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Principal or Rate Finding

Introduction

कई aptitude problems में compound interest (या final amount) दिया होता है और आपको principal या rate निकालना होता है। मुख्य तरीका है compound-interest के formula को rearrange करके unknown को isolate करना (P या R)। यह pattern clear, repeatable steps सिखाता है ताकि आप इन reverse problems को भरोसेमंद तरीके से हल कर सकें।

Pattern: Principal or Rate Finding

Pattern

Key concept: A = P(1 + R/100)^{T} या CI = P[(1 + R/100)^{T} - 1] को rearrange करके P या R निकालेँ।

Useful forms:
A = P × (1 + R/100)^{T} → T वर्षों के बाद amount।
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 → T-th root लें:
R = [ (CI/P + 1)^{1/T} - 1 ] × 100

Step-by-Step Example

Question

किसी राशि पर 2 साल के लिए 8% प्रति वर्ष की दर पर (annual compounding) compound interest ₹832 आया है। Principal निकालें।

Solution

  1. Step 1: मान पहचानें

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

  2. Step 2: CI formula से P isolate करें

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

  3. Step 3: संख्याएँ substitute करें

    (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

₹10,000 निवेश करने पर 2 साल में compound interest ₹2,100 मिलता है। वार्षिक दर (annual compounding) क्या है?

Solution

  1. Step 1: मान पहचानें

    P = ₹10,000; CI = ₹2,100; T = 2 साल।

  2. Step 2: factor के लिए equation लिखें

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

  3. Step 3: substitute और solve करें

    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. अगर compounding half-yearly है: R को R/n और T को nT से बदलें (n = 2)। प्रयोग करें P = CI / [(1 + R/(100·n))^{nT} - 1]।

2. अगर compounding quarterly या monthly है: n = 4 या 12 रखें और उसी तरह आगे बढ़ें।

3. जब T fractional हो, तो fractional exponent लागू करें और R निकालने के लिए roots या logs का प्रयोग करें: R = [ (CI/P + 1)^{1/T} - 1 ] × 100।

Trick to Always Use

  • Step 1 → समस्या को CI = P[(1 + r)^{T} - 1] में बदलें जहाँ r = R/100 (जब n ≠ 1 तो r और T adjust करें)।
  • Step 2 → unknown को isolate करें: P के लिए bracket से divide करें; R के लिए T-th root लें और 1 घटाएँ।
  • Step 3 → T = 2 के लिए square root, T = 3 के लिए cube root, non-integer T के लिए logs इस्तेमाल करें; मान निकालते समय 4-6 दशमलव रखें, और पैसे के उत्तर को 2 dp पर round करें।

Summary

Summary

  • अगर Principal (P) निकालना है: P = CI / [(1 + R/100)^{T} - 1]।
  • अगर Rate (R) निकालना है: R = [(CI/P + 1)^{1/T} - 1] × 100।
  • Half-yearly या quarterly compounding के लिए R को R/n और T को nT से बदलें।
  • Fractional years के लिए fractional powers या roots प्रयोग करें ताकि सही उत्तर मिले।
  • हमेशा अपने उत्तर की जांच के लिए amount फिर से निकालकर verify करें।

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.

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