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Principal or Rate Finding

Introduction

பல aptitude problems இல் Compound Interest (அல்லது final amount) கொடுக்கப்பட்டு, principal அல்லது rate ஐ கண்டுபிடிக்க கேட்கப்படும். முக்கியமானது, compound interest formula ஐ rearrange செய்து, தெரியாத மதிப்பை (P அல்லது R) isolate செய்வதே. இந்த pattern, இப்படியான reverse problems ஐ நம்பகமாக தீர்க்க தெளிவான, மீண்டும் பயன்படுத்தக்கூடிய steps ஐ கற்றுத்தரும்.

Pattern: Principal or Rate Finding

Pattern

Key concept: A = P(1 + R/100)^{T} அல்லது CI = P[(1 + R/100)^{T} - 1] ஆகியவற்றை rearrange செய்து P அல்லது R ஐ கண்டறிதல்.

Useful forms:
A = P × (1 + R/100)^{T} → T years கழித்த amount.
CI = A - P = P × [(1 + R/100)^{T} - 1]
P ஐ காண (CI கொடுக்கப்பட்டால்):
P = CI / [(1 + R/100)^{T} - 1]
R ஐ காண (CI மற்றும் P கொடுக்கப்பட்டால்):
(1 + R/100)^{T} = CI/P + 1 → T-th root எடுக்கவும்:
R = [ (CI/P + 1)^{1/T} - 1 ] × 100

Step-by-Step Example

Question

2 years க்கு, வருடத்திற்கு 8% வட்டியில் (annual compounding), ஒரு தொகையின் compound interest ₹832 ஆக உள்ளது. Principal ஐ காண்க.

Solution

  1. Step 1: Values ஐ கண்டறிதல்

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

  2. Step 2: CI formula பயன்படுத்தி P ஐ isolate செய்தல்

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

  3. Step 3: Values substitute செய்தல்

    (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 முதலீடு செய்யப்பட்டுள்ளது; 2 years இல் கிடைக்கும் compound interest ₹2,100. Annual compounding ஆக இருக்கும் போது annual rate ஐ காண்க.

Solution

  1. Step 1: Values ஐ கண்டறிதல்

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

  2. Step 2: Growth factor க்கான equation எழுதுதல்

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

  3. Step 3: Substitute செய்து solve செய்தல்

    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. Half-yearly compounding இருந்தால்: R ஐ R/n ஆகவும், T ஐ nT ஆகவும் மாற்றவும் (n = 2). P = CI / [(1 + R/(100·n))^{nT} - 1] பயன்படுத்தவும்.

2. Quarterly அல்லது monthly compounding இருந்தால்: n = 4 அல்லது 12 வைத்து அதே முறையில் தொடரவும்.

3. T fractional ஆக இருந்தால், fractional exponent பயன்படுத்தி roots அல்லது logs மூலம் R ஐ கண்டறியவும்: R = [ (CI/P + 1)^{1/T} - 1 ] × 100.

Trick to Always Use

  • Step 1 → Problem ஐ CI = P[(1 + r)^{T} - 1] என்ற வடிவத்திற்கு மாற்றுங்கள்; இங்கு r = R/100 (n ≠ 1 என்றால் r மற்றும் T ஐ adjust செய்யவும்).
  • Step 2 → Unknown ஐ isolate செய்யுங்கள்: P க்கு bracket ஆல் divide செய்யவும்; R க்கு T-th root எடுத்து 1 ஐ கழிக்கவும்.
  • Step 3 → T = 2 என்றால் square root, T = 3 என்றால் cube root, non-integer T க்கு logs பயன்படுத்தவும்; calculation போது 4-6 decimals வைத்துக் கொண்டு, money values ஐ 2 decimal places வரை round செய்யவும்.

Summary

Summary

  • Principal (P) ஐ காண: P = CI / [(1 + R/100)^{T} - 1].
  • Rate (R) ஐ காண: R = [(CI/P + 1)^{1/T} - 1] × 100.
  • Half-yearly அல்லது quarterly compounding க்கு, R ஐ R/n ஆகவும் T ஐ nT ஆகவும் மாற்றவும்.
  • Fractional years இருந்தால், accuracy காக fractional powers அல்லது roots பயன்படுத்தவும்.
  • எப்போதும் amount ஐ மீண்டும் கணக்கிட்டு verify செய்து உங்கள் answer சரியா என உறுதி செய்யுங்கள்.

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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