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Finding Principal (P)

Introduction

Many simple interest problems give the interest earned (SI), the rate (R), and the time (T), and ask for the original sum - the principal (P). Knowing how to reverse the SI formula is essential for loan, investment, and backward-calculation problems.

Pattern: Finding Principal (P)

Pattern

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

Formula:
P = (SI × 100) / (R × T)

Step-by-Step Example

Question

The simple interest earned on a sum for 3 years at 6% per annum is ₹540. Find the principal.

Options:
A. ₹2,500
B. ₹3,000
C. ₹3,500
D. ₹4,000

Solution

  1. Step 1: Identify the given values

    SI = 540, R = 6%, T = 3 years.

  2. Step 2: Write the rearranged formula

    P = (SI × 100) / (R × T)

  3. Step 3: Substitute values

    P = (540 × 100) / (6 × 3)

  4. Step 4: Simplify numerator and denominator

    Denominator = 6 × 3 = 18
    Numerator = 540 × 100 = 54,000

  5. Step 5: Divide to get the principal

    P = 54,000 ÷ 18 = 3,000

  6. Final Answer:

    ₹3,000 → Option B

  7. Quick Check:

    Yearly SI = (3,000 × 6)/100 = 180; for 3 years → 180 × 3 = 540 ✅

Quick Variations

1. SI given for fractional time (e.g., 2.5 years) - use T as decimal in formula.

2. SI given for months - convert months to years before substituting (e.g., 9 months = 0.75 year).

3. SI for multiple rates/time segments - compute total SI, then use overall R×T equivalent if applicable or split into parts and sum.

Trick to Always Use

  • Step 1 → Always convert time to years (months → divide by 12).
  • Step 2 → Rearrange formula clearly: P = (SI × 100) / (R × T).
  • Step 3 → Simplify numerator and denominator stepwise before final division to avoid arithmetic errors.

Summary

Summary

  • Use P = (SI × 100) / (R × T) to find the principal when SI, rate, and time are known.
  • Ensure rate is in percent and time is converted to years.
  • Always verify by recalculating SI using the found principal.
  • Simplify numerator and denominator before division to avoid mistakes.

Example to remember:
SI = ₹540 at 6% for 3 years → Principal = ₹3,000.

Practice

(1/5)
1. The simple interest earned on a sum for 3 years at 6% per annum is ₹540. Find the principal.
easy
A. 3000
B. 2500
C. 3500
D. 4000

Solution

  1. Step 1: Write the given values

    SI = 540, R = 6%, T = 3 years.

  2. Step 2: Apply the rearranged formula

    P = (SI × 100) / (R × T).

  3. Step 3: Substitute and simplify

    P = (540 × 100) / (6 × 3) = 54000 / 18 = 3000.

  4. Final Answer:

    ₹3000 → Option A

  5. Quick Check:

    Yearly SI = (3000 × 6)/100 = 180; ×3 = 540 ✅
Hint: Rearrange: P = (SI×100)/(R×T).
Common Mistakes: Forgetting to multiply R and T in the denominator or misplacing a zero.
2. A sum yields a simple interest of ₹375 at 5% per annum for 2 years. What is the principal?
easy
A. 3750
B. 3500
C. 4000
D. 3000

Solution

  1. Step 1: Note the given values

    SI = 375, R = 5%, T = 2 years.

  2. Step 2: Use the formula

    P = (SI × 100) / (R × T).

  3. Step 3: Substitute correctly

    P = (375 × 100) / (5 × 2) = 37500 / 10 = 3750.

  4. Final Answer:

    ₹3750 → Option A

  5. Quick Check:

    Yearly SI = (3750 × 5)/100 = 187.5; ×2 = 375 ✅
Hint: Compute (SI×100) first, then divide by (R×T).
Common Mistakes: Treating percentage 5% as 0.5 or forgetting decimal SI in quick check.
3. The simple interest on a certain sum for 2 years at 4.5% per annum is ₹225. Find the principal.
easy
A. 2400
B. 2500
C. 2600
D. 2250

Solution

  1. Step 1: Write SI, R, and T

    SI = 225, R = 4.5%, T = 2 years.

  2. Step 2: Apply formula carefully

    P = (225 × 100) / (4.5 × 2) = 22500 / 9.

  3. Step 3: Divide to find P

    P = 2500.

  4. Final Answer:

    ₹2500 → Option B

  5. Quick Check:

    Yearly SI = (2500 × 4.5)/100 = 112.5; ×2 = 225 ✅
Hint: When R is decimal (4.5), multiply R×T carefully (4.5×2=9).
Common Mistakes: Treating 4.5 as 45 or forgetting to convert months/years.
4. The simple interest on a sum of money for 4 years at 7% per annum amounts to ₹840. Find the principal.
medium
A. 2800
B. 3200
C. 3000
D. 3500

Solution

  1. Step 1: Identify the values

    SI = 840, R = 7%, T = 4 years.

  2. Step 2: Apply formula

    P = (840 × 100) / (7 × 4) = 84000 / 28.

  3. Step 3: Compute the division

    P = 3000.

  4. Final Answer:

    ₹3000 → Option C

  5. Quick Check:

    Yearly SI = (3000 × 7)/100 = 210; ×4 = 840 ✅
Hint: Multiply R and T first to get denominator (7×4=28).
Common Mistakes: Dividing by 4 then by 7 separately causing rounding errors.
5. A borrower paid simple interest of ₹612 on a loan for 2 years at 6% per annum. What was the principal borrowed?
medium
A. 5000
B. 5200
C. 4900
D. 5100

Solution

  1. Step 1: Note SI, R, and T

    SI = 612, R = 6%, T = 2 years.

  2. Step 2: Apply formula

    P = (612 × 100) / (6 × 2) = 61200 / 12.

  3. Step 3: Compute P

    P = 5100.

  4. Final Answer:

    ₹5100 → Option D

  5. Quick Check:

    Yearly SI = (5100 × 6)/100 = 306; ×2 = 612 ✅
Hint: Compute (R×T) first to simplify denominator.
Common Mistakes: Using incorrect denominator (e.g., 6×12 instead of 6×2) or misplacing zeros.

Mock Test

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