year you’d have £110, after two years you’d have £120, …
  • It can be bad: for example, owing £100 to a friend and they charge you simple interest of 10% for every year you don’t pay them back
    • After the first year you’d owe them £110, after the second year you’d owe them £120, …
• If £P is your initial amount of money and simple interest is added to it at a rate of R% per year for T years, then the total amount of interest gained, £I, is given by the formula 

• Remember that this formula calculates the amount of simple interest added over T years, not the total amount of money after T years
  • To find the total amount of money after T years, add the interest £I to the original amount £P

What is compound interest?

• Compound interest is where interest is paid on the interest from the previous year (or whatever time frame is being used), as well as on the original amount
• This is different from simple interest where interest is only paid on the original amount
  • Simple interest goes up by the same amount each time whereas compound interest goes up by an increasing amount each time

How do you work with compound interest?

• Keep multiplying by the decimal equivalent of the percentage you want (the multiplier, p)
• A 25% increase (p = 1 + 0.25) each year for 3 years is the same as multiplying by 1.25 × 1.25 × 1.25
  • Using powers, this is the same as × 1.25³
• In general, the multiplier p applied n times gives an overall multiplier of pⁿ
• If the percentages change varies from year to year, multiply by each one in order
  • a 5% increase one year followed by a 45% increase the next year is 1.05 × 1.45
• In general, the multiplier p₁ followed by the multiplier p₂ followed by the multiplier p₃... etc gives an overall multiplier of p₁p₂p₃... 
• Alternative method: A formula for the final ("after") amount is  where...
  • ...P is the original ("before") amount, r is the % increase, and n is the number of years
  • Note that  is the same value as the multiplier

What is meant by depreciation?

• Depreciation is where an item loses value over time
  • For example: cars, game consoles, etc
• Depreciation is usually calculated as a percentage decrease at the end of each year
  • This works the same as compound interest, but with a percentage decrease

How do I calculate a depreciation?

• You would calculate the new value after depreciation using the same method as compound interest
  • Identify the multiplier, p (1 - "% as a decimal")
    • 10% depreciation would have a multiplier of p = 1 - 0.1 = 0.9
    • 1% depreciation would have a multiplier of p = 1 - 0.01 = 0.99
  • Raise the multiplier to the power of the number of years (or months etc)
    • 
  • Multiply by the starting value
• New value is 
  • A is the starting value
  • p is the multiplier for the depreciation 
  • n is the number of years
• Alternative method: A formula for the final ("after") amount is where...   
  • ...P is the original ("before") amount, r is the % decrease, and n is the number of years
• If you are asked to find the amount the value has depreciated by:
  • Find the difference between the starting value and the new value