Unit 2: Compound Interest
Definitions, formulae and complete worked exercises from the supplied 38-page handwritten notes.
Original Scanned PDF – View Notes
Source Note
Compound Interest – Basic Concepts
1. Compound Interest
Compound interest is the interest which is calculated by adding the interest to the principal and calculating the interest on the new principal at the end of a certain time.
2. Compound Amount (C.A.)
Compound amount is the sum of the principal and compound interest.
3. Yearly Compound Interest
Yearly compound interest is the interest calculated by adding the interest to the principal and calculating interest on the new principal at the end of every year.
4. Half-Yearly Compound Interest
Half-yearly compound interest is calculated by adding the interest to the principal and calculating interest on the new principal at the end of every six months.
5. Quarterly Compound Interest
Quarterly compound interest is calculated by adding the interest to the principal and calculating interest on the new principal at the end of every three months.
Important Formulae
The first three pages also show mixed-period formulae for a whole number of years plus extra months, using the yearly/half-yearly/quarterly compound factor first and then the simple-interest factor for the remaining months.
Exercise – Compound Interest Without Using the Direct Formula
(a) P = Rs 10,000, T = 2 years, R = 6% p.a.
First year interest:
S.I. = (10,000 × 1 × 6) / 100 = Rs 600
Amount after first year = 10,000 + 600 = Rs 10,600
Second year interest = (10,600 × 1 × 6) / 100 = Rs 636
Compound Interest = 600 + 636 = Rs 1,236
Compound Amount = 10,000 + 1,236 = Rs 11,236
(b) P = Rs 64,000, T = 3 years, R = 6% p.a.
First year interest = (64,000 × 1 × 6)/100 = Rs 3,840
Principal for second year = 64,000 + 3,840 = Rs 67,840
Second year interest = (67,840 × 1 × 6)/100 = Rs 4,070.40
Principal for third year = 67,840 + 4,070.40 = Rs 71,910.40
Third year interest = (71,910.40 × 1 × 6)/100 = Rs 4,314.624
Compound Interest = 3,840 + 4,070.40 + 4,314.624 = Rs 12,225.024
Compound Amount = 64,000 + 12,225.024 = Rs 76,225.024
(c) P = Rs 20,000 for 2 years; first-year rate = 10%, second-year rate = 12%
First year interest = (20,000 × 1 × 10)/100 = Rs 2,000
Principal for second year = 20,000 + 2,000 = Rs 22,000
Second year interest = (22,000 × 1 × 12)/100 = Rs 2,640
Compound Interest = 2,000 + 2,640 = Rs 4,640
Compound Amount = 20,000 + 4,640 = Rs 24,640
Finding the Rate of Compound Interest
(a) Compound interest on Rs 100 for 2 years is Rs 12.
P = Rs 100, C.I. = Rs 12, T = 2 years.
The handwritten source substitutes these values in the compound-interest formula and records:
R = 12%
(b) Compound interest on Rs 200 for 2 years is Rs 42. Find the rate.
Compound Amount = 200 + 42 = Rs 242
242 = 200(1 + R/100)2
242/200 = (1 + R/100)2
1.21 = (1 + R/100)2
1.1 = 1 + R/100
R/100 = 0.1
∴ R = 10%
Direct Compound Interest Problems
(a) A farmer borrowed Rs 20,000 for 3 years at 15% p.a.
C.A. = 20,000(1 + 15/100)3
= 20,000(1.15)3
= Rs 30,417.50
C.I. = 30,417.50 − 20,000
∴ C.I. = Rs 10,417.50
(b) A teacher deposited Rs 50,000 for 3 years at 10% yearly compound interest.
C.A. = 50,000(1.10)3
= Rs 66,550
C.I. = 66,550 − 50,000
∴ C.I. = Rs 16,550
Half-Yearly and Quarterly Compound Interest
(a) Rs 50,000 at 8% p.a., compounded half-yearly for 2 years
C.I. = 50,000[(1 + 8/200)4 − 1]
= 50,000[(1.04)4 − 1]
Source result: Rs 8,492.928
Compound Amount = 50,000 + 8,492.928 = Rs 58,492.928
(b) Quarterly compound-interest problem at 12% p.a. for 2 years
Pages 12–13 contain a bank-deposit problem solved using:
Difference Between Simple Interest and Compound Interest
(a) Rs 80,000 at 8% p.a. for 2 years
Simple Interest:
S.I. = (80,000 × 2 × 8)/100 = Rs 12,800
Compound Interest:
C.I. = 80,000[(1.08)2 − 1]
= Rs 13,312
Difference = 13,312 − 12,800 = Rs 512
(b) Rs 7,500 at 12% p.a. for 3 years
S.I. = (7,500 × 3 × 12)/100 = Rs 2,700
C.I. = 7,500[(1.12)3 − 1]
Source result = Rs 3,036.96
Difference = 3,036.96 − 2,700 = Rs 336.96
Comparing Yearly, Half-Yearly and Quarterly Compounding
(a) Difference between yearly and half-yearly compound interest
Pages 15–16 compare yearly and half-yearly compound interest on a deposit at 6% p.a. for 2 years.
(b) Difference between semi-annual and quarterly compound interest
Pages 16–17 use Rs 18,000 for 1 year at 12% p.a. and compare:
(c) Two-bank / two-account comparison
Pages 17–19 compare interest on Rs 60,000 for 2 years under different compounding/rate conditions.
For one account, the source uses half-yearly compounding at 10%:
C.I. = 60,000[(1 + 10/200)4 − 1]
= Rs 12,930.375
For the other account, the source uses yearly compounding at 12%:
C.I. = 60,000[(1.12)2 − 1]
= Rs 15,264
Difference = 15,264 − 12,930.375 = Rs 2,333.625
Finding the Principal / Sum from Interest Conditions
(a) Yearly C.I. exceeds S.I. by Rs 180 for 2 years at 15% p.a.
Let the principal = P.
S.I. = P × 2 × 15 / 100 = 0.30P
C.I. = P[(1.15)2 − 1] = 0.3225P
0.3225P − 0.30P = 180
0.0225P = 180
∴ P = Rs 8,000
(b) Half-yearly C.I. exceeds yearly C.I. by Rs 40 for 1 year at 10% p.a.
Half-yearly C.I. = P[(1.05)2 − 1] = 0.1025P
Yearly C.I. = 0.10P
0.1025P − 0.10P = 40
0.0025P = 40
∴ P = Rs 16,000
(c) Another amount-comparison problem
Pages 21–22 contain an additional compound-amount comparison at 10% for 2 years. The handwritten question statement is incomplete in the scan, so only the source method is retained:
Finding Time and Rate
(a) Find the time when C.I. = Rs 2,11,000 on P = Rs 11,00,000 at 10% p.a.
C.I. = P[(1 + R/100)T − 1]
2,11,000 = 11,00,000[(1.10)T − 1]
1 + 2,11,000/11,00,000 = (1.10)T
1.21 = (1.10)T
(1.10)2 = (1.10)T
∴ T = 2 years
(b) Amount = Rs 847, Principal = Rs 700, Time = 2 years. Find the rate.
847 = 700(1 + R/100)2
847/700 = (1 + R/100)2
1.21 = (1 + R/100)2
1.1 = 1 + R/100
∴ R = 10%
(c) Three-year problem giving the ratio 1.331
The source reduces the compound-amount equation to:
1.331 = (1 + R/100)3
1.1 = 1 + R/100
∴ R = 10%
Principal and Rate from Compound Amounts at Different Years
(a) Amount after 2 years = Rs 6,050; after 3 years = Rs 6,655
P(1 + R/100)2 = 6,050 … (i)
P(1 + R/100)3 = 6,655 … (ii)
Dividing (ii) by (i):
1 + R/100 = 6,655 / 6,050 = 1.1
∴ R = 10%
Putting R = 10% in (i):
P(1.1)2 = 6,050
1.21P = 6,050
∴ P = Rs 5,000
(b) Amount after 2 years = Rs 10,580; after 3 years = Rs 12,167
P(1 + R/100)2 = 10,580 … (i)
P(1 + R/100)3 = 12,167 … (ii)
Dividing (ii) by (i):
1 + R/100 = 12,167 / 10,580 = 1.15
∴ R = 15%
Putting R = 15% in (i):
P(1.15)2 = 10,580
1.3225P = 10,580
∴ P = Rs 8,000
Compound Interest with Tax Deduction
(a) Principal Rs 5,00,000, rate 10%, first-year interest with 5% tax deduction
The source first calculates half-yearly compound interest for one year:
C.I. = 5,00,000[(1 + 10/200)2 − 1]
= Rs 51,250
Interest after deducting 5% tax:
51,250 − (5/100 × 51,250)
= Rs 48,687.50
Principal for the next year = 5,00,000 + 48,687.50 = Rs 5,48,687.50
Pages 30–31 continue with the second-year compound-interest calculation and compare the two yearly interests. The source finally records a percentage change of approximately 11.19%.
(b) Principal Rs 80,000, rate 15%, tax = 5%
First-year C.I. = 80,000 × 15/100 = Rs 12,000
Tax = 5% of 12,000 = Rs 600
Interest after tax = 12,000 − 600 = Rs 11,400
Principal for second year = 80,000 + 11,400 = Rs 91,400
Second-year C.I. = 91,400 × 15/100 = Rs 13,710
Difference = 13,710 − 11,400 = Rs 2,310
Percentage increase = (2,310 / 11,400) × 100
Source result ≈ 20.26%
Division of Principal so that Compound Amounts Become Equal
(a) Total sum = Rs 21,000, rate = 10%; one part for 3 years and the other for 2 years
Let the first part = x.
First amount:
C.A.1 = x(1.10)3 = 1.331x
Second part = 21,000 − x
C.A.2 = (21,000 − x)(1.10)2
= (21,000 − x)(1.21)
= 25,410 − 1.21x
Since the compound amounts are equal:
1.331x = 25,410 − 1.21x
2.541x = 25,410
∴ x = Rs 10,000
Second part = 21,000 − 10,000 = Rs 11,000
(b) Total sum = Rs 41,000, rate = 5%; first part for 2 years and second part for 3 years
Let the first part = x.
C.A.1 = x(1.05)2 = 1.1025x
Second part = 41,000 − x
C.A.2 = (41,000 − x)(1.05)3
= (41,000 − x)(1.157625)
= 47,462.625 − 1.157625x
Equating the amounts, the source obtains:
First part = Rs 21,000
Second part = Rs 20,000
Finding Principal and Rate from Compound Interest for 1 Year and 2 Years
(a) C.I. for 1 year = Rs 950 and C.I. for 2 years = Rs 1,995
For 1 year:
P[(1 + R/100) − 1] = 950
PR/100 = 950 … (i)
For 2 years:
P[(1 + R/100)2 − 1] = 1,995
PR/100(2 + R/100) = 1,995 … (ii)
Dividing (ii) by (i):
2 + R/100 = 1,995 / 950 = 2.1
R/100 = 0.1
∴ R = 10%
Using (i):
P × 10/100 = 950
∴ P = Rs 9,500
(b) C.I. for 1 year = Rs 1,800 and C.I. for 2 years = Rs 3,816
For 1 year:
PR/100 = 1,800 … (i)
For 2 years:
PR/100(2 + R/100) = 3,816 … (ii)
Dividing (ii) by (i):
2 + R/100 = 3,816 / 1,800 = 2.12
R/100 = 0.12
∴ R = 12%
Putting R = 12% in (i):
P × 12/100 = 1,800
∴ P = Rs 15,000
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