Compound interest (extension)

Starting investment

\(\displaystyle \$860\)

Annual rate

\(\displaystyle 5\%\)

First year's interest

\(\displaystyle \$43\)

Interest is added at the end of each year. There are no other payments or fees. How much interest is earned in the second year?

Both accounts start with $1,280 at an annual rate of 2.5%. There are no other payments or fees. Compare the interest added in the second and third years. The compound account adds interest annually

From the second year to the third year, simple interest stays the same. How much does the compound account's interest for the year increase?

Repeat the annual multiplier

The principal is the starting amount. Annual compounding adds each year's interest to the balance before the next year's interest is calculated. With a constant annual rate and no other payments or fees, apply the same multiplier once for each complete year. Keep calculator precision until the final answer

Define the quantities

P is the principal, A is the final balance and I is the total interest, all in dollars. The annual percentage number is r, and n counts complete years. The annual multiplier is q

Keep the balance and add the interest

Starting principal: $1,280; annual rate: 2.5%; interest compounded annually

The multiplier includes the whole opening balance as well as the percentage increase

Use the new balance each year

Each closing balance becomes the next year's opening balance

Write repeated multiplication as a power

The original principal has been multiplied by the annual multiplier three times. The exponent records those three yearly updates

Separate the balance and total interest

The final balance includes the principal. Subtract the principal when the question asks for total interest

Work backwards or measure total growth

Challenge questions change what is unknown. Use the same annual model. Reverse the growth by division when the starting amount is missing. To find total percentage growth, compare the total increase with the starting amount

Recover the principal

The final balance is the principal multiplied by the whole growth factor. Divide by that factor to undo the growth

Find the growth proportion

Subtract the original whole from the growth factor. The principal cancels, so the annual rate and term are enough

Convert the proportion to a percentage

Let g be the percentage number. Report the total increase as g per cent. It is the increase over the whole term, not an annual rate

Compare two arrangements

Compare $1,800 over two complete years: annual compound interest at 3%, or simple interest at 3.5% p.a. There are no other payments or fees. Use each arrangement's own rate. Keep unrounded values until the comparison and subtraction are complete, then round the requested amounts to cents

Calculate the compound balance

Apply the compound multiplier once for each year

Calculate the simple balance

Simple interest uses the original principal each year. Add that interest to the principal

Compare both balances and their difference

Here the balances are exact to cents. Simple interest gives the higher balance, so it favours an investor. The compound arrangement leaves less owed, so it favours a borrower. The difference is positive whichever arrangement is preferred

Principal

\(\displaystyle \$540\)

Annual rate

\(\displaystyle 4\%\)

Term

\(\displaystyle 3\text{ years}\)

Interest is compounded annually, with no other payments or fees. A student is finding the final balance. Which statement identifies the first error?

A loan has a final balance of $2,675 after four years at 3.5% p.a., compounded annually. There were no repayments or fees. Use the stated final balance to find the original principal to the nearest cent

The multiplier keeps the opening balance and adds the annual interest

Apply the annual multiplier once for each year

Divide by the whole growth factor to recover the starting amount

Round only the final answer to the nearest cent