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Finding Time (T)

Introduction

Simple interest problems में अक्सर principal (P), rate (R) और total interest (SI) दिया होता है, और पूछा जाता है कि time (T) कितना लगेगा। यह real-life situations में बहुत उपयोगी होता है, जैसे किसी fixed rate पर पैसा double होने में कितना समय लगेगा।

Pattern: Finding Time (T)

Pattern

Key concept: SI = (P × R × T) / 100 को rearrange करके T निकालना।

Formula:
T = (SI × 100) / (P × R)

Step-by-Step Example

Question

₹5,000 पर 10% per annum की दर से simple interest ₹1,500 मिलता है। Time period निकालें।

Options:
A. 2 years
B. 3 years
C. 4 years
D. 5 years

Solution

  1. Step 1: Given values पहचानें

    P = 5,000, SI = 1,500, R = 10%.

  2. Step 2: Time का formula लिखें

    T = (SI × 100) / (P × R)

  3. Step 3: Values substitute करें

    T = (1,500 × 100) / (5,000 × 10)

  4. Step 4: Numerator और denominator simplify करें

    Numerator = 1,500 × 100 = 150,000
    Denominator = 5,000 × 10 = 50,000

  5. Step 5: Divide करके time निकालें

    T = 150,000 ÷ 50,000 = 3

  6. Final Answer:

    3 years → Option B

  7. Quick Check:

    Yearly SI = (5,000 × 10) / 100 = 500; 3 years → 500 × 3 = 1,500 ✅

Quick Variations

1. Time months या days में हो तो उसे years में convert करें (जैसे 18 months = 1.5 years)।

2. कभी-कभी SI fractional time के लिए दिया होता है - formula ठीक से rearrange करें।

3. “Doubling time” वाले questions में SI = principal होता है, वहाँ भी यही formula use होता है।

Trick to Always Use

  • Step 1 → Months/days हों तो time को years में convert करें।
  • Step 2 → Formula apply करें: T = (SI×100) ÷ (P×R).
  • Step 3 → Verify करें: SI = (P×R×T)/100 से।

Summary

Summary

  • जब P, R और SI दिए हों, time निकालने के लिए T = (SI × 100) / (P × R) का उपयोग करें।
  • Months/days को हमेशा years में convert करें।
  • Arithmetic mistakes से बचने के लिए पहले SI×100 और P×R अलग-अलग simplify करें।
  • Verification के लिए yearly SI = (P×R)/100 निकालें और T से multiply करके check करें।

याद रखने लायक example:
P = ₹5,000, R = 10%, SI = ₹1,500 → T = 3 years.

Practice

(1/5)
1. A sum of ₹6000 earns a simple interest of ₹900 at 5% per annum. How long was the money invested?
easy
A. 2 years
B. 3 years
C. 4 years
D. 5 years

Solution

  1. Step 1: Identify given values

    P = 6000, SI = 900, R = 5%.

  2. Step 2: Apply the time formula

    T = (SI × 100) / (P × R).

  3. Step 3: Substitute the values

    T = (900 × 100) / (6000 × 5) = 90000 / 30000 = 3.

  4. Final Answer:

    3 years → Option B

  5. Quick Check:

    Yearly SI = (6000 × 5)/100 = 300; ×3 years = 900 ✅
Hint: Divide total SI by yearly SI to get time quickly.
Common Mistakes: Forgetting to multiply P and R in the denominator.
2. A principal of ₹2000 yields ₹200 as simple interest at a rate of interest of 5% per annum. For how long was the money lent?
easy
A. 1 year
B. 2 years
C. 3 years
D. 2.5 years

Solution

  1. Step 1: Identify given values

    P = 2000, SI = 200, R = 5%.

  2. Step 2: Apply the formula

    T = (200 × 100) / (2000 × 5) = 20000 / 10000 = 2.

  3. Final Answer:

    2 years → Option B

  4. Quick Check:

    Yearly SI = (2000 × 5)/100 = 100; ×2 = 200 ✅
Hint: Calculate SI×100 first, then divide by (P×R).
Common Mistakes: Mixing up percent decimals or skipping the ×100 factor.
3. A sum of ₹3000 yields a simple interest of ₹450 when invested at 6% per annum. For how many years was it invested?
easy
A. 2 years
B. 1.5 years
C. 2.5 years
D. 3 years

Solution

  1. Step 1: Identify given values

    P = 3000, SI = 450, R = 6%.

  2. Step 2: Use the formula for time

    T = (450 × 100) / (3000 × 6) = 45000 / 18000 = 2.5.

  3. Final Answer:

    2.5 years → Option C

  4. Quick Check:

    Yearly SI = (3000 × 6)/100 = 180; ×2.5 = 450 ✅
Hint: Multiply R×P first in denominator to simplify division.
Common Mistakes: Treating 2.5 as 25 or misplacing decimal points.
4. The simple interest on a sum at 8% per annum for a certain period is ₹640. If the principal is ₹2000, how long was the money invested?
medium
A. 2 years
B. 3 years
C. 4 years
D. 5 years

Solution

  1. Step 1: Identify given values

    P = 2000, SI = 640, R = 8%.

  2. Step 2: Apply formula

    T = (640 × 100) / (2000 × 8) = 64000 / 16000 = 4.

  3. Final Answer:

    4 years → Option C

  4. Quick Check:

    Yearly SI = (2000 × 8)/100 = 160; ×4 = 640 ✅
Hint: Compute (P×R) then divide SI×100 by that product.
Common Mistakes: Dividing by R only or forgetting to include time factor correctly.
5. A sum of ₹1200 earns a simple interest of ₹90 at a rate of 3.75% per annum. For how long was the money invested?
medium
A. 2 years
B. 1.5 years
C. 3 years
D. 2.5 years

Solution

  1. Step 1: Identify given values

    P = 1200, SI = 90, R = 3.75%.

  2. Step 2: Apply formula

    T = (90 × 100) / (1200 × 3.75) = 9000 / 4500 = 2.

  3. Final Answer:

    2 years → Option A

  4. Quick Check:

    Yearly SI = (1200 × 3.75)/100 = 45; ×2 = 90 ✅
Hint: Expect decimal rates; keep precision (e.g., 3.75) intact during division.
Common Mistakes: Rounding 3.75 too early or misreading months as years.

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