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Mixed SI & CI Problems

Introduction

Mixed SI & CI problems में Simple Interest और Compound Interest दोनों एक ही question में मिलते हैं। कभी interest का कुछ हिस्सा SI पर होता है और बाकी CI पर, या कभी principal के अलग-अलग हिस्सों पर SI और CI लागू किए जाते हैं। इस pattern को सीखने से आप question को stages में तोड़कर हर stage में सही formula apply कर सकते हैं।

Pattern: Mixed SI & CI Problems

Pattern

Key concept: हर period/portion को अलग treat करें - SI के लिए formula SI = P·R·T / 100 और CI के लिए formula A = P × (1 + R/100)^n use करें; फिर results को combine करें।

Typical approaches:

  • Stage-wise time split: पहले कुछ समय तक SI apply करें, फिर बाकी समय पर amount को CI से grow करें (या उल्टा)।
  • Principal split: Principal के एक हिस्से पर SI और दूसरे हिस्से पर CI apply करें; फिर दोनों amounts/interest को जोड़ें।
  • Mixed rates: अगर compounding frequency अलग हो, तो rate को per-period में convert करें और time units align करें।

Step-by-Step Example

Question

₹50,000 को 5% प्रति वर्ष simple interest पर 2 साल के लिए invest किया गया। 2 साल बाद मिला amount 6% प्रति वर्ष compound interest पर 1 साल के लिए invest किया गया। Final amount find करें।

Solution

  1. Step 1: Stage 1 (SI) के values identify करें

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

  2. Step 2: Stage 1 (SI) compute करें

    SI = 50,000 × 5 × 2 / 100 = 50,000 × 0.10 = ₹5,000. 2 साल बाद amount = 50,000 + 5,000 = ₹55,000.

  3. Step 3: Stage 2 (CI) के values identify करें

    अब A₁ = ₹55,000 को 6% p.a. compound पर n = 1 year के लिए invest करें। Use A = P × (1 + R/100)^n.

  4. Step 4: Stage 2 (CI) compute करें

    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 के बाद amount ₹55,000; उसका 6% = ₹3,300 → final ₹58,300 ✅

Quick Variations

1. SI पहले, फिर resulting amount पर CI (stage-wise)।

2. पहले CI, फिर नए amount पर SI (reverse stage order)।

3. Principal को SI और CI में split करके algebraic equations बनाएं।

4. Compounding frequency अलग हो तो rate को per-period में convert करें और time alignment करें।

Trick to Always Use

  • Step 1: Problem को independent parts (stages या portions) में बांटें।
  • Step 2: जहाँ जरूरत हो SI formula SI = P·R·T / 100 और जहाँ जरूरत हो CI formula A = P(1 + R/100)^n apply करें।
  • Step 3: अगर compounding frequency अलग है तो convert करें (r = R/n, periods = nT).
  • Step 4: अंत में amounts या interests को combine करें और quick check करें (एक stage forward/backward recompute करके)।

Summary

Summary

  • Mixed problems को stages या parts में break करें; हर part पर सही formula apply करें।
  • Simple Interest: SI = P·R·T / 100. Compound Interest: A = P × (1 + R/100)^n.
  • Principal split होने पर दोनों हिस्सों की interest expressions बनाकर algebraically solve करें।
  • हमेशा time units और compounding frequency align करें; अंत में एक quick sanity check ज़रूर करें।

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.

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