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Different Time Periods Comparison

Introduction

कई SI questions में अलग-अलग time periods की तुलना करनी होती है - जैसे months और years में मिले interest की तुलना, या unequal time spans पर same principal का interest compare करना। Time units को सही तरीके से convert करना और SI formula को ठीक से apply करना इन problems को बहुत आसान बना देता है।

Pattern: Different Time Periods Comparison

Pattern

Key concept: सभी time periods को एक ही unit (years) में convert करें और SI = (P × R × T)/100 का उपयोग करें - जहाँ T हमेशा years में होगा।

Useful conversions and reminders:
• Months → years: months ÷ 12 (जैसे 9 months = 9/12 = 0.75 years)
• Days → years: days ÷ 365 (जब तक question कुछ अलग न बताए)
• दो interests compare करते समय या तो SI equations equate करें या दोनों को calculate करके numeric values compare करें

Step-by-Step Example

Question

किसी राशि पर 6% per annum की दर से 9 months का interest ₹90 है। Principal निकालें।

Options:

  1. ₹2,000
  2. ₹1,800
  3. ₹2,200
  4. ₹2,500

Solution

  1. Step 1: Convert और given values पहचानें

    R = 6%, SI = ₹90, T = 9 months = 9/12 = 0.75 years.

  2. Step 2: SI formula years में apply करें

    SI = (P × R × T) / 100

  3. Step 3: Values substitute करें

    90 = (P × 6 × 0.75) / 100

  4. Step 4: Simplify

    6 × 0.75 = 4.5 → 90 = (P × 4.5) / 100 → P × 4.5 = 9,000

  5. Step 5: Solve

    P = 9,000 / 4.5 = 2,000

  6. Final Answer:

    ₹2,000 → Option A

  7. Quick Check:

    Yearly SI = (2000 × 6) / 100 = 120 → For 0.75 years → 120 × 0.75 = 90 ✅

Quick Variations

1. Same principal पर अलग-अलग times के लिए SI compare करना हो → yearly SI निकालें और time से multiply करें।

2. Rates अलग हों तो दोनों SI अलग-अलग calculate करें और फिर compare करें।

3. एक time months में हो और दूसरा years में → दोनों को years में convert करें।

4. Multiple segments हों → times को decimals में add करें (जैसे 1 + 0.5 = 1.5 years) या segment-wise SI निकालकर जोड़ें।

Trick to Always Use

  • Step 1 → सभी time values years में convert करें।
  • Step 2 → SI = (P × R × T)/100 apply करें।
  • Step 3 → Interests को numeric या algebraic तरीके से compare करें।

Summary

Summary

  • Time periods को हमेशा years में बदलें।
  • SI = (P × R × T)/100 में T को years में रखें।
  • Compare करने से पहले दोनों interests को uniform time unit में convert करें।
  • Quick check: yearly SI = (P×R)/100 निकालें और time से multiply करें।

Example to remember:
6% पर 9 months में SI = ₹90 → Principal = ₹2,000

Practice

(1/5)
1. A sum of ₹2000 is lent at 6% per annum for 9 months. Find the simple interest.
easy
A. ₹90
B. ₹100
C. ₹110
D. ₹120

Solution

  1. Step 1: Convert months to years

    Given P = 2000, R = 6% per annum, T = 9 months = 9/12 = 0.75 years.

  2. Step 2: Apply simple interest formula

    SI = (P × R × T) / 100 = (2000 × 6 × 0.75) / 100 = (2000 × 4.5) / 100 = 9000 / 100 = 90.

  3. Final Answer:

    SI = ₹90 → Option A.

  4. Quick Check:

    Yearly SI = (2000 × 6)/100 = 120; for 0.75 year → 120 × 0.75 = 90 ✅
Hint: Convert months to years (9/12 = 0.75) before using SI formula.
Common Mistakes: Using months as if they were years (e.g., treating 9 as 9 years).
2. If the SI on a sum of money for 15 months at 8% per annum is ₹1200, find the principal.
easy
A. ₹9600
B. ₹10,000
C. ₹11,000
D. ₹12,000

Solution

  1. Step 1: Convert months to years

    Given SI = 1200, R = 8% per annum, T = 15 months = 15/12 = 1.25 years.

  2. Step 2: Substitute in SI formula

    Use SI = (P × R × T)/100 → 1200 = (P × 8 × 1.25)/100.

  3. Step 3: Simplify the factor

    (8 × 1.25)/100 = 10/100 = 0.10, so 1200 = 0.10 × P.

  4. Step 4: Solve for principal

    P = 1200 / 0.10 = 12000.

  5. Final Answer:

    Principal = ₹12,000 → Option D.

  6. Quick Check:

    (12000 × 8 × 1.25)/100 = 1200 ✅
Hint: Convert months to years (15/12 = 1.25) and simplify the factor (R×T)/100 before dividing.
Common Mistakes: Forgetting to convert months to years or miscomputing (R×T)/100.
3. A sum of ₹5000 is lent at 10% per annum for 2 years. The same sum at the same rate is lent for 18 months. Find the difference in the interests.
easy
A. ₹250
B. ₹300
C. ₹350
D. ₹400

Solution

  1. Step 1: Write given values

    Given P = 5000, R = 10% per annum.

  2. Step 2: Compute SI for 2 years

    SI₁ = (5000 × 10 × 2)/100 = 1000.

  3. Step 3: Convert 18 months to years and compute SI

    18 months = 1.5 years → SI₂ = (5000 × 10 × 1.5)/100 = 750.

  4. Step 4: Compare the two interests

    Difference = SI₁ - SI₂ = 1000 - 750 = 250.

  5. Final Answer:

    Difference = ₹250 → Option A.

  6. Quick Check:

    Extra 0.5 year interest = (5000 × 10 × 0.5)/100 = 250 ✅
Hint: Compute yearly SI and scale by time (2.0 vs 1.5 years) to find the difference quickly.
Common Mistakes: Not converting 18 months to 1.5 years.
4. A sum of ₹8000 earns simple interest of ₹600 in 9 months at a certain rate. What will be the interest for 2 years at the same rate?
medium
A. ₹1200
B. ₹1600
C. ₹1400
D. ₹1800

Solution

  1. Step 1: Convert given time to years

    Given P = 8000, SI₁ = 600 for T₁ = 9 months = 0.75 years.

  2. Step 2: Find rate using SI formula

    R = (SI × 100)/(P × T) → R = (600 × 100)/(8000 × 0.75) = 60000/6000 = 10% per annum.

  3. Step 3: Use rate to compute new interest

    SI₂ = (8000 × 10 × 2)/100 = 1600.

  4. Final Answer:

    Interest for 2 years = ₹1600 → Option B.

  5. Quick Check:

    Yearly interest = (8000 × 10)/100 = 800; ×2 = 1600 ✅
Hint: First find the annual rate from the short period, then reuse it for the longer period.
Common Mistakes: Forgetting to convert 9 months to 0.75 years when finding R.
5. If ₹10,000 earns simple interest of ₹800 in 16 months, find the rate of interest per annum.
medium
A. 9%
B. 8%
C. 6%
D. 12%

Solution

  1. Step 1: Convert months to years

    Given P = 10000, SI = 800, T = 16 months = 16/12 = 4/3 years.

  2. Step 2: Apply rate formula

    R = (SI × 100)/(P × T) = (800 × 100)/(10000 × 4/3).

  3. Step 3: Simplify the fraction

    Denominator = 10000 × 4/3 = 40000/3 → R = 80000 / (40000/3) = 80000 × (3/40000) = 6.

  4. Final Answer:

    Rate = 6% → Option C.

  5. Quick Check:

    SI = (10000 × 6 × 4/3)/100 = 800 ✅
Hint: Use fraction simplification: 16/12 = 4/3 to keep arithmetic exact.
Common Mistakes: Rounding intermediate steps or not simplifying the fraction 16/12 to 4/3.

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