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31. The true discount on a certain sum of money due 6 years hence is Rs 200 and the simple interest on the same sum for the same time and at the same rate is Rs 300.Find the rate per cent per annum.
  A.   8
3 / 7
%
  B.   8
1 / 3
%
  C.   7
5 / 8
%
  D.   6
8 / 5
%
     
   
View Answer

Shortcut:
To find the rate (R) when simple interest (SI), time (T) and True Discount (TD) are given. R =
100 / T
x [
SI / TD
− 1]
Here, SI = 300, TD = 200, T = 6
Using these values in the shortcut, we get:
R =
100 / 6
x [
300 / 200
− 1]
=
25 / 3
= 8
1 / 3

Hence, the rate is 8
1 / 3
%.

32. If Rs 60 be the true discount on Rs 240 for a certain time, what is the discount on the same sum for double the time, the rate being the same in both the cases.
  A.  Rs 90
  B.  Rs 100
  C.  Rs 96
  D.  Rs 95
     
   
View Answer

Shortcut:
To find the ratio between two True Discount (TD), when two Present Worth (PW), two Rates (R) and two Time (T) are given.
(TD)1 / (TD)2
=
PW1 x R1 x T1 / PW2 x R2 x T2

PW1 = A − TD1
PW2 = A − TD2
Here, (TD)1 = 60, (TD)2 = ?, T = 6, A = 240, R1 = R2 = R, T1 = T, T2 = 2T
Using these values in the shortcut, we get:
60 / (TD)2
=
(240 − 60) x R x T / (240 − TD2) x R x 2T

=
60 / (TD)2
=
180 / (240 − TD2) x 2

=
(240 − TD2) / (TD)2
=
3 / 2

2(240 − TD2) = 3(TD)2
480 − 2(TD)2 = 3(TD)2
5(TD)2 = 480
(TD)2 =
480 / 5
= 96.
Hence, the true discount on the same sum for double the time is Rs 96.

33. The present worth of a bill due 6 months hence is Rs 1500 and if the bill were due at the end of 2 years, its present worth would be Rs 1200. Find the rate percent per annum.
  A.  16
11 / 5
%
  B.   18
5 / 9
%
  C.   17
3 / 7
%
  D.   18
2 / 11
%
     
   
View Answer

Shortcut:
To find the ratio between two Present Worth (PW), When Rates (R) and two Time (T) are given.
(PW)1 / (PW)2
=
100 + R x T2 / 100 + R x T1

PW1 = Present Worth of an amount due in T1 year
PW2 = Present Worth of the same amount due in T2 year
Here, (PW)1 = 1500, (PW)2 = 1200, R = ?, T1 = 1/2, T2 = 2
Using these values in the shortcut, we get:
1500 / 1200
=
(100 + R) x 2 / (100 + R) x 1/2

=
5 / 4
=
100 + 2R / (200 + R)/2

5(200 + R) = 4(100 + 2R) x 2
1000 + 5R = 8(100 + 2R)
1000 + 5R = 800 + 16R
1000 − 800 = 16R − 5R
200 = 11R
or, 11R = 200
r =
200 / 11
= 18
2 / 11

Hence, the rate is 18
2 / 11
%.

34. The present worth of a bill due 7 months hence is Rs 1200, and if the bill were due at the end of 2 ½ years, its present worth would be Rs 1016.Find the rate per cent and the sum of the bill.
  A.   10%, Rs 1270
  B.   12%, Rs 1200
  C.   11%, Rs 1180
  D.   14%, Rs 1290
     
   
View Answer

Shortcut:
To find the ratio between two Present Worth (PW), When Rates (R) and two Time (T) are given.
(PW)1 / (PW)2
=
100 + R x T2 / 100 + R x T1

PW1 = Present Worth of an amount due in T1 year
PW2 = Present Worth of the same amount due in T2 year
Here, (PW)1 = 1200, (PW)2 = 1016, R = ?, T1 = 7/12, T2 = 5/2
Using these values in the shortcut, we get:
1200 / 1016
=
100 + R x 5/2 / 100 + R x 7/12

=
1200 / 1016
=
100 + (5R/2) / 100 + (7R/12)

=
1200 / 1016
=
(200 + 5R)6 / 1200 + 7R

1200(1200 + 7R) = 1016(200 + 5R) x 6
200(1200 + 7R) = 1016(200 + 5R)
100(1200 + 7R) = 508(200 + 5R)
50(1200 + 7R) = 254(200 + 5R)
25(1200 + 7R) = 127(200 + 5R)
30000 + 175R = 25400 + 635R 635R − 175R = 25400 − 14400
460R = 4600
46R = 460
R =
460 / 46
= 10
Hence, the rate is 10%.
∴ Sum due = 1200 + (SI on Rs 1200 for 7 months at 10%)
= 1200 + (1200 x 7/12 x 10/100)
= 1200 + 100 x 7 x 1/10
= 1200 + 70 = 1270
Hence, the sum of the bil is Rs 1270.

35. If the simple interest on Rs 600 for 5 years be equal to the true discount on Rs 720 for the same time and at the same rate, find the rate per cent per annum.
  A.   7%
  B.   5%
  C.   4%
  D.   3%
     
   
View Answer

Shortcut:
If the simple interest on Rs Z for T years be equal to the true discount on Rs M for the same time and at the same rate, then the rate per cent per annum is given by
100 / Z x T
[M − Z] per cent.
Here, Z = 600, T = 5, M = 720
Using these values in the shortcut, we get:
=
100 / 600 x 5
[720 − 600] =
1 / 30
x 120 = 4 Hence, the rate is 4%.

36. If the interest on Rs 50 at 4
1 / 2
be equal to the discount on Rs 59 for the same time and at the same rate when is the latter sum due?
  A.  7 years
  B.  4 years
  C.  5 years
  D.  2 years
     
   
View Answer

Shortcut:
If the simple interest on Rs Z for T years be equal to the true discount on Rs M for the same time and at the same rate, then the rate per cent per annum is given by
100 / Z x T
[M − Z] per cent.
Here, Z = 50, T = ?, M = 59, R = 9/2
Using these values in the shortcut, we get:
R =
100 / 50 x T
[59 − 50]
9 / 2
=
2 / T
x 9
1 / 2
=
2 / T

T = 4
Hence, the time is 4 years.

37. If the discount on Rs 3050 be equal to the simple interest on Rs 3000 for the same time, find the time, the rate of interest being 5 % per annum.
  A.  7 months
  B.  6 months
  C.  5 months
  D.  4 months
     
   
View Answer

Shortcut:
If the simple interest on Rs Z for T years be equal to the true discount on Rs M for the same time and at the same rate, then the rate per cent per annum is given by
100 / Z x T
[M − Z] per cent.
Here, Z = 3050, T = ?, M = 3000, R = 5
Using these values in the shortcut, we get:
R =
100 / 3000 x T
[3050 − 3000]
5 =
1 / 30 x T
x 50
5 =
50 / 30 x T

1 =
1 / 3 x T

3 x T = 1
T =
1 / 3
years Hence, the time is 4 months.

38. Find the present worth and the discount of Rs 1722.25 due in 2 years reckoning compound interest at 3
3 / 4
%.
  A.   Rs 1610,130.45
  B.   Rs 1580,120.25
  C.   Rs 1600,122.25
  D.   Rs 1620,125.25
     
   
View Answer

Shortcut:
To find the present worth of A rupees due 'n' years hence at 'r' per cent compounded interest payable annually, (i) A = Present Worth x (1 +
r / 100
)n
(ii) Present Worth =
A / (1 + r/100)n
)
(iii) True Discount = Amount − Present Worth
Here, A = 1722.25, n = 2, R = 15/4
Using these values in the shortcut, we get:
Present Worth =
1722.25 / (1 + (15/4)/ 100)2
)
=
1722.25 / (1 + 15/ 400)2
)
=
1722.25 / [(400 + 15)/ 400]2
)
=
1722.25 x 400 x 400 / 415 x 415
)
=
172225 x 4 x 400 / 415 x 415
= 4 x 400 = 1600
True Discount = A − PW = 1722.25 − 1600 = Rs 122.25
Hence, the present worth is Rs 1600 and true discount is Rs 122.25

39. Find the present worth of a bill of Rs 2420 due 2 years hence at 10% compound interest. Also find the true discount.
  A.   Rs2000,Rs 420
  B.   Rs2300,Rs 450
  C.   Rs2120,Rs 425
  D.   Rs2180,Rs 430
     
   
View Answer

Shortcut:
To find the present worth of A rupees due 'n' years hence at 'r' per cent compounded interest payable annually, (i) A = Present Worth x (1 +
r / 100
)n
(ii) Present Worth =
A / (1 + r/100)n
)
(iii) True Discount = Amount − Present Worth
Here, A = 2420, n = 2, R = 10
Using these values in the shortcut, we get:
Present Worth =
2420 / [1 + (10/ 100)]2
)
=
2420 / [1 + (1/ 10)]2
)
=
2420 / [(10 + 1)/ 10]2
)
=
2420 / (11/10)2
)
=
2420 x 10 x 10 / 11 x 11
= 20 x 10 x 10 = 2000
True Discount = A − PW = 2420 − 2000 = Rs 420
Hence, the present worth is Rs 2000 and true discount is Rs 420

40. Find the true discount on Rs 39.69 due in 2 years reckoning compound interest at 5 %
  A.  Rs 3.20
  B.  Rs 3.69
  C.  Rs 3.50
  D.  Rs 4.69
     
   
View Answer

Shortcut:
To find the present worth of A rupees due 'n' years hence at 'r' per cent compounded interest payable annually, (i) A = Present Worth x (1 +
r / 100
)n
(ii) Present Worth =
A / (1 + r/100)n
)
(iii) True Discount = Amount − Present Worth
Here, A = 39.69, n = 2, R = 5
Using these values in the shortcut, we get:
Present Worth =
39.69 / [1 + (5/ 100)]2
)
=
39.69 / [1 + 1/ 20)]2
)
=
39.69 / [(20 + 1)/ 20]2
)
=
39.69 / (21/20)2
)
=
3969 x 2 x 2 / 21 x 21
= 9 x 2 x 2 = 36
True Discount = A − PW = 39.69 − 36 = Rs 3.69
Hence, the present worth is Rs 36 and true discount is Rs 3.69

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