Simple interest · Reviewer

Simple interest

A concise reviewer for Slater Chapter 16.

Annual rates use time in years: months/12, actual days/360 for ordinary interest, or actual days/365 for exact interest. Exclude the first date and include the last. Keep full precision except for U.S. Rule interval interest, which is rounded to cents before adjusting principal. Round fractional days up where requested.

Interest & maturity value

Simple interest is the price of using money. Maturity value is the full amount repaid.

Principal (P) is the original amount borrowed. Interest (I) is the charge for using it. With simple interest and no partial payments, the interest base stays the same.

The rate R is annual and must be a decimal. Time T must be in years: 9 months is 9/12, and 7.5% is 0.075.

• I = P × R × T
• M = P + I = P(1 + RT)

Worked example: A nine-month loan

Borrow $4,800 at 7.5% annual simple interest for 9 months. Find interest and maturity value.

1. Convert time: T = 9/12 = 0.75 year.
2. Interest: I = 4,800 × 0.075 × 0.75 = $270.00.
3. Maturity value: M = 4,800 + 270 = $5,070.00.

$270.00 interest · $5,070.00 at maturity

Watch out: Do not report the interest as the maturity value. The final payment contains both the principal and the interest.

Slater Ch. 16, pp. 425–427

Ordinary vs. exact interest

The same calendar interval can produce slightly different interest, depending on the year denominator.

Ordinary interest uses actual days divided by 360. The chapter calls this the Banker’s Rule. Exact interest uses actual days divided by 365.

Exclude the loan’s starting date and include the repayment date. Actual day counts include February 29 when it occurs; this study set keeps the chapter’s 365-day exact-interest denominator. A 360-day year does not mean replacing every month with 30 days.

• Ordinary: T = actual days / 360
• Exact: T = actual days / 365

Worked example: One loan, two conventions

Find interest on $10,000 at 6% for 90 actual days under both conventions.

1. Ordinary: 10,000 × 0.06 × 90/360 = $150.00.
2. Exact: 10,000 × 0.06 × 90/365 = $147.95.
3. The smaller denominator produces the larger time fraction, so ordinary interest is higher.

Ordinary: $150.00 · Exact: $147.95

Watch out: Keep the time fraction unrounded. Rounding 90/365 before multiplying can change the final answer.

Slater Ch. 16, pp. 427–428

Find principal, rate, or time

Start with I = PRT and isolate the missing quantity.

When solving for principal, rate, or time, interest goes in the numerator. Multiply the other two known factors in the denominator.

T = I/(PR) gives years. Multiply the full result by 12 for total months, by 360 for ordinary-interest days, or by 365 for exact-interest days. Round a genuine fractional day upward when the question requests a whole-day term.

• P = I / (RT)
• R = I / (PT)
• T = I / (PR)
• If M is known: P = M / (1 + RT)

Worked example: Recover the annual rate

A $5,000 loan earns $112.50 in ordinary interest over 90 days. Find the annual rate.

1. Use T = 90/360 = 0.25 year.
2. R = 112.50 / (5,000 × 0.25) = 0.09.
3. Convert the decimal to a percent: 0.09 × 100 = 9.00%.

9.00% per year

Watch out: Interest divided only by principal gives the rate for the term. Divide by time as well to get the annual rate.

Slater Ch. 16, pp. 429–431

Partial payments: the U.S. Rule

Each payment covers accrued interest first. Only the remainder reduces principal.

Calculate interest for the time since the last payment. Round that interval’s interest to cents, subtract it from the payment, and use the remaining payment to reduce principal.

Repeat on the adjusted principal. At maturity, add interest since the final partial payment. Total interest is the sum of every interval, including interest covered by earlier payments.

• Interval I = current P × R × interval days / 360
• Principal reduction = payment − interval I
• Final payoff = remaining principal + last interval I

Worked example: Two payments before maturity

An $8,000 loan at 6% runs 120 days. Pay $1,500 on day 30 and $2,000 on day 75. Use ordinary interest.

1. Days 0–30: I = 8,000 × 0.06 × 30/360 = $40.00. Payment reduces principal by $1,460.00. New P = $6,540.00.
2. Days 30–75: I = 6,540 × 0.06 × 45/360 = $49.05. Payment reduces principal by $1,950.95. New P = $4,589.05.
3. Days 75–120: I = 4,589.05 × 0.06 × 45/360 = $34.42. Final payoff = $4,623.47.
4. Total interest = 40.00 + 49.05 + 34.42 = $123.47.

Final payoff: $4,623.47 · Total interest: $123.47

Watch out: The three intervals are 30, 45, and 45 days. Using 30, 75, and 120 would count some days more than once.

Slater Ch. 16, pp. 431–433

Check your work

Use units, substitution, and a cash-flow check to catch mistakes before submitting.

For positive interest, maturity value must exceed principal. With the same positive principal, rate, and actual days, ordinary interest must exceed exact interest.

Verify an unknown by putting it back into I = PRT. For a partial-payment loan, total payments plus final payoff minus original principal must equal total interest. Earlier principal reductions usually save interest.

• I = M − P
• Total interest = all cash repaid − original principal

Worked example: Check the savings from early payments

The $8,000, 6%, 120-day loan above has payments of $1,500 and $2,000 and a final payoff of $4,623.47. Verify the interest and savings.

1. Cash-flow check: 1,500 + 2,000 + 4,623.47 − 8,000 = $123.47.
2. Without early payments: I = 8,000 × 0.06 × 120/360 = $160.00.
3. Interest saved: 160.00 − 123.47 = $36.53.

$123.47 total interest · $36.53 saved

Watch out: The final payoff is neither the original principal nor the total interest. It is the remaining balance plus the last interest interval.

Slater Ch. 16, pp. 425–433