You are the accounting manager for Kool Ragz, Inc., a manufacturer of men's and women's clothing. The company needs to borrow \( \$ 1,500,000 \) for 90 days in order to purchase a large quantity of material at "closeout" prices. The interest rate for such loans at your bank, Rimrock Bank, is \( 13 \% \) using ordinary interest. (Challenge) If Southside National wants to compete with Rimrock's last offer 46,875 by charging \( \$ 1,875 \) less interest, what rate (as a \%), rounded to the nearest hundredths of a percent, must it quote using exact interest?

We start by noting that Rimrock Bank’s ordinary‐interest loan is stated as 13%. However, when we compute interest on a $1,500,000 loan for 90 days at 13% using the ordinary interest formula (which uses a 360‑day year), we get   Interest = Principal × Rate × (Days/360)        = 1,500,000 × 0.13 × (90/360)        = 1,500,000 × 0.13 × 0.25        = 1,500,000 × 0.0325        = $48,750. On the other hand, the “last offer” mentioned is for an interest amount of $46,875. (Notice that $46,875 is exactly what you would get if the rate were 12.5% because 1,500,000 × 0.125 × 0.25 = 46,875.) Thus, it appears the competing offer is based on a 12.50% rate using ordinary (360‐day) interest. Now, Southside National wants to charge $1,875 less interest than this offer. So, the interest they would collect is   $46,875 – $1,875 = $45,000. However, Southside National will use “exact interest,” meaning interest is computed using the actual number of days in the year (365 days). For an exact‐interest calculation over 90 days, the interest is computed as   Interest = Principal × (Exact Annual Rate) × (Days/365). We set this equal to $45,000 and solve for the rate (r):   1,500,000 × r × (90/365) = 45,000. Solve for r:   r = 45,000 / [1,500,000 × (90/365)]. It is often easiest to rearrange the fraction:   r = (45,000 × 365) / (1,500,000 × 90). Now calculate step‐by‐step: 1. First, simplify 45,000/1,500,000:   45,000/1,500,000 = 0.03. 2. Then, compute 365/90:   365/90 ≈ 4.05556. 3. Multiply the results:   r ≈ 0.03 × 4.05556 ≈ 0.12167. Expressed as a percentage, this is approximately 12.17%. Therefore, if Southside National wants to charge $1,875 less interest (i.e. $45,000 total interest on a $1,500,000 loan for 90 days), it must quote an exact annual interest rate of about 12.17% (rounded to the nearest hundredth of a percent).