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Mixed SI & CI Problems

Introduction

Mixed SI & CI problems combine Simple Interest and Compound Interest within the same question. You may see interest applied as SI for one period and CI for another, or different portions of money treated under SI and CI. Learning this pattern helps you break the question into stages and apply the correct formula for each stage.

Pattern: Mixed SI & CI Problems

Pattern

Key concept: Treat each period/portion separately - use SI formula SI = P·R·T / 100 for simple interest and CI formula A = P × (1 + R/100)^n for compound interest; then combine results.

Typical approaches:

  • Stage-wise time split: Apply SI over the first part of time, then compute CI on the amount for the remaining time (or vice versa).
  • Principal split: Apply SI on one part of principal and CI on the other part, then add amounts/interests.
  • Mixed rates: Convert rates to per-period if compounding frequency differs, and align time units.

Step-by-Step Example

Question

₹50,000 is invested at 5% per annum simple interest for 2 years. The amount after 2 years is then invested at 6% per annum compound interest for 1 year. Find the final amount.

Solution

  1. Step 1: Identify values for Stage 1 (SI)

    Stage 1 principal P₁ = ₹50,000; R₁ = 5% p.a.; T₁ = 2 years. Use SI = P·R·T / 100.

  2. Step 2: Compute Stage 1 (SI)

    SI = 50,000 × 5 × 2 / 100 = 50,000 × 0.10 = ₹5,000 interest. Amount after 2 yrs = 50,000 + 5,000 = ₹55,000.

  3. Step 3: Identify values for Stage 2 (CI)

    Now invest A₁ = ₹55,000 at R₂ = 6% p.a. compound for n = 1 year. Use A = P × (1 + R/100)^n.

  4. Step 4: Compute Stage 2 (CI)

    Final amount A = 55,000 × (1 + 0.06)^1 = 55,000 × 1.06 = ₹58,300.

  5. Final Answer:

    Final amount = ₹58,300.

  6. Quick Check:

    SI gave ₹55,000 after 2 years; 6% of 55,000 is ₹3,300 → final ₹58,300 ✅

Quick Variations

1. SI first then CI on the resulting amount (stage-wise).

2. CI first then SI on the new amount (reverse stage order).

3. Principal split between SI and CI investments - set up algebraic equations.

4. Different compounding frequencies - convert rates to per-period and align time units.

Trick to Always Use

  • Step 1: Split the problem into independent parts (stages or portions).
  • Step 2: Apply SI formula SI = P·R·T / 100 where appropriate and CI formula A = P(1 + R/100)^n where appropriate.
  • Step 3: Convert compounding frequency if needed (r = R/n, periods = nT).
  • Step 4: Combine amounts or interests at the end and perform a quick check by recomputing one stage forward/backward.

Summary

Summary

  • Break mixed problems into stages or parts; treat each independently with the correct formula.
  • Simple Interest: SI = P·R·T / 100. Compound Interest: A = P × (1 + R/100)^n.
  • When principals are split, write interest expressions for each part and solve algebraically.
  • Always align time units and compounding frequency; finish with a quick sanity check.

Practice

(1/5)
1. ₹10,000 is invested at 6% p.a. simple interest for 2 years and the amount is then invested at 5% p.a. compound interest for 1 year. Find the final amount.
easy
A. ₹11,760.00
B. ₹11,236.50
C. ₹11,200.00
D. ₹11,150.00

Solution

  1. Step 1: Compute simple interest for stage 1

    Stage 1 (SI): P₁ = ₹10,000; R₁ = 6%; T₁ = 2 years → SI = (10,000 × 6 × 2) / 100 = ₹1,200.

  2. Step 2: Find amount after simple interest stage

    Amount after 2 years = 10,000 + 1,200 = ₹11,200.

  3. Step 3: Apply compound interest on the new principal

    Stage 2 (CI): P₂ = ₹11,200; R₂ = 5%; n = 1 year → A = 11,200 × (1 + 0.05) = 11,200 × 1.05 = ₹11,760.00.

  4. Final Answer:

    Final amount = ₹11,760.00 → Option A.

  5. Quick Check:

    SI gave ₹11,200; 5% of 11,200 = ₹560; 11,200 + 560 = 11,760 ✅
Hint: Compute SI first, add to principal, then apply CI on the new amount.
Common Mistakes: Applying compound interest for the simple-interest period or vice versa.
2. ₹5,000 is invested at 8% p.a. compound interest for 1 year and then at 10% p.a. simple interest for 2 years. Find the total amount after 3 years.
easy
A. ₹6,100.00
B. ₹6,480.00
C. ₹6,000.00
D. ₹6,050.00

Solution

  1. Step 1: Compute stage 1 compound amount

    Stage 1 (CI): P₁ = ₹5,000; R₁ = 8%; n = 1 year → A₁ = 5,000 × 1.08 = ₹5,400.

  2. Step 2: Compute simple interest on the new principal

    Stage 2 (SI): Principal for SI = ₹5,400; R₂ = 10%; T₂ = 2 years → SI = 5,400 × 10 × 2 / 100 = ₹1,080.

  3. Step 3: Add SI to the CI amount

    Total amount = 5,400 + 1,080 = ₹6,480.00.

  4. Final Answer:

    Total amount = ₹6,480.00 → Option B.

  5. Quick Check:

    CI gave ₹5,400; SI on that for 2 years at 10% gives ₹1,080 → total ₹6,480 ✅
Hint: Apply the correct formula for each stage in sequence, then add results.
Common Mistakes: Using SI for the period that is compound or vice versa.
3. Out of ₹20,000, one part is lent at 5% simple interest and the other at 10% compound interest for 2 years. If the total interest earned is ₹2,200, find the amount lent at simple interest.
easy
A. ₹10,000.22
B. ₹12,000.42
C. ₹18,181.82
D. ₹9,000.62

Solution

  1. Step 1: Write SI for the portion at simple interest

    Let x = amount at 5% SI. SI interest = x × 5% × 2 = 0.10x.

  2. Step 2: Write CI interest for the remainder

    Amount at 10% CI for 2 years has interest = (20,000 - x)[(1.1)^2 - 1] = (20,000 - x) × 0.21 = 4,200 - 0.21x.

  3. Step 3: Form and solve the total-interest equation

    Total interest: 0.10x + (4,200 - 0.21x) = 2,200 → 4,200 - 0.11x = 2,200 → -0.11x = -2,000 → x = 18,181.818... = ₹18,181.82.

  4. Final Answer:

    Amount at SI = ₹18,181.82 → Option C.

  5. Quick Check:

    SI interest ≈ 0.10×18,181.82 = 1,818.18; CI interest ≈ 0.21×1,818.18 ≈ 381.82; sum ≈ 2,200 ✅
Hint: Write SI + CI expressions in x, solve the linear equation for x.
Common Mistakes: Ignoring compound factor (using 10%×2 for CI part) or mixing up parts.
4. ₹12,000 earns simple interest for 1 year at 8% and then compound interest for 2 years at 10%. Find the total amount (rounded to 2 d.p.).
medium
A. ₹15,972.00
B. ₹15,973.00
C. ₹15,680.60
D. ₹15,681.60

Solution

  1. Step 1: Compute simple interest for year 1

    Stage 1 (SI): P₁ = ₹12,000; R₁ = 8%; T₁ = 1 year → SI = 12,000 × 0.08 = ₹960 → Amount after 1 year = 12,960.

  2. Step 2: Apply compound interest for the next 2 years

    Stage 2 (CI): P₂ = ₹12,960; R₂ = 10%; n = 2 years → A = 12,960 × (1.1)^2 = 12,960 × 1.21 = ₹15,681.60.

  3. Final Answer:

    Total amount = ₹15,681.60 → Option D.

  4. Quick Check:

    21% of 12,960 ≈ 2,721 → 12,960 + 2,721 = 15,681 ✅
Hint: Do SI stage first, then apply CI on the new principal for subsequent years.
Common Mistakes: Applying CI for the first year when SI is specified.
5. A total of ₹30,000 is invested: part at 8% simple interest and the rest at 10% compound interest for 2 years. If the total interest is ₹5,220, find the amount invested at compound interest.
medium
A. ₹8,400.00
B. ₹12,000.00
C. ₹8,000.00
D. ₹9,000.00

Solution

  1. Step 1: Express SI on the simple-interest part

    Let x = amount at 10% CI. Then (30,000 - x) at 8% SI for 2 years → SI = (30,000 - x) × 0.16 = 4,800 - 0.16x.

  2. Step 2: Express CI interest on x

    CI interest on x for 2 years = x × [(1.1)^2 - 1] = x × 0.21.

  3. Step 3: Form and solve the total-interest equation

    Total interest: 4,800 - 0.16x + 0.21x = 5,220 → 4,800 + 0.05x = 5,220 → 0.05x = 420 → x = ₹8,400.00.

  4. Final Answer:

    Amount at CI = ₹8,400.00 → Option A.

  5. Quick Check:

    CI interest = 8,400 × 0.21 = 1,764; SI interest = 21,600 × 0.16 = 3,456; sum = 1,764 + 3,456 = 5,220 ✅
Hint: Form equation: SI(part) + CI(part) = total interest, solve for x.
Common Mistakes: Treating CI part as simple interest or vice versa.

Mock Test

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