Simple interest, unknowns and repayments

An investment earns simple interest at a fixed annual rate. Which amount is used to calculate each year's interest?

Predict: with fixed principal and rate, three months earns $19.50 in simple interest. What should six months earn?

This model uses a fixed principal and a fixed annual rate. Move the term through several positions and watch how the interest changes at each step

What happens to the interest at each three-month step?

Interest and final amount

The principal is the original amount invested or borrowed. Interest is the extra money earned or owed. With simple interest, equal periods add equal interest. An annual rate is a rate per year; p.a. means per annum

1. 1Name the quantities P is the principal in dollars, I is the interest in dollars and A is the final amount in dollars. The letter r is the percentage number in the annual rate. The letter t counts years
2. 2Match the time unit For an annual rate, divide the number of months by twelve. Keep the fraction exact until the calculation is finished \(\displaystyle t=\frac{\text{months}}{12}\)
3. 3Choose the amount requested For interest, use the original principal throughout the term. For the final investment amount or total loan amount owed, add the principal. These examples assume no fees or changes to the principal

Principal

\(\displaystyle \$2{,}340\)

Annual rate

\(\displaystyle 5\%\text{ p.a.}\)

Term

\(\displaystyle 8\text{ months}\)

Calculate the simple interest earned

The annual rate needs a time measured in years. Keep the month conversion as a fraction

Principal

\(\displaystyle \$2{,}750\)

Annual rate

\(\displaystyle 4.8\%\text{ p.a.}\)

Term

\(\displaystyle 15\text{ months}\)

Calculate the final amount of this simple-interest investment. There are no fees or changes to the principal

Find the interest first. The final amount also includes the money originally invested

Find a missing quantity

Put the known values into the interest rule. Then divide by the factors attached to the unknown. For time, dividing the total interest by one year's interest tells you how many years it took

1. 1Find time For this example, the annual interest is found from the principal and rate. Dividing the total interest by that annual amount gives years \(\displaystyle \begin{aligned}163.8&=2{,}600\times\frac{4.2}{100}\times t\\163.8&=109.2t\\t&=\frac{163.8}{109.2}\\t&=1.5\end{aligned}\)
2. 2Find the principal Multiply the rate factor by the time in years. Divide the interest by that factor to recover the original amount \(\displaystyle \begin{aligned}94.5&=P\times\frac{3.6}{100}\times1.25\\94.5&=0.045P\\P&=\frac{94.5}{0.045}\\P&=2{,}100\end{aligned}\)
3. 3Find the annual rate This gives the percentage number. Attach the percent sign when you state the annual rate \(\displaystyle r=\frac{100I}{Pt}\)

Principal

\(\displaystyle \$2{,}880\)

Annual rate

\(\displaystyle 4.5\%\text{ p.a.}\)

Interest

\(\displaystyle \$291.60\)

Find the term of this simple-interest investment in years

Find the interest earned in one year, then use it to find the number of years

Interest

\(\displaystyle \$147\)

Annual rate

\(\displaystyle 4.2\%\text{ p.a.}\)

Term

\(\displaystyle 14\text{ months}\)

Find the original principal of this simple-interest investment

Convert the term to years. Divide the interest by the rate factor and the time

Principal

\(\displaystyle \$3{,}250\)

Interest

\(\displaystyle \$312\)

Term

\(\displaystyle 16\text{ months}\)

Find the annual simple-interest rate. Enter the percentage number

Convert the term to years before finding the annual rate

A savings model assumes a fixed principal, a fixed annual rate and no fees. Halfway through the term, some of the principal is withdrawn. What should happen to the model?

Principal

\(\displaystyle \$1{,}950\)

Annual rate

\(\displaystyle 6.4\%\text{ p.a.}\)

Term

\(\displaystyle 10\text{ months}\)

The student is finding the final amount of a simple-interest investment. Where is the first error?

A fixed principal must earn a fixed interest target. Predict what happens to the required time if its annual simple-interest rate doubles

Keep the principal $3,000 and interest target $300 fixed. Test rates 2%, 5% and 10% p.a. Watch the required term. Readouts round to two decimals; the calculation keeps the exact fraction

Which statement fits the experiment?

A flat-rate repayment model

For these fictional loan exercises, interest is charged on the original principal for the whole term, even while payments are made. This is a flat-rate model, not a reducing-balance calculation. Assume a fixed rate and no fees

1. 1Find the whole amount Add original principal P and full-term interest I to obtain total A. \(\displaystyle A=P+I\)
2. 2Count the payments For a term t in years, use n=12t monthly payments or n=26t fortnightly payments. The 26-fortnight year is a stated scheduling approximation \(\displaystyle n=12t\quad\text{or}\quad n=26t\)
3. 3Average payment R is the average payment before rounding. Divide the whole amount by the payment count, not by the years \(\displaystyle R=A/n\)
4. 4Check the cents Rounding each payment may leave a small difference. A final payment can be adjusted so the payments total A.

Original principal

\(\displaystyle \$2{,}400\)

Annual flat rate

\(\displaystyle 5\%\)

Term

\(\displaystyle 2\text{ years}\)

Payments per year

\(\displaystyle 12\)

Find the average monthly payment under a flat-rate loan model. Interest uses the original principal for the full term; no fees apply

Find full-term interest, add the principal, then divide by the number of monthly payments

Term

\(\displaystyle 1.5\text{ years}\)

Fortnightly payments per model year

\(\displaystyle 26\)

Use the stated scheduling model to count fortnightly payments over the term

Multiply years by the number of payments per model year

Original principal

\(\displaystyle \$2{,}600\)

Annual flat rate

\(\displaystyle 6\%\)

Term

\(\displaystyle 2\text{ years}\)

Payments per model year

\(\displaystyle 26\)

Find the average fortnightly payment under the flat-rate model. Interest uses the original principal for the entire term; no fees apply

Calculate interest and total first. Divide by the number of fortnightly payments across the whole term

After all flat-rate interest is included, a fictional total of $1,000 is split into three payments. The first two are $333.33 each. What final payment settles the total exactly?

A loan charges interest on the unpaid balance, which decreases with each payment. Can you apply one original-principal simple-interest calculation to get its exact total?

Choose and defend a plan

Use a fictional simple-interest model with no fees. A $1,500 principal at 4% p.a. must earn $120. Find the required time. If the target must instead be reached in 18 months at the same rate, find the required principal. Check both plans and explain why the required principal changes

1. 1Formulate Choose the unknown for each plan and write a formula before calculating
2. 2Evaluate and report State the fixed-rate, no-fee and unchanged-principal assumptions. Explain what would change if interest were reinvested or money withdrawn. Your written reasoning needs human review