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Finding Rate (R)

Introduction

कई simple interest questions में principal (P), time (T) और earned interest (SI) दिया होता है, और पूछा जाता है कि annual rate of interest (R) कितना है। Investments और loans compare करने के लिए SI formula को rearrange करके rate निकालना एक जरूरी skill है।

Pattern: Finding Rate (R)

Pattern

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

Formula:
R = (SI × 100) / (P × T)

Step-by-Step Example

Question

₹4,000 पर 2 years में simple interest ₹480 मिलता है। Rate of interest per annum निकालें।

Options:
A. 5% per annum
B. 6% per annum
C. 6.5% per annum
D. 7% per annum

Solution

  1. Step 1: Given values पहचानें

    P = 4,000, SI = 480, T = 2 years.

  2. Step 2: Rearranged formula लिखें

    R = (SI × 100) / (P × T)

  3. Step 3: Values substitute करें

    R = (480 × 100) / (4,000 × 2)

  4. Step 4: Numerator और denominator simplify करें

    Numerator = 480 × 100 = 48,000
    Denominator = 4,000 × 2 = 8,000

  5. Step 5: R निकालें

    R = 48,000 ÷ 8,000 = 6

  6. Final Answer:

    6% per annum → Option B

  7. Quick Check:

    Yearly SI = (4,000 × 6) / 100 = 240; 2 years → 240 × 2 = 480 ✅

Quick Variations

1. Fractional time दिया हो (जैसे 9 months) → T को years में बदलें (9/12 = 0.75) और formula use करें।

2. SI और P दिए हों लेकिन time months/days में हो → पहले time को years में convert करें।

3. Different rates/time segments हों → पहले total SI निकालें, फिर effective P×T लेकर rate निकालें या parts को अलग-अलग solve करें।

Trick to Always Use

  • Step 1 → Time को years में convert करें (months ÷ 12).
  • Step 2 → पहले (SI × 100) compute करें ताकि arithmetic simple रहे, फिर P × T से divide करें।
  • Step 3 → Result decimal आए तो उसे सही precision में rate लिखें (जैसे 6.25%).

Summary

Summary

  • जब P, SI और T दिए हों, तो rate निकालने के लिए R = (SI × 100) / (P × T) use करें।
  • Time को हमेशा years में बदलें, फिर formula में substitute करें।
  • Arithmetic आसान बनाने के लिए पहले SI×100 compute करें, फिर P×T से divide करें।
  • Verify करने के लिए yearly SI = (P × R)/100 निकालें और उसे T से multiply करें।

याद रखने लायक example:
P = ₹4,000, SI = ₹480 in 2 years → R = 6% per annum.

Practice

(1/5)
1. A sum of ₹2000 yields a simple interest of ₹240 in 2 years. Find the annual rate of interest.
easy
A. 6%
B. 5%
C. 7%
D. 4%

Solution

  1. Step 1: Write the given values

    Given P = 2000, SI = 240, T = 2 years.

  2. Step 2: Apply the rate formula

    Use R = (SI × 100) / (P × T).

  3. Step 3: Calculate the rate

    R = (240 × 100) / (2000 × 2) = 24000 / 4000 = 6.

  4. Final Answer:

    Rate = 6% → Option A.

  5. Quick Check:

    Yearly SI = (2000 × 6)/100 = 120; for 2 years → 120 × 2 = 240 ✅
Hint: Compute (SI×100) first, then divide by (P×T).
Common Mistakes: Forgetting to multiply time with principal in the denominator.
2. Simple interest of ₹135 is earned on ₹1500 in 3 years. What is the yearly rate of interest?
easy
A. 4%
B. 3%
C. 5%
D. 2.5%

Solution

  1. Step 1: Write the given values

    Given P = 1500, SI = 135, T = 3 years.

  2. Step 2: Apply formula

    R = (135 × 100) / (1500 × 3).

  3. Step 3: Calculate the rate

    R = 13500 / 4500 = 3.

  4. Final Answer:

    Rate = 3% → Option B.

  5. Quick Check:

    Yearly SI = (1500 × 3)/100 = 45; ×3 years = 135 ✅
Hint: Multiply R×T in denominator to avoid extra steps later.
Common Mistakes: Dividing by 3 (time) before applying the 100 factor incorrectly.
3. A principal of ₹3000 earns a simple interest of ₹450 in 3 years. Find the annual rate of interest.
easy
A. 5%
B. 6%
C. 4%
D. 7%

Solution

  1. Step 1: Write the given values

    Given P = 3000, SI = 450, T = 3 years.

  2. Step 2: Apply formula

    R = (450 × 100) / (3000 × 3).

  3. Step 3: Calculate the rate

    R = 45000 / 9000 = 5.

  4. Final Answer:

    Rate = 5% → Option A.

  5. Quick Check:

    Yearly SI = (3000 × 5)/100 = 150; ×3 = 450 ✅
Hint: After computing R, verify by computing yearly interest (P×R/100).
Common Mistakes: Using total years incorrectly (e.g., using 2 instead of 3).
4. A sum produces a simple interest of ₹150 in 1.5 years when invested at an annual rate of interest. If the principal is ₹2000, find the rate (per annum).
medium
A. 4%
B. 3.5%
C. 5%
D. 6%

Solution

  1. Step 1: Write the given values

    Given P = 2000, SI = 150, T = 1.5 years.

  2. Step 2: Apply formula

    R = (150 × 100) / (2000 × 1.5).

  3. Step 3: Calculate the rate

    R = 15000 / 3000 = 5.

  4. Final Answer:

    Rate = 5% → Option C.

  5. Quick Check:

    Yearly SI = (2000 × 5)/100 = 100; for 1.5 years → 100 × 1.5 = 150 ✅
Hint: Convert fractional years to decimal (e.g., 1.5) before substitution.
Common Mistakes: Forgetting to convert months/halves to decimal years.
5. Simple interest of ₹90 is earned on ₹1200 in 2 years. What is the annual rate of interest?
medium
A. 4%
B. 3.5%
C. 4.5%
D. 3.75%

Solution

  1. Step 1: Write the given values

    Given P = 1200, SI = 90, T = 2 years.

  2. Step 2: Apply formula

    R = (90 × 100) / (1200 × 2).

  3. Step 3: Calculate the rate

    R = 9000 / 2400 = 3.75.

  4. Final Answer:

    Rate = 3.75% → Option D.

  5. Quick Check:

    Yearly SI = (1200 × 3.75)/100 = 45; for 2 years → 45 × 2 = 90 ✅
Hint: Expect decimal rates; keep two decimal places if needed (e.g., 3.75%).
Common Mistakes: Rounding 3.75 to 4% prematurely or misdividing 9000 ÷ 2400.

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