Comparing Simple and Compound Interest

Interest & Investment 11th-12th Grade
PROBLEM
Sam invested $16,000 in an account paying r% simple interest. Colin invested $16,000 in an account paying 3% compound interest. They have the same amount after two years. Calculate r.
sam invested 16000 in an account paying r% simple interest colin invested 16000 in an account paying 3% compound...

What You Will Learn

  • How to set up and solve equations where simple and compound interest produce equal returns
  • The relationship between simple interest rates and compound interest rates for equivalent outcomes
  • Why simple interest requires a higher rate than compound interest to achieve the same final amount
  • Practical applications of interest rate comparisons in financial planning
  • Working backwards from equal outcomes to find unknown parameters

Solution: Method 1 — Calculate and Equate

Step 1 — Calculate Colin's compound interest amount

Since Colin's investment details are completely known, we can find his final amount using the compound interest formula:

A = P(1 + r)^t
A = 16000(1 + 0.03)^2
A = 16000(1.03)^2
A = 16000(1.0609)
A = $16,974.40

Step 2 — Set up Sam's simple interest equation

For simple interest, the formula is A = P(1 + rt). Since Sam must have the same final amount as Colin:

16000(1 + r × 2) = 16974.40

Note that we use r as the decimal form of the percentage (so r% = r/100 in decimal form, but we'll solve for the percentage directly).

Step 3 — Solve for r

Divide both sides by 16000:

1 + 2r = 16974.40 ÷ 16000
1 + 2r = 1.06090
2r = 0.06090
r = 0.03045

Step 4 — Convert to percentage

Since the problem asks for r%, we convert:

r = 0.03045 × 100% = 3.045%

Solution: Method 2 — Interest Earned Comparison

Step 1 — Find Colin's total interest earned

Instead of working with final amounts, we can compare the interest earned by each person:

Colin's final amount = 16000(1.03)^2 = $16,974.40
Colin's interest earned = $16,974.40 - $16,000 = $974.40

Step 2 — Set up Sam's interest equation

For simple interest, Interest = Prt. Since Sam earns the same total interest:

Sam's interest = 16000 × (r/100) × 2 = 320r

Step 3 — Equate the interest amounts

Setting Sam's interest equal to Colin's:

320r = 974.40
r = 974.40 ÷ 320
r = 3.045%
r = 3.045%

Verification

Let's check our answer by calculating both final amounts:

Sam's amount = 16000(1 + 0.03045 × 2)
= 16000(1 + 0.0609)
= 16000(1.0609)
= $16,974.40 ✓

Colin's amount = 16000(1.03)^2
= 16000(1.0609)
= $16,974.40 ✓

Both amounts are equal, confirming our answer is correct.

Does This Seem Reasonable?

Our answer makes intuitive sense. Colin's compound interest rate is 3%, and we found Sam needs 3.045% simple interest to match it. This small difference tells us something important:

Key Insight: Simple interest requires a slightly higher rate than compound interest to achieve the same return over a given period. This happens because compound interest earns "interest on interest" while simple interest only earns on the principal.

The difference is small here (0.045%) because we're only looking at 2 years. Over longer periods, the gap would be much larger.

Common Pitfalls

✗ Using the wrong formula for compound interest:

A = P(1 + rt) = 16000(1 + 0.03 × 2) = $16,960

Why it's wrong: This is actually the simple interest formula! Compound interest requires the exponent: A = P(1 + r)^t.

✗ Forgetting to convert between percentage and decimal:

Setting up: 16000(1 + 2r) = 16974.40 and solving to get r = 0.03045, then reporting "r = 0.03045%"

Why it's wrong: If you use r as a decimal in the formula, then r = 0.03045 means the percentage is 3.045%, not 0.03045%.

✗ Mixing up who gets which type of interest:

Calculating 16000(1 + 0.03 × 2) for Sam and 16000(1 + r)^2 for Colin

Why it's wrong: The problem clearly states Sam gets simple interest (r%) and Colin gets compound interest (3%). Double-check which person gets which type.

The Underlying Pattern

This problem illustrates a fundamental relationship in finance:

When simple and compound interest produce equal amounts:
P(1 + r_simple × t) = P(1 + r_compound)^t

Solving for the simple rate:
r_simple = [(1 + r_compound)^t - 1] ÷ t

In our problem: r_simple = [(1.03)² - 1] ÷ 2 = [1.0609 - 1] ÷ 2 = 0.03045 = 3.045%

This formula works for any time period and shows why simple interest rates must be higher: they need to "catch up" to the compounding effect by earning more per year on the principal alone.

Real Applications

Investment comparison: When choosing between a simple interest bond and a compound interest savings account, this calculation helps determine equivalent rates.

Loan analysis: Some loans quote "simple" interest while others compound. Understanding the equivalent rates helps borrowers make informed decisions.

Financial planning: This type of calculation appears in retirement planning when comparing different types of accounts with varying compounding schedules.

Four "What-If?" Problems

1
Different Principal
Sam invests $20,000 at r% simple interest. Colin invests $16,000 at 3% compound interest. After 2 years, their amounts are equal. Find r.
Step 1 — Calculate Colin's amount

Colin: A = 16000(1.03)² = 16000(1.0609) = $16,974.40

Step 2 — Set up Sam's equation

Sam: 20000(1 + 2r) = 16974.40

Step 3 — Solve for r

1 + 2r = 16974.40 ÷ 20000 = 0.84872
2r = -0.15128
r = -0.07564 = -7.564%

Step 4 — Interpretation

This is impossible! A negative interest rate would mean Sam loses money. When Sam invests more principal ($20,000 vs $16,000), no positive simple interest rate can make his final amount equal Colin's smaller final amount.

2
Finding Time
Sam invests $16,000 at 4% simple interest. Colin invests $16,000 at 3% compound interest. After how many years will their amounts be equal?
Step 1 — Set up the equation

Sam's amount = Colin's amount
16000(1 + 0.04t) = 16000(1.03)^t

Step 2 — Simplify

Divide by 16000: 1 + 0.04t = (1.03)^t

Step 3 — Test values

At t = 0: 1 = 1 ✓ (both start equal)
At t = 10: 1.4 ≈ 1.344 (simple > compound)
At t = 20: 1.8 ≈ 1.806 (very close!)

Step 4 — Exact solution

This requires logarithms to solve exactly, but the answer is approximately 20 years. After this point, simple interest (higher rate) overtakes compound interest permanently.

3
Monthly Compounding
Colin's 3% interest is now compounded monthly instead of annually. Sam still earns simple interest. They invest $16,000 and have equal amounts after 2 years. Find Sam's rate r.
Step 1 — Calculate Colin's amount with monthly compounding

Formula: A = P(1 + r/n)^(nt) where n = 12
A = 16000(1 + 0.03/12)^(12×2)
A = 16000(1.0025)^24 = 16000(1.0617) = $16,987.84

Step 2 — Set up Sam's equation

16000(1 + 2r) = 16987.84

Step 3 — Solve for r

1 + 2r = 16987.84 ÷ 16000 = 1.0617
2r = 0.0617
r = 0.03085 = 3.085%

Step 4 — Verify

Sam: 16000(1 + 0.03085 × 2) = $16,987.20
Answer: r = 3.085% (slightly higher due to monthly compounding effect)

4
Reverse the Unknown
Sam invests $16,000 at 3.2% simple interest. Colin invests $16,000 at c% compound interest, compounded annually. After 2 years, their amounts are equal. Find c.
Step 1 — Calculate Sam's amount

A = 16000(1 + 0.032 × 2) = 16000(1.064) = $17,024

Step 2 — Set up Colin's equation

16000(1 + c)² = 17024

Step 3 — Solve for c

(1 + c)² = 17024 ÷ 16000 = 1.064
1 + c = √1.064 = 1.0315
c = 0.0315 = 3.15%

Step 4 — Verify

Colin: 16000(1.0315)² = 16000(1.064) = $17,024 ✓
Answer: c = 3.15%

Frequently Asked Questions

What's the difference between simple and compound interest formulas? +
Simple interest uses A = P(1 + rt) where interest is calculated only on the principal. Compound interest uses A = P(1 + r)^t where interest is calculated on principal plus accumulated interest. In this problem, Sam's simple interest formula gives 16000(1 + 2r/100) while Colin's compound gives 16000(1.03)².
How do you solve when simple and compound interest amounts are equal? +
Set the two formulas equal and solve for the unknown rate. Calculate the compound interest amount first, then set the simple interest formula equal to that value. Here, 16000(1.03)² = $16,974.40, so 16000(1 + 2r/100) = 16974.40, giving r = 3.05%.
Why is simple interest rate higher than compound interest rate for equal returns? +
Simple interest only earns on the original principal, while compound interest earns on both principal and accumulated interest. To match compound interest returns, simple interest needs a higher rate to compensate for not earning interest on interest. In this example, 3.05% simple matches 3% compound over 2 years.
DN

Dr. Neven Jurkovic

Professor of Mathematics with 15+ years teaching experience in algebra, calculus, and financial mathematics. Passionate about making complex concepts accessible through clear, step-by-step explanations.

NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-08-02

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