Sally can buy some furniture for $11,000 cash or monthly payments of $1000 for 1 year,with the first $1000 payment made today.What range contains the interest rate j₁₂ that Sally is being charged?

Question

Quizplus · Accepted Answer

The monthly payments total $12,000 ($1,000 x 12 months), which is $1,000 more than the cash price of $11,000. This extra $1,000 is the interest charged over the year. To find the interest rate, we can use the formula for the present value of an annuity since the first payment is made immediately (which makes it an annuity due). The formula for the present value of an annuity due is $PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \times (1 + r)$, where PV is the present value, PMT is the payment amount, r is the monthly interest rate, and n is the number of payments. Given the complexity of solving for r directly, financial calculators or software are typically used to find the exact rate. However, since the total interest paid over the year is significantly higher than the principal (over 9% of the $11,000), and considering the effect of compounding, the interest rate is likely to be in the higher range of the options provided. The correct range is determined by calculating or estimating the effective annual rate given the structure of the payments and comparing it to the options provided.

Sally Can Buy Some Furniture for $11,000 Cash or Monthly