Compound interest & loans

Compound interest grows using the formula A = P(1 + r)ⁿ, where A is the final amount, P is the principal, r is the interest rate per period (as a decimal), and n is the number of periods. Unlike simple interest, each period's interest is calculated on the growing balance, not just the original amount — meaning growth accelerates over time. This same formula underlies both savings growth and loan balances.

Key terms

Compound interest:

Interest calculated on the principal plus previously earned interest.

Principal:

The original amount invested or borrowed.

Compounding period:

How often interest is calculated and added (e.g. annually, monthly).

Questions

1. 1. The compound interest formula is:

   • A = P(1 + r)ⁿ
   • A = P + r + n
   • A = P × r × n (simple interest formula)
   • A = P − r × n

2. 2. In A = P(1 + r)ⁿ, P represents:

   • The principal (starting amount)
   • The number of years
   • The interest rate
   • The final amount

3. 3. In A = P(1 + r)ⁿ, n represents:

   • The number of compounding periods
   • The principal
   • The interest rate
   • The final amount

4. 4. Compound interest is calculated on:

   • The principal plus previously earned interest
   • Only the original principal, forever
   • Nothing at all
   • A random amount each time

5. 5. Compared to simple interest at the same rate, compound interest grows:

   • Faster over time
   • Slower over time
   • At an identical rate always
   • Not at all

6. 6. A rate of 5% as a decimal (for use in the formula) is:

   • 0.05
   • 5.0
   • 50
   • 0.5

7. 7. A compounding period could be:

   • Annually or monthly
   • Only ever exactly one day
   • Never applicable to real accounts
   • Always exactly 100 years

8. 8. You invest $1,000 at 5% compound interest annually for 2 years. Using A = P(1+r)ⁿ, the final amount is approximately:

   • $1,102.50
   • $1,100
   • $1,050
   • $1,200

9. 9. You invest $2,000 at 4% compound interest annually for 3 years. The final amount is approximately:

   • $2,249.73
   • $2,240
   • $2,000
   • $2,400

10. 10. Why does the compound interest formula use (1 + r) instead of just r?

    • It represents the original amount (1) plus the growth (r) together each period
    • It has no real mathematical meaning
    • It always doubles the interest rate
    • It replaces the need for a principal amount

11. 11. A $5,000 loan at 6% compound interest annually for 2 years accrues a final amount of approximately:

    • $5,618
    • $5,600
    • $5,300
    • $6,000

12. 12. If you leave an investment for longer at the same compound interest rate, the total growth:

    • Accelerates, growing faster in later years
    • Stays exactly linear throughout
    • Decreases over time
    • Has no relationship to time at all

13. 13. A $10,000 investment at 3% compound interest annually for 4 years grows to approximately:

    • $11,255
    • $11,200
    • $10,300
    • $12,000

14. 14. A loan's balance growing under compound interest, if left unpaid, will:

    • Grow increasingly faster the longer it is left
    • Stay exactly the same forever
    • Shrink automatically over time
    • Have no connection to how long it is left unpaid

15. 15. You invest $3,000 at 4.5% compound interest annually for 5 years. Approximately how much interest is earned in total?

    • About $780
    • Exactly $675 (simple interest amount)
    • About $150
    • About $3,000

16. 16. Comparing $5,000 at 6% simple interest for 5 years versus 6% compound interest for 5 years, which earns more?

    • Compound interest earns more, since interest builds on itself
    • Simple interest always earns more over any period
    • They earn exactly identical amounts
    • Neither earns any interest after 5 years

17. 17. Why might a loan with monthly compounding cost more in total interest than one with annual compounding, at the same stated annual rate?

    • More frequent compounding means interest is calculated and added more often, accelerating growth
    • Compounding frequency never affects the total interest paid
    • Monthly compounding always results in lower total interest
    • Compounding frequency only matters for savings, never loans

18. 18. A $4,000 investment grows to approximately $4,864 after 5 years of compound interest. Using A = P(1+r)ⁿ, this suggests an approximate annual rate of:

    • 4%
    • 10%
    • 20%
    • 1%

19. 19. Why is understanding compound interest important when comparing loan offers with different interest rates and compounding frequencies?

    • The true cost of a loan depends on both the rate and how often interest compounds, not just the headline rate
    • All loans with the same stated rate always cost exactly the same regardless of compounding
    • Compounding frequency has no bearing on the true cost of a loan
    • Only the headline interest rate matters, with nothing else relevant

20. 20. You invest $8,000 at 5% compound interest annually for 3 years. The approximate final amount is:

    • $9,261
    • $9,200
    • $8,400
    • $9,600

21. 21. A "rule of thumb" approximation says money doubles roughly every 72÷r years under compound interest (the "Rule of 72"). At 8% annual compound interest, money would roughly double in about:

    • 9 years
    • 72 years
    • 8 years
    • 2 years