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41. The sides of two cubes are in the ratio 3 : 1. Find the ratio of their volumes.
  A.  21 : 1
  B.  21 : 3
  C.  27 : 1
  D.  29 : 2
     
   
View Answer

In any two 3-dimensional figure, if the corresponding sides or other measuring lengths are in the ratio of a:b, then their volumes are in the ratio a3:b3 Let ratio of the sides of two cubes be a:b = 3:1 then, by using the short cut method, the ratio of volumes of two cubes be a3:b3 = 33:13 = 27:1

42. Each side of a parallelopipe is thriced. Find the ratio of volume of old to new parallelopipe.
  A.  1 : 9
  B.  3 : 9
  C.  2 : 8
  D.  1 : 27
     
   
View Answer

Let ratio of the sides of two parallelopipe be a:b = 1:3 then, by using the short cut method, the ratio of volumes of two parallelopipe be a3:b3 = 13:33 = 1:27

43. The ratio between two numbers is 3 : 4. If each number be increased by 3, the ratio becomes 7 : 9. Find the numbers.
  A.  18 and 24
  B.  9 and 12
  C.  24 and 18
  D.  15 and 20
     
   
View Answer

short cut: The ratio between two numbers is a:b. if each number be increased by x, the ratio becomes c:d. Then, the two numbers are given as

xa(c-d) / ad-bc
and
xb(c-d) / ad-bc
where c-a ≠ d-b Here, a:b = 3:4 c:d = 7:9 x=3 putting these values in the above short cut, we get ∴ Numbers are 3x3(7-9)/3x9 - 4x7 and 3x4(7-9)/(3x9)-(4x7) or 18 and 24 Note: This question also be asked as "The ratio between two numbers is 7:9. If each number be decreased by 3 the ratio becomes 3:4. Find the numbers."

44. The ratio of two numbers is 4 : 5. If each of them decreased by 6, the ratio becomes 3 : 4. Find the numbers.
  A.  15 and 20
  B.  24 and 30
  C.  28 and 35
  D.  30 and 24
     
   
View Answer

short cut: The ratio between two numbers is a:b. if each number be increased by x, the ratio becomes c:d. Then, the two numbers are given as

xa(c-d) / ad-bc
and
xb(c-d) / ad-bc
where c-a ≠ d-b Here, a:b = 4:5 c:d = 3:4 x=6 putting these values in the above short cut, we get ∴ Numbers are
6x4(3-4) / 4x4 - 5x3
and
6x5(3-4) / (4x4)-(5x3)
or 24 and 30

45. The clients in three rooms are in the ratio 2 : 3 : 5. If 30 clients are increased in each room, the ratio changes to 4 : 5 : 7. What was the total number clients in the three rooms before the increased.
  A.  160 clients
  B.  130 clients
  C.  120 clients
  D.  150 clients
     
   
View Answer

In this question, the increase by the same value, i.e, 4-2 = 5-3 = 7-5 = 2 Thus we have 2 &equi; 30 ∴ (2+3+5) &equi;

30 / 2
x 10 = 150 clients

46. An amount is to divided between two men in the ratio of 3 : 5. If the share of one person is Rs 30 less than that of the other, find the amount.
  A.  120
  B.  30
  C.  50
  D.  80
     
   
View Answer

Short cut method:

Total sum / 10
=
3 + 5 / 5-3
∴ Total sum =
8 / 2
x 30 = Rs 120

47. The income of Amit and Babbi are in the ratio 3 : 2 and their expenditures are in the ratio 5 : 3. If each saves Rs 1000, what is their income?
  A.  Amit's income: Rs 3,000; Babbi's income: Rs 1,000
  B.  Amit's income: Rs 9,000; Babbi's income: Rs 3,000
  C.  Amit's income: Rs 6,000; Babbi's income: Rs 4,000
  D.  Amit's income: Rs 12,000; Babbi's income: Rs 8,000
     
   
View Answer

Shortcut: The income of two persons are in the ratio a:b and their expenditure are in the ratio c:d. If each of them saves Rs Y, then their incomes are given by

Ya(d-c) / ad-bc
and
Yb(d-c) / ad-bc
Here, a:b = 3:2 (Income) c:d=5:3(Expenditure) Y=1000 (savings) Putting these values in shortcut method, we get Amit's income =
Ya(d-c) / ad-bc
=
1000 x 3(3-5) / 3x3-2x5
= Rs 6000 Babbi's income =
Yb(d-c) / ad-bc
=
1000 x2(3-5) / 3x3-2x5
= Rs 4,000

48. The incomes of Amit and Beena are in the ratio 3 : 2 and their expenditures are in the ratio 5 : 3. if each saves Rs 1000, what are their expenditures?
  A.  Amit's expenditure: Rs 3,000; Beena's expenditure: Rs 5,000
  B.  Amit's expenditure: Rs 5,000; Beena's expenditure: Rs 3,000
  C.  Amit's expenditure: Rs 6,000; Beena's expenditure: Rs 2,000
  D.  Amit's expenditure: Rs 7,000; Beena's expenditure: Rs 3,000
     
   
View Answer

The income of two persons are in the ratio a:b and their expenditure are in the ratio c:d. If each of them saves Rs Y, then their expenditure are given by

Yc(b-a) / ad-bc
and
Yd(b-a) / ad-bc
Here, a:b=3:2 (income) c:d = 5:3 (expenditure) Y=1000 (savings) Putting these values in shortcut, we get Amit's expenditure=
Yc(b-a) / ad-bc
=
1000x5(3-2) / 3x3-2x5
= Rs 5000 Beena's expenditure=
Yd(b-a) / ad-bc
=
1000x3(3-2) / 3x3-2x5
= Rs 3000

49. The ratio between two numbers is 3 : 4. If each number be increased by 6, the ratio becomes 18 : 23. Find the sum of the numbers.
  A.  90
  B.  60
  C.  40
  D.  70
     
   
View Answer

Shortcut method: The ratio between two numbers is a:b, if each number be increased by x, the ratio beccomes c:d. Then sum of two numbers =

x(a+b)(c-d) / ad-bc
Here, a:b=3:4 c:d=18:23 Y=6 putting these values in shortcut, we get sum of two numbers =
6(3+4)(18-23) / 3x23 - 4x18
=
6x7(-5) / -3
= 70

50. The ratio between two numbers is 3 : 4. If each number be increased by 3, the ratio becomes 7 : 9. Find the difference of the two numbers.
  A.  5
  B.  4
  C.  6
  D.  8
     
   
View Answer

Shortcut: The ratio between two numbers is a:b. If each number be increased by y, the ratio becomes c:d. Then, difference of the two nnumbers

y(a-b)(c-d) / ad-bc
Here, a:b=3:4 c:d = 7:9 y=3 Using these values in shortcut, we get difference of the two numbers =
3(3-4)(7-9) / 9x3-4x7
= -6 Hence, the difference between two numbers = 6

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