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31. An amount of money grows upto Rs 210 in 2 years and upto Rs 270 in 3 years on compound interest. Find the sum.
  A.  Rs 131
  B.  Rs 127
  C.  Rs 123
  D.  Rs 120
     
   
View Answer

Shortcut:
If an amount of money grows upto Rs Z1, in t years and upto Rs Z2 in (t + 1)years on compound interest, then the sum is given by Rs
[
Z1
(
Z1 / Z2
)
t
]

Here, Z1 = 210, t = 2, Z2 = 270
Using these values in shortcut, we get:
Sum =
[
210
(
210 / 270
)
2
]

=
[
210
(
7 / 9
)
2
]

=
70 x 7 x 7 / 3 x 9
= 127
Hence, the sum is Rs 127.

32. Find the ratio of CI to SI on a certain sum at 5% per annum for 2 years.
  A.  
37 / 42
  B.  
45 / 41
  C.  
41 / 40
  D.  
43 / 39
     
   
View Answer

Shortcut:
The ratio of C.I to SI on a certain sum at r% for 2 years is given by
r / 200
+ 1
Here, r = 5, t = 2
Using these values in shortcut, we get:
CI : SI =
5 / 200
+ 1 =
1 / 40
+ 1 =
1 + 40 / 40
=
41 / 40

Hence, the ratio is 41 : 40

33. Rs 8,000 becomes Rs 12,000 in 4 years at a certain rate of compound interest. What will be the sum after 12 years?
  A.  Rs 28100
  B.  Rs 27300
  C.  Rs 26200
  D.  Rs 27000
     
   
View Answer

Shortcut:
If a sum 'Y' becomes 'Z' in t1 years at compound rate of interest, then after t2 years the sum becomes
Zt2/t1 / Y(t2/t1 − 1)

Here, Y = 8000, Z = 12000, t1 = 4, t2 = 12
Using these values in shortcut, we get:
Sum =
(12000)12/4 / (8000)(12/4 − 1)

=
(12000)3 / (8000)(3 − 1)

=
12000 x 12000 x 12000 / (8000)2

=
12000 x 12000 x 12000 / 8000 x 8000

=
12 x 12 x 12000 / 8 x 8
= 27000
Hence, the sum is Rs 27,000.

34. Find the compound interest on Rs 10,000 in 2 years, the rate of interest being 4% for the first yer and 8% for the second year.
  A.  Rs 1232
  B.  Rs 1122
  C.  Rs 1252
  D.  Rs 1032
     
   
View Answer

Shortcut:
If the compound rate of interest for the first t1 years is r1%, for the next t2 years is r2%, for the next t3 years is r3% and so on.... for the last tn years is rn%, then compound interest on Rs Z for (t1 + t2 + t3 +.....tn) years is
[
Z
(
1 +
r1 / 100
)
t1
(
1 +
r2 / 100
)
t2 .........
(
1 +
rn / 100
)
tn
]
− Z
Here, Z = 10000, r1 = 4, r2 =8
Using these values in shortcut, we get:
CI =
[
10000
(
1 +
4 / 100
)
(
1 +
8 / 100
)
]
− 10000
=
[
10000
(
100 + 4 / 100
)
(
100 + 8 / 100
)
]
− 10000
=
[
10000
(
104 / 100
)
(
108 / 100
)
]
− 10000
= 11232 − 10000 = 1232
Hence, the compound interest is Rs 1232.

35. Find the compound interest on Rs 8,000 for 3 years if the rate of interest is 4% for the first year, 5% for the second year and 6% for the third year.
  A.  Rs 1220.25
  B.  Rs 1160.56
  C.  Rs 1260.16
  D.  Rs 1280.36
     
   
View Answer

Shortcut:
If the compound rate of interest for the first t1 years is r1%, for the next t2 years is r2%, for the next t3 years is r3% and so on.... for the last tn years is rn%, then compound interest on Rs Z for (t1 + t2 + t3 +.....tn) years is
[
Z
(
1 +
r1 / 100
)
t1
(
1 +
r2 / 100
)
t2 .........
(
1 +
rn / 100
)
tn
]
− Z
Here, Z = 8000, r1 = 4, r2 = 5, r3 = 6
Using these values in shortcut, we get:
CI =
[
8000
(
1 +
4 / 100
)
(
1 +
5 / 100
)
(
1 +
6 / 100
)
]
− 8000
=
[
8000
(
100 + 4 / 100
)
(
100 + 5 / 100
)
(
100 + 6 / 100
)
]
− 8000
=
[
8000
(
104 / 100
)
(
105 / 100
)
(
106 / 100
)
]
− 8000
=
8000 x 104 x 105 x 106 / 100 x 100 x 100
− 8000
=
4 x 104 x 21 x 106 / 100
− 8000
= 9260.16 − 8000 = 1260.16
Hence, the compound interest is Rs 1260.16

36. What sum of money at compound interest will amount to Rs 11575.2 in 3 years, if the rate of interest is 4% for the first year, 5% for the second year and 6% for the third year?
  A.  Rs 9,500
  B.  Rs 10,000
  C.  Rs 11,100
  D.  Rs 10,200
     
   
View Answer

Shortcut:
Certain sum of money at compound interest will amount to Rs Z in (t1 + t2 + t3+ .... + tn) years. If the rate of interest for the first t1 years is r1%, for the next t2 years is r2%, for the next t3 years it is r3% and so on ....... the last tn years is rn%, then the sum is given by
[
Z
(
100 / 100 + r1
)
(
100 / 100 + r2
)
(
100 / 100 + r3
)
]

Here, Z = 11575.2, r1 = 4, r2 = 5, r3 = 6
Using these values in shortcut, we get:
Sum =
[
11575.2
(
100 / 100 + 4
)
(
100 / 100 + 5
)
(
100 / 100 + 6
)
]

=
[
11575.2
(
100 / 104
)
(
100 / 105
)
(
100 / 106
)
]

=
11575.2 x 100 x 100 x 100 / 104 x 105 x 106

Hence, the sum is Rs 10,000

37. Ajay borrows Rs 2000 at 10% compound rate of interest. At the end of each year he pays back Rs 1000. How much amount should he pay at the end of the third year to clear all his dues?
  A.  Rs 341
  B.  Rs 358
  C.  Rs 348
  D.  Rs 352
     
   
View Answer

Shortcut:
If a person borrows Rs Z at r% compound interest and pays back Rs M at the end of each year, then at the end of the nth year he should pay
Rs Z
[
1 +
r / 100
]
n − M
[
(
1 +
r / 100
)
n − 1 +
(
1 +
r / 100
)
n − 2 + ...... +
(
1 +
r / 100
)
1
]

Here, Z = 2000, r = 10, M = 1000, n = 3
Using these values in shortcut, we get:
Amount = Rs 2000
[
1 +
10 / 100
]
3 − 1000
[
(
1 +
10 / 100
)
3 − 1 +
(
1 +
10 / 100
)
3 − 2
]

= Rs 2000
[
1 +
1 / 10
]
3 − 1000
[
(
1 +
1 / 10
)
2 +
(
1 +
1 / 10
)
1
]

= Rs 2000
[
10 + 1 / 10
]
3 − 1000
[
(
10 + 1 / 10
)
2 +
(
10 + 1 / 10
)
t1
]

= Rs 2000
[
11 / 10
]
3 − 1000
[(
11 / 10
)
2 +
(
11 / 10
)
1
]

= Rs 2000
[
11 x 11 x 11 / 10 x 10 x 10
]
− 1000
[(
11 x 11 / 10 x 10
)
+
(
11 / 10
)
]

= Rs 2000
[
1331 / 1000
]
− 1000
[(
121 / 100
)
+
(
11 / 10
)]

= 2 x 1331 − 1000
(
121 / 100
)
− 1000
(
11 / 10
)]

= 2662 − 10 x 121 − 100 x 11
= 2662 − 1210 − 1100 = 352
Hence, the he has to pay Rs 352.

38. Divide Rs 2602 between Ajay and Bihu, so that Ajay's share at the end of 7 years may equal Bihu's share at the end of 9 years, compound interest being at 4%.
  A.  Ajay share = Rs 1,350; Bihu share = Rs 1,252
  B.  Ajay share = Rs 1,150; Bihu share = Rs 1,452
  C.  Ajay share = Rs 1,250; Bihu share = Rs 1,352
  D.  Ajay share = Rs 1,260; Bihu share = Rs 1,342
     
   
View Answer

Shortcut:
If a sum of money say Rs Z is divided among n parts in such a manner that when placed at compound interest, amount obtained in each case remains equal while the rate of interest on each part is r1, r2,r3,.....rn respectively and time period for each part is t1,t2,t3, .....tn respectively, then the divided parts of the sum will in the ratio of
1 / (1 + r1/100)t1
:
1 / (1 + r2/100)t2
:
1 / (1 + r3/100)t3
: ........:
1 / (1 + rn/100)tn

Here, Z = 2602, t1 = 7, t2 = 9, r1 = r2 = 4
Using these values in shortcut, we get:
Ajay : Bihu =
[
1 / (1 + 4/100)7
]
:
[
1 / (1 + 4/100)9
]

=
[
1 / (104/100)7)
]
:
[
1 / (104/100)9)
]

=
[
100 / 104
]
7 :
[
100 / 104
]
9
= 1:
[
100 / 104
]
2
= 1:
[
100 x 100 / 104 x 104
]

= 1 :
[
25 x 25 / 26 x 26
]

= 1 :
[
625 / 676
]

Divide 2602 in the ratio of 676 : 625
Ajay share =
625 / 676 + 625
x 2602
=
625 / 1301
x 2602 = 625 x 2 = 1250
Bihu share = 2602 − 1250 = Rs 1352
Hence, Ajay share is RS 1250 and Bihu share is Rs 1352.

39. The difference between the SI and CI compounded every six months at the rate of 10% per annum at the end of 2 years is Rs 124.05. What is the sum?
  A.  Rs 8000
  B.  Rs 7600
  C.  Rs 8500
  D.  Rs 7000
     
   
View Answer

Let the sum is Rs y.
r = 10%
t = 2 years
Then according to the question
124.05 = C.I − S.I
124.05 =
[
y
(
1 +
10/2 / 100
)
2 x 2 − y
]
−
[
y x 10 x 2 / 100
]

124.05 =
[
y
(
1 +
5 / 100
)
4 − y
]
−
[
y / 5
]

124.05 =
[
y
(
1 +
1 / 20
)
4 − y
]
−
[
y / 5
]

124.05 =
[
y
(
20 + 1 / 20
)
4 − y
]
−
[
y / 5
]

124.05 =
[
y
(
21 / 20
)
4 − y
]
−
[
y / 5
]

124.05 =
[
y
(
21 x 21 x 21 x 21 / 20 x 20 x 20 x 20
)
− y
]
−
[
y / 5
]

124.05 =
[
y
(
194481 / 160000
)
− y
]
−
[
y / 5
]

124.05 =
[
y
(
194481 − 160000 / 160000
)
−
[
y / 5
]

124.05 =
[
34481y − 32000y / 160000
]

124.05 =
[
2481y / 160000
]

124.05 x 160000 = 2481y
y =
124.05 x 160000 / 2481
= 8000
Hence, the sum is Rs 8000.

40. A person invested a certain amoutnat SI at the rate of 6% per annum earning Rs 900 as an interest at the end of 3 years. ha dthe interest been compounded every year, how much more interest would he have earned on the same amount with the same interest rate after 3 years?
  A.  Rs 53.18
  B.  Rs 55.08
  C.  Rs 58.22
  D.  Rs 60.04
     
   
View Answer

Certain sum for the person =
900 x 100 / 18
= Rs 5000
∴ Interest on Rs 5000 by Compound interest is:
=
[
5000
(
1 +
6 / 100
)
3 − 5000
]

=
[
5000
(
100 + 6 / 100
)
3 − 5000
]

=
[
5000
(
106 / 100
)
3 − 5000
]

=
[(
106 x 106 x 106 / 2 x 100
)
− 5000
]

=
[(
1191016 / 200
)
− 5000
]

=
[(
1191016 − 1000000 / 200
)]

=
[(
91016 / 200
)]
= 955.08
Hence, more interest is 955.08 − 900 = Rs 55.08
Hence, the sum is Rs 8000.

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