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Principal or Rate Finding

Introduction

Many aptitude problems give the compound interest (or final amount) and ask you to find the principal or the rate. The key is to rearrange the compound-interest formula and solve for the unknown (isolate P or R). This pattern teaches clear, repeatable steps so you can handle these reverse problems reliably.

Pattern: Principal or Rate Finding

Pattern

Key concept: Rearrange A = P(1 + R/100)^{T} or CI = P[(1 + R/100)^{T} - 1] to solve for P or R.

Useful forms:
A = P × (1 + R/100)^{T} → amount after T years.
CI = A - P = P × [(1 + R/100)^{T} - 1]
To find P when CI is given:
P = CI / [(1 + R/100)^{T} - 1]
To find R when CI and P are given:
(1 + R/100)^{T} = CI/P + 1 → take the T-th root:
R = [ (CI/P + 1)^{1/T} - 1 ] × 100

Step-by-Step Example

Question

The compound interest on a sum for 2 years at 8% per annum (compounded annually) is ₹832. Find the principal.

Solution

  1. Step 1: Identify values

    CI = ₹832; R = 8% p.a.; T = 2 years.

  2. Step 2: Use CI formula to isolate P

    CI = P × [ (1 + R/100)^{T} - 1 ] → P = CI / [ (1 + R/100)^{T} - 1 ].

  3. Step 3: Substitute numbers

    (1 + R/100)^{T} = (1.08)^{2} = 1.1664 → (1.08)^{2} - 1 = 0.1664.
    P = 832 / 0.1664 = ₹5,000.00.

  4. Final Answer:

    Principal = ₹5,000.00

  5. Quick Check:

    Amount = 5,000 × 1.1664 = 5,832 → CI = 5,832 - 5,000 = ₹832 ✅

Question

A sum of ₹10,000 is invested and yields a compound interest of ₹2,100 in 2 years. Find the annual rate of interest (compounded annually).

Solution

  1. Step 1: Identify values

    P = ₹10,000; CI = ₹2,100; T = 2 years.

  2. Step 2: Write equation for factor

    CI/P + 1 = (A/P) = (1 + R/100)^{T}.

  3. Step 3: Substitute and solve

    CI/P + 1 = 2,100/10,000 + 1 = 0.21 + 1 = 1.21.
    (1 + R/100)^{2} = 1.21 → 1 + R/100 = √1.21 = 1.1 → R/100 = 0.1 → R = 10%.

  4. Final Answer:

    Rate = 10% per annum

  5. Quick Check:

    Amount = 10,000 × 1.1² = 10,000 × 1.21 = 12,100 → CI = 12,100 - 10,000 = ₹2,100 ✅

Quick Variations

1. If compounding is half-yearly: replace R by R/n and T by nT (n = 2). Use P = CI / [(1 + R/(100·n))^{nT} - 1].

2. If compounding is quarterly or monthly: set n = 4 or 12 and proceed similarly.

3. When T is fractional, apply fractional exponent and use roots or logs to solve for R: R = [ (CI/P + 1)^{1/T} - 1 ] × 100.

Trick to Always Use

  • Step 1 → Convert the problem into CI = P[(1 + r)^{T} - 1] where r = R/100 (adjust r and T when n ≠ 1).
  • Step 2 → Isolate the unknown: for P divide CI by the bracket; for R take the T-th root then subtract 1.
  • Step 3 → Use square root for T = 2, cube root for T = 3, or logs for non-integer T; keep at least 4-6 decimals while computing, round money to 2 dp.

Summary

Summary

  • To find Principal (P): P = CI / [(1 + R/100)^{T} - 1].
  • To find Rate (R): R = [(CI/P + 1)^{1/T} - 1] × 100.
  • For half-yearly or quarterly compounding, replace R with R/n and T with nT.
  • For fractional years, use fractional powers or roots for accuracy.
  • Always verify by recalculating the amount to ensure your answer is correct.

Practice

(1/5)
1. The compound interest on a sum for 1 year at 8% per annum (compounded annually) is ₹540. Find the principal.
easy
A. ₹6,750
B. ₹7,000
C. ₹6,500
D. ₹6,250

Solution

  1. Step 1: Note given values

    CI = ₹540; R = 8% p.a.; T = 1 year.

  2. Step 2: Set up CI formula

    CI = P[(1 + R/100)^T - 1] → bracket = (1.08)^1 - 1 = 0.08.

  3. Step 3: Compute principal

    P = CI / 0.08 = 540 / 0.08 = ₹6,750.00.

  4. Final Answer:

    ₹6,750 → Option A.

  5. Quick Check:

    Amount = 6,750 × 1.08 = 7,290 → CI = 7,290 - 6,750 = ₹540 ✅
Hint: Divide CI by the growth bracket [(1 + r)^T - 1] to get P.
Common Mistakes: Forgetting to subtract 1 from the growth factor before dividing CI.
2. A sum of ₹12,000 yields a compound interest of ₹2,520 in 2 years. Find the annual rate of interest (compounded annually).
easy
A. 9%
B. 10%
C. 11%
D. 12%

Solution

  1. Step 1: Identify given values

    P = ₹12,000; CI = ₹2,520; T = 2 years.

  2. Step 2: Compute overall growth factor

    1 + CI/P = 1 + 2,520/12,000 = 1.21.

  3. Step 3: Take T-th root to find rate

    (1 + R/100)^2 = 1.21 → 1 + R/100 = √1.21 = 1.1 → R = 10%.

  4. Final Answer:

    10% → Option B.

  5. Quick Check:

    Amount = 12,000 × 1.1² = 12,000 × 1.21 = 14,520 → CI = 14,520 - 12,000 = ₹2,520 ✅
Hint: Compute (1 + CI/P) then take the T-th root to find 1 + R/100.
Common Mistakes: Using simple-interest logic instead of taking roots for compound-rate problems.
3. Compound interest on a sum for 1.5 years at 6% per annum (compounded half-yearly) is ₹1,020. Find the principal.
easy
A. ₹10,500
B. ₹10,800
C. ₹11,000
D. ₹10,200

Solution

  1. Step 1: Record given data

    CI = ₹1,020; R = 6% p.a.; T = 1.5 years; n = 2 (half-yearly).

  2. Step 2: Convert rate and periods

    Rate per half-year = 6/2 = 3% = 0.03; total periods = nT = 2 × 1.5 = 3.

  3. Step 3: Compute growth bracket and principal

    Growth bracket = (1 + 0.03)^3 - 1 = 1.092727 - 1 = 0.092727 (approx).
    P = CI / growth_bracket = 1,020 / 0.092727 ≈ ₹11,000.00.

  4. Final Answer:

    ₹11,000.00 → Option C.

  5. Quick Check:

    Amount = 11,000 × (1.03)^3 ≈ 11,000 × 1.092727 = ₹12,020.00 → CI = 12,020 - 11,000 = ₹1,020 ✅
Hint: When n ≠ 1 compute bracket = (1 + R/(100·n))^{nT} - 1, then divide CI by bracket to get P.
Common Mistakes: Using annual rate directly instead of rate per period for n > 1 or rounding too early.
4. A principal of ₹8,000 yields a compound interest of ₹1,103.95 in 1.5 years. Find the annual rate of interest (compounded annually).
medium
A. 8.5%
B. 9%
C. 9.5%
D. 10%

Solution

  1. Step 1: Note the known values

    P = ₹8,000; CI = ₹1,103.95; T = 1.5 years.

  2. Step 2: Find the growth factor

    1 + CI/P = 1 + 1,103.95/8,000 = 1.13799375.

  3. Step 3: Take T-th root to find rate

    (1 + R/100)^{1.5} = 1.13799375 → 1 + R/100 = (1.13799375)^{1/1.5} ≈ 1.09 → R ≈ 9%.

  4. Final Answer:

    9% → Option B.

  5. Quick Check:

    (1.09)^{1.5} ≈ 1.15399; 8,000 × 1.15399 ≈ 9,103.95 → CI ≈ 1,103.95 ✅
Hint: Take T-th root of (1 + CI/P) to find 1 + R/100, then subtract 1 and multiply by 100.
Common Mistakes: Using simple-interest formulas for compound-rate problems with fractional years.
5. A sum invested for 1 year at 12% per annum (compounded quarterly) yields a compound interest of ₹1,506.11. Find the principal.
medium
A. ₹11,900
B. ₹12,000
C. ₹12,100
D. ₹11,800

Solution

  1. Step 1: Record given values

    CI = ₹1,506.11; R = 12% p.a.; T = 1 year; n = 4 (quarterly).

  2. Step 2: Convert rate and compute periods

    Rate per quarter = 12/4 = 3% ; total periods = 4 × 1 = 4.

  3. Step 3: Compute growth bracket and principal

    Growth bracket = (1 + 0.03)^4 - 1 = 1.12550881 - 1 = 0.12550881. P = CI / 0.12550881 = 1,506.11 / 0.12550881 = ₹12,000.00.

  4. Final Answer:

    ₹12,000 → Option B.

  5. Quick Check:

    Amount = 12,000 × 1.12550881 = ₹13,506.11 → CI = 13,506.11 - 12,000 = ₹1,506.11 ✅
Hint: For quarterly compounding, use bracket = (1 + R/400)^{4T} - 1 before dividing CI.
Common Mistakes: Not converting R and T properly for quarterly compounding.

Mock Test

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