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51. The ratio of two numbers is 7 : 9. If each number is decreased by 4, the ratio becomes 3 : 4. Find the sum of the two numbers.
  A.  64
  B.  60
  C.  66
  D.  58
     
   
View Answer

Shortcut method: The ratio of two numbers is a:b. If each number be decreased by z, the ratio becomes c:d. Then, sum of the two numbers =

z(a+b)(d-c) / ad-bc
Here, a:b=7:9 c:d=3:4 z=4 putting these values in shortcut, we get sum of two numbers =
4(7+9)(4-3) / 7x4 - 9x3
=
4x16 / 1
= 64

52. The ratio of two numbers is 7 : 9. If each number is decreased by 3 the ratio becomes 3 : 4. Find the difference of the numbers.
  A.  8
  B.  6
  C.  9
  D.  3
     
   
View Answer

Shortcut method: The ratio of two numbers is a:b. If each number be decreased by z, the ratio becomes c:d. Then, difference of the two numbers =

z(a-b)(d-c) / ad-bc
Here, a:b=7:9 c:d=3:4 z=3 putting these values in shortcut, we get sum of two numbers =
3(7-9)(4-3) / 7x4 - 9x3
=
3x(-2) / 1
= -6 ∴ the difference of the numbers = 6

53. The speed of three bikes are in the ratio 4 : 3 : 2. What is the ratio among the times taken by these bikes to travel the same distance?
  A.  6 : 4 : 3
  B.  4 : 3 : 6
  C.  3 : 4 : 6
  D.  6 : 4 : 2
     
   
View Answer

Short cut If the ratio of any quantities be a:b:c:d, then the ratio of other quantities which are inversely proportional to that is given by

1 / a
:
1 / b
:
1 / c
:
1 / d
Here a:b:c = 4:3:2 Ratio of time taken =
1 / 4
:
1 / 3
:
1 / 2
Taking L.C.M of these ratio and solving, we get time taken ratio = 3:4:6

54. A job is assigned to three groups of labours. The ratio of labours in the groups is 3 : 4 : 5. Find the ratio of days in which they will complete the job.
  A.  13 : 12 : 10
  B.  10 : 12 : 15
  C.  20 : 15 : 12
  D.  12 : 16 : 15
     
   
View Answer

Apply the shortcut o Q51: Ratio of persons in the groups is 3:4:5 then ratio of days in which they wil complete the work will be

1 / 3
:
1 / 4
:
1 / 5
Taking L.C.M of these ratio and solving, we get time taken ratio = 20:15:12

55. The sum of two numbers is 30 and their difference is 5. What is the ratio of the two numbers?
  A.  7 : 5
  B.  7 : 8
  C.  5 : 7
  D.  3 : 5
     
   
View Answer

Shortcut: The sum of two numbers is 'Z' and their difference is 'z', then the ratio of numbers is given by Z+z : Z-z here Z = 30 z = 5 Applying these in shortcut, we get ratio = 30+5 : 30-5 = 35 : 25 = 7:5

56. Find the number which, when added to the terms of the ratio 1 : 2 makes it equal to the ratio 4 : 5.
  A.  9
  B.  12
  C.  5
  D.  3
     
   
View Answer

shortcut: A number which, when added to the terms of the ratio a:b makes it equal to the ratio c:d is

ad-bc / c-d
a:b = 1:2 c:d = 4:5 Applying these values in shortcut, we get
1x5-2x4 / 4-5
=
-3 / -1
= 3 Hence, 3 is to be added to the terms.

57. A cane contains a mixture of milk and wine in the proportion 7 : 5. If 9 litres of the mixture is replaced by 9 litres of wine, then the ratio of the two liquids becomes 7 : 9. How much of the milk was there in the cane?
  A.  21 litres
  B.  17 litres
  C.  25 litres
  D.  11 litres
     
   
View Answer

1x5-2x4 / 4-5
Let the Milk and Juice are 7x and 5x liters respectively. Now, when 9 litres of mixture are taken out, Milk remains 7x-9(
7 / 7+5
= 7x -
9 x 7 / 12
= 7x -
21 / 4
Juice remains 5x - 9(
5 / 7+5
= 5x -
9x5 / 12
= 5x -
15 / 4
Now, when 9 litres of juice are added, 7x -
21 / 4
: 5x -
15 / 4
+ 9 = 7 : 9
7x-21/4 / 5x -15/4 + 9
=
7 / 9
63x -189/4=35x-105/4 +63 28x=189/4-105/4+6 28x=21+63 28x=84 x=84/28=3 ∴ 7x = 7 x 3 = 21 litres Shortcut method: first ratio = 7:5 second ratio= 7:9 Difference of cross products of ratios (D) = 7x9 - 7x5 = 63 - 35 = 28 Now, the short cut is Common factor of first ratio =
Quantity replaced / Sum of terms in first ratio
+
Quantity replaced x term A in second ratio / D
=
9 / 7+5
+
9 x 7 / 28
=
9 / 12
+
9 / 4
=
36 / 12
= 3 ∴ Quantity of milk = 7 x 3 = 21 litres. Similarly, Quantity of Juice = 5 x 3 = 15 litres.

58. A vessel contains milk and water in ratio 5 : 3. if 16 litres of the mixture are removed and the same quantity of water is added, the ratio becomes 3 : 5. What quantity does the vessel hold?
  A.  40 litres
  B.  50 litres
  C.  25 litres
  D.  15 litres
     
   
View Answer

Shortcut method: When the ratio is getting reversed i.e, from 5:3 to 3:5), We use the formula Total quantity =

(5+3)2 / 52-32
x quantity of milk in the removed mixture =
64 / 16
x 10 = 40 litres Remarks: Since, the liquid water is added here, the quantity of milk is used in the formula. If liquid milk is to be added, then the quantity of water is used in the shortcut.

59. A company reduces the number of its employees in the ratio 5 : 4 and increases their salary in the ratio 6 : 7. Determine whether its bill of total salary increases or decreases and in what ratio.
  A.  11 : 16
  B.  15 : 14
  C.  12 : 17
  D.  14 : 15
     
   
View Answer

Reduction in the number of employees in the ratio 5 : 4 Increases of salary in the ratio 6 : 7 We know that the total ill = wage per person x number of total employees. Therefore, the ratio of change in bill = 5x6 : 4x7 = 30:28 = 15:14 The ratio shows that there is a decrease in the bill.

60. Two candles of the same height are lighted at the same time. The first is consumed in 4 hours and the second in 3 hours. Assuming theat each candle burns at a constant rate, in how many hours after being lighted, was the first candle twice the height of the second?
  A.  1 hours 34 minutes
  B.  3 hours 25 minutes
  C.  2 hours 30 minutes
  D.  2 hours 24 minutes
     
   
View Answer

shortcut: Two candels of the same height are lighted at the same time. The first is consumed in T1 hours and the second in T2 hours. Assuming that each candle burns at a constant rate, the time after which the ratio of first candle to second candle becomes a:b is given by

T1T2(a/b - 1) / (a/b)T1 - T2
Here, T1=4 T2=3 a:b = 2:1 Applying these values in shortcut, we get =
4x3(2/1 - 1) / (2/1)4 - 3
=
12 / 5
hours = 2 hours 24 minutes

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