25. Difference between simple and compound interest problems

25. Difference between simple and compound interest problems - step by step solution procedure and detailed explanation for each problem.

Below are 10 problems that illustrate the difference between simple interest (SI) and compound interest (CI), with step-by-step solutions and detailed explanations for each. The problems are designed to cover various scenarios, including different principal amounts, interest rates, time periods, and compounding frequencies, to highlight how SI and CI differ in calculation and outcome. Each problem includes a comparison of the interest earned and an explanation of why the results differ.
Key Concepts
  • Simple Interest (SI): Interest is calculated only on the initial principal amount throughout the entire period.
    • Formula: SI = (P × R × T) / 100, where:
      • P = Principal
      • R = Rate of interest per annum (%)
      • T = Time period (in years)
    • Total Amount = P + SI
  • Compound Interest (CI): Interest is calculated on the initial principal and also on the accumulated interest from previous periods.
    • Formula: A = P × (1 + R/(100 × n))^(n × T), where:
      • A = Amount after time T
      • P = Principal
      • R = Rate of interest per annum (%)
      • T = Time period (in years)
      • n = Number of times interest is compounded per year
    • CI = A - P
  • Key Difference: SI remains constant over time, while CI grows exponentially because interest is added to the principal periodically.
Problem 1: Basic SI vs CI Comparison
Question: A principal of $1000 is invested at 5% per annum for 2 years. Calculate the simple interest, compound interest (compounded annually), and compare the results.
Solution:
  • Step 1: Identify the given values.
    • Principal (P) = $1000
    • Rate (R) = 5%
    • Time (T) = 2 years
    • For CI, n = 1 (compounded annually)
  • Step 2: Calculate Simple Interest.
    • SI = (P × R × T) / 100 = (1000 × 5 × 2) / 100 = $100
    • Total Amount (SI) = P + SI = 1000 + 100 = $1100
  • Step 3: Calculate Compound Interest.
    • A = P × (1 + R/100)^T = 1000 × (1 + 5/100)^2 = 1000 × (1.05)^2
    • (1.05)^2 = 1.1025
    • A = 1000 × 1.1025 = $1102.5
    • CI = A - P = 1102.5 - 1000 = $102.5
  • Step 4: Compare the results.
    • SI = $100
    • CI = $102.5
    • Difference = CI - SI = 102.5 - 100 = $2.5
Answer:
  • Simple Interest = $100
  • Compound Interest = $102.5
  • Difference = $2.5
Explanation:
  • SI is calculated only on the initial $1000, so it’s straightforward: 5% of $1000 per year for 2 years.
  • CI includes interest on the first year’s interest in the second year. After year 1, interest is $50, making the principal $1050 for year 2, and 5% of $1050 = $52.5, leading to a slightly higher total interest.
Problem 2: Longer Time Period
Question: A sum of $5000 is invested at 4% per annum for 3 years. Find SI, CI (compounded annually), and the difference.
Solution:
  • Step 1: Identify the given values.
    • P = $5000, R = 4%, T = 3 years, n = 1
  • Step 2: Calculate SI.
    • SI = (5000 × 4 × 3) / 100 = $600
    • Amount = 5000 + 600 = $5600
  • Step 3: Calculate CI.
    • A = 5000 × (1 + 4/100)^3 = 5000 × (1.04)^3
    • (1.04)^3 = 1.04 × 1.04 × 1.04 = 1.124864
    • A = 5000 × 1.124864 = $5624.32
    • CI = 5624.32 - 5000 = $624.32
  • Step 4: Compare.
    • SI = $600
    • CI = $624.32
    • Difference = 624.32 - 600 = $24.32
Answer:
  • SI = $600
  • CI = $624.32
  • Difference = $24.32
Explanation:
  • The longer time period amplifies the effect of compounding. Each year, CI is calculated on a slightly larger principal, leading to a greater difference compared to SI, which remains linear.
Problem 3: Higher Interest Rate
Question: A principal of $2000 is invested at 10% per annum for 2 years. Calculate SI and CI (compounded annually).
Solution:
  • Step 1: P = $2000, R = 10%, T = 2 years, n = 1
  • Step 2: Calculate SI.
    • SI = (2000 × 10 × 2) / 100 = $400
    • Amount = 2000 + 400 = $2400
  • Step 3: Calculate CI.
    • A = 2000 × (1 + 10/100)^2 = 2000 × (1.1)^2
    • (1.1)^2 = 1.21
    • A = 2000 × 1.21 = $2420
    • CI = 2420 - 2000 = $420
  • Step 4: Compare.
    • SI = $400
    • CI = $420
    • Difference = 420 - 400 = $20
Answer:
  • SI = $400
  • CI = $420
  • Difference = $20
Explanation:
  • A higher interest rate increases the gap between SI and CI because the interest earned on interest (in CI) is more significant at higher rates.
Problem 4: Semi-Annual Compounding
Question: $3000 is invested at 6% per annum for 2 years. Find SI and CI (compounded semi-annually).
Solution:
  • Step 1: P = $3000, R = 6%, T = 2 years, n = 2 (semi-annually)
  • Step 2: Calculate SI.
    • SI = (3000 × 6 × 2) / 100 = $360
    • Amount = 3000 + 360 = $3360
  • Step 3: Calculate CI.
    • A = P × (1 + R/(100 × n))^(n × T) = 3000 × (1 + 6/(100 × 2))^(2 × 2)
    • A = 3000 × (1 + 0.03)^4 = 3000 × (1.03)^4
    • (1.03)^4 = 1.12550881
    • A = 3000 × 1.12550881 = $3376.53
    • CI = 3376.53 - 3000 = $376.53
  • Step 4: Compare.
    • SI = $360
    • CI = $376.53
    • Difference = 376.53 - 360 = $16.53
Answer:
  • SI = $360
  • CI = $376.53
  • Difference = $16.53
Explanation:
  • Semi-annual compounding means interest is added twice a year, increasing the frequency of interest accrual, which results in higher CI compared to annual compounding for the same period.
Problem 5: Quarterly Compounding
Question: $4000 is invested at 8% per annum for 1 year. Calculate SI and CI (compounded quarterly).
Solution:
  • Step 1: P = $4000, R = 8%, T = 1 year, n = 4 (quarterly)
  • Step 2: Calculate SI.
    • SI = (4000 × 8 × 1) / 100 = $320
    • Amount = 4000 + 320 = $4320
  • Step 3: Calculate CI.
    • A = 4000 × (1 + 8/(100 × 4))^(4 × 1) = 4000 × (1 + 0.02)^4
    • (1.02)^4 = 1.08243216
    • A = 4000 × 1.08243216 = $4329.73
    • CI = 4329.73 - 4000 = $329.73
  • Step 4: Compare.
    • SI = $320
    • CI = $329.73
    • Difference = 329.73 - 320 = $9.73
Answer:
  • SI = $320
  • CI = $329.73
  • Difference = $9.73
Explanation:
  • Quarterly compounding (4 times a year) results in more frequent interest additions, increasing CI even for a short period like 1 year, compared to SI’s linear calculation.
Problem 6: Finding Principal Given SI and CI
Question: For a certain principal, the SI for 2 years at 5% per annum is $200, and the CI (compounded annually) is $205. Find the principal.
Solution:
  • Step 1: Use SI to find P.
    • SI = (P × R × T) / 100
    • 200 = (P × 5 × 2) / 100
    • P = 200 × 100 / 10 = $2000
  • Step 2: Verify CI.
    • A = 2000 × (1 + 5/100)^2 = 2000 × (1.05)^2 = 2000 × 1.1025 = $2205
    • CI = 2205 - 2000 = $205 (matches given CI)
  • Step 3: Compare.
    • SI = $200
    • CI = $205
    • Difference = 205 - 200 = $5
Answer:
  • Principal = $2000
  • SI = $200, CI = $205, Difference = $5
Explanation:
  • The principal is found using the SI formula, and the CI calculation confirms the given data. The difference arises because CI includes interest on the first year’s interest.
Problem 7: Finding Time Period
Question: A principal of $10,000 yields $2000 SI at 10% per annum. How long does it take, and what is the CI for the same period (compounded annually)?
Solution:
  • Step 1: Find T using SI.
    • SI = (P × R × T) / 100
    • 2000 = (10000 × 10 × T) / 100
    • T = 2000 / 1000 = 2 years
  • Step 2: Calculate CI for T = 2 years.
    • A = 10000 × (1 + 10/100)^2 = 10000 × (1.1)^2 = 10000 × 1.21 = $12100
    • CI = 12100 - 10000 = $2100
  • Step 3: Compare.
    • SI = $2000
    • CI = $2100
    • Difference = 2100 - 2000 = $100
Answer:
  • Time = 2 years
  • SI = $2000, CI = $2100, Difference = $100
Explanation:
  • The time is derived from the SI formula, and CI is higher because it compounds the interest earned in the first year.
Problem 8: Finding Rate of Interest
Question: A principal of $5000 yields $1000 SI in 2 years. Find the rate and the CI (compounded annually) for the same period.
Solution:
  • Step 1: Find R using SI.
    • SI = (P × R × T) / 100
    • 1000 = (5000 × R × 2) / 100
    • R = 1000 × 100 / (5000 × 2) = 10%
  • Step 2: Calculate CI.
    • A = 5000 × (1 + 10/100)^2 = 5000 × (1.1)^2 = 5000 × 1.21 = $6050
    • CI = 6050 - 5000 = $1050
  • Step 3: Compare.
    • SI = $1000
    • CI = $1050
    • Difference = 1050 - 1000 = $50
Answer:
  • Rate = 10%
  • SI = $1000, CI = $1050, Difference = $50
Explanation:
  • The rate is found using SI, and CI is higher due to the compounding effect over 2 years.
Problem 9: Large Principal and Long Period
Question: $50,000 is invested at 7% per annum for 5 years. Calculate SI and CI (compounded annually).
Solution:
  • Step 1: P = $50,000, R = 7%, T = 5 years, n = 1
  • Step 2: Calculate SI.
    • SI = (50000 × 7 × 5) / 100 = $17500
    • Amount = 50000 + 17500 = $67500
  • Step 3: Calculate CI.
    • A = 50000 × (1 + 7/100)^5 = 50000 × (1.07)^5
    • (1.07)^5 ≈ 1.40255173
    • A = 50000 × 1.40255173 ≈ $70127.59
    • CI = 70127.59 - 50000 = $20127.59
  • Step 4: Compare.
    • SI = $17500
    • CI = $20127.59
    • Difference = 20127.59 - 17500 = $2627.59
Answer:
  • SI = $17500
  • CI = $20127.59
  • Difference = $2627.59
Explanation:
  • Over a longer period and with a large principal, the compounding effect is significantly pronounced, leading to a substantial difference between SI and CI.
Problem 10: Finding Amount for Equal SI and CI
Question: For what principal and time period will SI equal CI at 10% per annum, if CI is compounded annually for 1 year?
Solution:
  • Step 1: Assume T = 1 year, as CI for 1 year is often equal to SI.
    • Let P = Principal, R = 10%, T = 1, n = 1
  • Step 2: Calculate SI.
    • SI = (P × 10 × 1) / 100 = 0.1P
  • Step 3: Calculate CI.
    • A = P × (1 + 10/100)^1 = P × 1.1
    • CI = A - P = 1.1P - P = 0.1P
  • Step 4: Compare.
    • SI = 0.1P, CI = 0.1P
    • SI equals CI for T = 1 year, regardless of P.
  • Step 5: Verify for any P, e.g., P = $1000.
    • SI = (1000 × 10 × 1) / 100 = $100
    • A = 1000 × 1.1 = $1100, CI = 1100 - 1000 = $100
    • Difference = 0
Answer:
  • SI equals CI for any principal when T = 1 year.
  • Example: For P = $1000, SI = $100, CI = $100, Difference = $0
Explanation:
  • For 1 year with annual compounding, SI and CI are equal because there’s no time for interest to accumulate and compound further. This is a special case highlighting that differences arise only after the first compounding period.
Summary of Differences
  • Calculation Basis: SI is calculated only on the initial principal, while CI is calculated on the principal plus accumulated interest.
  • Growth Pattern: SI grows linearly, while CI grows exponentially due to compounding.
  • Impact of Time: The difference between SI and CI increases with longer time periods.
  • Impact of Rate: Higher interest rates amplify the CI advantage.
  • Compounding Frequency: More frequent compounding (e.g., semi-annually, quarterly) increases CI further compared to SI.
  • Special Case: For 1 year with annual compounding, SI equals CI.
General Tips
  • Use the SI formula for quick, linear calculations.
  • For CI, ensure the compounding frequency (n) is clear, and use the appropriate formula.
  • The difference between SI and CI is more significant for larger principals, higher rates, longer periods, and more frequent compounding.
  • Always double-check calculations, especially for CI, as small errors in exponents can lead to large discrepancies.
These problems demonstrate the practical and mathematical differences between SI and CI across various scenarios.

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