Compound Interest

⇒ CI for 2 years 6 months at the rate of 6, applying the net% effect for first 2 years
⇒ = 6 + 6 + (6 × 6)/100 = 12.36%
⇒ Rate of interest for 6 months = (6/12) × 6 = 3%
⇒ For next 6 months = 12.36 + 3 + (12.36 × 3) /100 = 15.36 + 0.37% = 15.73%
⇒ Here, we can see that in 2 years 6 months the given compound rate of interest is approximate 15.73%.
⇒ Now, 115.73% of 250000 = (115.73 × 250000)/100 = 289,325.
⇒ Hence, option D is correct.

Let the principal be 'x'.
Amount lent for 1 yr at 18% per annum compounding annually,
Amount = P (1 + r/100)ⁿ
Amount = x(1 + 18/100)
= 1.18x
When compounding been done half yearly,
r = 9%; n = 2 years
Amount = x( 1 + 9/100)²
= 1.1881x
We got from Q,
1.1881x - 1.18x = 324
0.0081x = 324
x = 324/0.0081
x = ₹ 40,000
Hence, the principal amount is ₹ 40,000.

Let each annual installment = ₹ X.
X[(100/(100 + R)) + (100/(100 + R))² + ...... + (100/(100 + R))^T] = Debt amount
X[(100/(100 + 5)) + (100/(100 + 5))² + (100/(100 + 5))³] = 50440
X[(100/105) + (100/105)² + (100/105)³] = 50440
X×(20/21)[1 + (20/21) + (20/21)²] = 50440
X×(20/21)[1 + (20/21) + (400/441)] = 50440
X×(20/21)[1261/441] = 50440
X = (50440×21×441)/(20×1261)
X = ₹ 18,522.

We know Let the principal of A and B be 'a' and 'b' respectively.
Amount = ₹ 39,030
r = 4%
Amount given to A for 7 yrs = Amount given to B for 9 yrs
Amount = P[1+ (r/100)]ⁿ
a[1 + (4/100)]⁷ = b[1 + (4/100)]⁹
a/b = [1 + (4/100)]⁹/[1 + (4/100)]⁷
a/b = [1 + (4/100)]²/1
a/b = [26/25]²/1
a : b = 676/625
a : b = 676 : 625
Part of A = 39030×(676/1301) = ₹ 20,280.

Given, Principal, P = ₹ 6000
No. of years, n = 3
Rate of interest, r = 5% per annum.
Amount repaid at first year = ₹ 2100
Amount repaid at second year = ₹ 2100
For 1st year, Compound Interest, C.I = P {[1 + (r/100)]ⁿ - 1}
⇒ C.I = 6000 × {[1 + (5/100)]¹ - 1}
= 6000 × {[1 + ($$\frac{1}{2}$$0)] - 1}
= 6000 × {[2$$\frac{1}{2}$$0] - 1}
= 6000 × {$$\frac{1}{2}$$0)}
= 300
At the end of 1st year, Amount payable = Principal + C.I
= 6000 + 300 = 6300
As the person paid ₹ 2100 at first year, the principal for 2nd year = 6300 - 2100 = 4200
For 2nd year, Compound Interest, C.I = 4200 × {[1 + (5/100)]¹ - 1}
= 4200 × {$$\frac{1}{2}$$0)}
= 210
At the end of 2nd year, Amount payable = Principal (for 2nd year) + C.I
= 4200 + 210
= 4410
As the person paid ₹ 2100 at second year, the principal for 3rd year = 4410 - 2100 = 2310
For 3rd year, Compound Interest, C.I = 2310 × {[1 + (5/100)]¹ - 1}
= 2310 × {$$\frac{1}{2}$$0)}
= 115.5
So, The amount that the person should pay at the end of 3rd to clear all his debts = Principal (for 3rd year) + C.I
= 2310 + 115.5
= ₹ 2425.5

Given, Compound Interest, rate = 5%
W.K.T: Compound Amount = P [1 + (R/100)]ⁿ
Here, Principal amount (P) = Present share of Sanjay/ Sekh
Given, Sanjay's share after 5 years = Sekh's share after 7 years
⇒ (Sanjay's present share) [1 + (5/100)]⁵ = (Sekh's present share) [1 + (5/100)]⁷
⇒ (Sanjay's present share) / (Sekh's present share) = [1 + (5/100)]⁷/ [1 + (5/100)]⁵
⇒ (Sanjay's present share) / (Sekh's present share) = [1 + (5/100)]^{(7 - 5)}
⇒ (Sanjay's present share) / (Sekh's present share) = [1 + (5/100)]²
⇒ (Sanjay's present share) / (Sekh's present share) = [1 + ($$\frac{1}{2}$$0)]²
⇒ (Sanjay's present share) / (Sekh's present share) = [2$$\frac{1}{2}$$0)]²
⇒ (Sanjay's present share) / (Sekh's present share) = [441/400)]
i.e, Sanjay's present share : Sekh's present share = 441 : 400
Since the total present amount is ₹ 3364,
Sanjay's present share = [441/(441+400)] × 3364
= [441/(841)] × 3364
= ₹ 1764
Sekh's present share = Total present amount - Sanjay's present share
= 3364 - 1764
= ₹ 1600

By using given condition,
P (1 + R /100)²⁰ = 4 P
(1 + R/100)²⁰ = 4
(1 + R/100)²⁰ = 2² --
Let P (1 + R/100)ⁿ = 16 P
(1 + R/100)ⁿ = 16 = 2⁴ ----
Using (1)
(1+R/ 100)ⁿ = (1+R /100)⁴⁰
n = 40
Thus required time 40 years.

The difference between simple interest and compound interest for 3years given by
C.I - S.I = P[(R/100)² × ((R/100) + 3)]
1 =×[(4/100)² × ((4/100) + 3)]
1 =x × ($$\frac{1}{2}$$5)² × (76/25)
1 = 76x/15625
x = 205.59.
Hence the required sum is ₹ 205.6

Amount = 4000
20% of 4000 = 800
At the end of 1st year = 4000 + 800 = 4800
Pays back = 1500
Amount = 4800 -1500 = 3300
20% of 3300 = 660
At the end of 2nd year = 3300 + 660 = 3960
Pays back = 1500
Amount = 3960 - 1500 = 2460
20% of 2460 = 492
Amount to be paid at the end of third year = 2460 + 492 = 2952

amount at the end of 2nd year
= ₹ 7500 (1 + 8/100) (1 + 10/100)
= ₹ 7500 × 1.08 × 1.10
= ₹ 8910
Thus C.I. for two years = amount - principal
= ₹ 8910 - ₹ 7500 = ₹ 1410