Compare the total cost of ways of paying, or the return on ways of investing

Price ($)

\(\displaystyle 1200\)

One-off fee ($)

\(\displaystyle 40\)

Monthly fee ($)

\(\displaystyle 15\)

Number of fees

\(\displaystyle 12\)

A $1,200 purchase has a $40 one-off fee and twelve month-end fees of $15. There is no interest. What is the total cost?

One payment plan advertises a lower interest rate than another. What establishes which costs less over the stated term?

Both plans finance the same $1,200 purchase, with no interest. Plan A has a $50 one-off fee and an $8 fee at each completed month-end. Plan B has no one-off fee and a $12 fee at each completed month-end. Find the first completed month at which Plan A costs less

At which completed month does Plan A first cost less?

Compare the complete cost

A headline fee or interest rate is not the total. Start with the amount being paid for, add each stated charge, and compare every option over the same term

1. 1Use a common term Use the same starting amount and elapsed time. Include every stated fee and payment, with its timing and frequency
2. 2Count fees separately A one-off fee is charged once. A monthly fee charged at each month-end is charged twelve times over twelve months \(\displaystyle C=P+F+mn\)

Plan A cost ($)

\(\displaystyle 1442\)

Plan B cost ($)

\(\displaystyle 1488\)

For those plans, at month 24 Plan A costs $1,442 and Plan B costs $1,488. How much less does Plan A cost?

Simple and compound interest

On $1,000 at 8% per year for two years, simple interest uses the original principal each year. Annual compound interest applies the second increase to the first-year balance. Assume no fees or cash flows and retain full precision

Final amount ($)

\(\displaystyle 1166.40\)

Initial amount ($)

\(\displaystyle 1000\)

That compound model finishes with $1,166.40 from an initial $1,000. How much interest was earned?

Two ideal investments have the same initial amount and term, no fees and no other cash flows. The question asks which produces the greater final amount. Which criterion applies?

Count the update periods

A rate of 2% per quarter is applied eight times in two years. It is already a quarterly rate. No annual-rate conversion is needed

1. 1Match the rate and period Use the rate for one update and the number of those updates. A rate already given per quarter is not divided by four again \(\displaystyle A=P(1+i)^n\)

Initial amount ($)

\(\displaystyle 1000\)

Annual rate

\(\displaystyle 5\%\)

Completed years

\(\displaystyle 2\)

An ideal investment of $1,000 earns 5% per year, compounded annually for two years, with no fees or cash flows. Find the final amount

Why can 1% per month and 2% per year not be ranked just by comparing 1 and 2?

State rounding and timing

If twelve fees are charged at the ends of months 1 through 12, include all twelve. Keep ideal-model values unrounded until the requested final report. If a question instead specifies rounding after each update, follow that different convention

1. 1Count fees separately A one-off fee is charged once. A monthly fee charged at each month-end is charged twelve times over twelve months \(\displaystyle C=P+F+mn\)
2. 2Keep the model explicit Use the stated rounding convention. In an ideal full-precision model, round a final money report once. A numerical ranking alone does not account for affordability, risk or unstated charges

Explain a comparison

Written task: compare paying $1,200 now with paying the same purchase price plus a $40 one-off fee and twelve $15 month-end fees. State the total costs, the difference, and one assumption needed before treating the cheaper model as a real purchasing decision

Model comparison

Paying now costs $1,200. The fee plan costs $1,420, so paying now is $220 cheaper under the given cost model. This conclusion assumes all relevant charges are stated; it does not establish that the buyer has $1,200 available now