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Finding Rate (R)

Introduction

Many simple interest questions give the principal (P), time (T), and the interest earned (SI), and ask for the annual rate of interest (R). Being able to rearrange the SI formula to find the rate is essential for comparing investments and loans.

Pattern: Finding Rate (R)

Pattern

Key concept: Rearrange SI = (P × R × T) / 100 to solve for R.

Formula:
R = (SI × 100) / (P × T)

Step-by-Step Example

Question

A sum of ₹4,000 yields a simple interest of ₹480 in 2 years. Find the rate of interest per annum.

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

Solution

  1. Step 1: Identify the given values

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

  2. Step 2: Write the rearranged formula

    R = (SI × 100) / (P × T)

  3. Step 3: Substitute the values

    R = (480 × 100) / (4,000 × 2)

  4. Step 4: Simplify numerator and denominator

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

  5. Step 5: Divide to find R

    R = 48,000 ÷ 8,000 = 6

  6. Final Answer:

    6% per annum → Option B

  7. Quick Check:

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

Quick Variations

1. SI given for fractional time (e.g., 9 months): convert T to years (9/12 = 0.75) then use formula.

2. SI and P given but time in months/days: convert time into years before substituting.

3. When SI is given for multiple segments with different rates/times, compute total SI and effective P×T equivalent or split into parts and find rates separately if asked.

Trick to Always Use

  • Step 1 → Convert time to years (months ÷ 12) before plugging into the formula.
  • Step 2 → Compute (SI × 100) first to keep integer arithmetic, then divide by (P × T).
  • Step 3 → If result is a decimal, express rate to required precision (e.g., 6.25%).

Summary

Summary

  • Use R = (SI × 100) / (P × T) to find the rate when principal, SI, and time are known.
  • Always ensure time is converted to years before substitution.
  • Compute SI×100 first and divide by P×T to keep arithmetic simple.
  • Quick-check by recalculating yearly SI = (P × R)/100 and scaling by T to verify.

Example to remember:
For 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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