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11. Ten years ago A was half of B in age. If the ratio of their present ages is 3 : 4, what will be the total of their present ages?
  A.  35 years
  B.  45 years
  C.  25 years
  D.  40 years
     
   
View Answer

Shortcut:
If t years earlier, the father's age was 'a' times that of his son. At present the father's age is 'b' times that of his son. Then the sum total of the age of the father and the son is
[{
t(a − 1) / a − b
}
(b + 1)
]
years
10 years ago A was 1/2 of B's age. At present A is 3/4 of B's age. So, t = 10, a = 1/2, b = 3/4
Using these values in the shortcut, we get:
Total sum of ages =
[{
10(1/2 − 1) / 1/2 − 3/4
}
(3/4 + 1)
]
years

=
20 x 7 / 4
= 35 years
Hence, the sum of ages of (A + B) is 35 years.

12. At present the age of the father is 5 times the age of his son, three years hence the fathers' age would be four times that of his son. What is the sum of the present ages of father and his son?
  A.  50 years
  B.  54 years
  C.  44 years
  D.  64 years
     
   
View Answer

Shortcut:
If the present age of the father is 'a' times that the age of his son. 't' years hence, the fathers' age becomes 'b' times the age of his son. Then the sum of the present ages of father and his son is
[{
t(b − 1) / a − b
}
(a + 1)
]
years
t = 3, a = 5, b = 4
Using these values in the shortcut, we get:
Total sum of ages =
[{
3(4 − 1) / 5 − 4
}
(5 + 1)
]
years
=
[{
3 x 3 / 1
}
x 6
]
years
= 9 x 6 = 54 years
Hence, the sum of their ages is 54 years.

13. The sum of the ages of a mother and her daughter is 50 years. Also, 5 years ago, the mohter's age was 7 times the age of the daughter. What are the present ages of the mother and the daughter?
  A.  Mother's age: 45 years; Daughter's age: 10 years
  B.  Mother's age: 40 years; Daughter's age: 10 years
  C.  Mother's age: 40 years; Daughter's age: 15 years
  D.  Mother's age: 50 years; Daughter's age: 12 years
     
   
View Answer

Shortcut:
If the sum of the present ages of X and Y is 'a' years. 't' years ago, the ages of X was 'b' times the age of the Y. Then the present ages of X and Y are
ab − t(b − 1) / b + 1

and
a + t(b − 1) / b + 1
years respectively.
t = 5, a = 50, b = 7
Using these values in the shortcut, we get:
Mother's ages =
50 x 7 − 5(7 − 1) / 7 + 1

=
350 − 5 x 6 / 8

=
350 − 30 / 8
=
320 / 8
= 40 years
Hence, the age of mother is 40 years.
Daughter's age =
50 + 5(7 − 1) / 7 + 1

=
50 + 5 x 6 / 8

=
50 + 30 / 8

=
80 / 8
= 10
Hence, the age of daughter is 10 years.

14. The sum of the ages of a son and father is 56 years. After 4 years, the age of the father will be three times that of the son. What is the age of the son?
  A.  12 years
  B.  14 years
  C.  16 years
  D.  18 years
     
   
View Answer

Shortcut:
If the sum of the present ages of X and Y is 'a' years. 't' years after, the ages of X will 'b' times the age of the Y. Then the ages of X and Y are
ab + t(b − 1) / b + 1

and
a − t(b − 1) / b + 1
years respectively.
t = 4, a = 56, b = 3
Using these values in the shortcut, we get:
Son's age =
56 − 4(3 − 1) / 3 + 1

=
56 − 4 x 2 / 4

=
56 − 8 / 4

=
48 / 4
= 12
Hence, the age of son is 12 years.

15. The ratio of the ages of the father and the son at present is 6 : 1. After 5 years the ratio will beocme 7 : 2. What are the present ages of the son and the father?
  A.  Son's age: 6 years; Father's age: 30 years
  B.  Son's age: 5 years; Father's age: 30 years
  C.  Son's age: 4 years; Father's age: 28 years
  D.  Son's age: 8 years; Father's age: 40 years
     
   
View Answer

Shortcut:
If the ratio of the ages of X and Y at present is a : b. After 't' years the raio will become c : d. Then the present ages of X and Y are a x
t(c − d) / a x d − b x c
and b x
t(c − d) / a x d − b x c
years respectively.
t = 5, a = 6, b = 1, c = 7, d = 2
Using these values in the shortcut, we get:
Father's age = 6 x
5(7 − 2) / 6 x 2 − 1 x 7

= 6 x
5 x 5 / 12 − 7

=
6 x 5 x 5 / 5
= 6 x 5 = 30
Hence, the age of father is 30 years. Son's age = 1 x
5(7 − 2) / 6 x 2 − 1 x 7

=
5 x 5 / 12 − 7

=
5 x 5 / 5
= 5
Hence, the age of son is 5 years.

16. The ratio of present ages of P and Q is 8 : 5. After 6 years their ages are in the ratio of 3 : 2. Find the ratio of the sum and difference of the present ages of P and Q.
  A.  13:3
  B.  15:4
  C.  12:5
  D.  14:3
     
   
View Answer

Shortcut:
If the ratio of the ages of X and Y at present is a : b. After 't' years the raio will become c : d. Then the present ages of X and Y are a x
t(c − d) / a x d − b x c
and b x
t(c − d) / a x d − b x c
years respectively.
t = 6, a = 8, b = 5, c = 3, d = 2
Using these values in the shortcut, we get:
P's age = 8 x
6(3 − 2) / 8 x 2 − 5 x 3

= 8 x
6 x 1 / 16 − 15

=
8 x 6 / 1
= 48
Hence, the age of P is 48 years. t = 6, a = 8, b = 5, c = 3, d = 2
Q's age = 5 x
6(3 − 2) / 8 x 2 − 5 x 3

= 5 x
6 x 1 / 16 − 15
= 5 x 6 = 30
∴ the age of Q is 30 years.
Now, the sum of ages of P & Q = 48 + 30 = 78
and, the difference of ages of P & Q = 48 - 30 = 18
The ratio (Sum of ages : Difference of ages) = 78 : 18 = 39 : 9 = 13 : 3
Hence, the ratio is 13 : 3.

17. The ratio of the ages of the father and the son at present is 3:1. Four year earlier, the ratio was 4:1. What are the present ages of the son and the father?
  A.  Son's age:11 years; Father's age:33 years
  B.  Son's age:13 years; Father's age:39 years
  C.  Son's age:10 years; Father's age:30 years
  D.  Son's age:12 years; Father's age:36 years
     
   
View Answer

Shortcut:
If the ratio of the ages of X and Y at present is a : b. 't' years earlier, the ratio was c : d. Then the present ages of X and Y are a x
t(c − d) / b x c − a x d
and b x
t(c − d) / b x c − a x d
years respectively.
t = 4, a = 3, b = 1, c = 4, d = 1
Using these values in the shortcut, we get:
Father's age = 3 x
4(4 − 1) / 1 x 4 − 3 x 1

= 3 x
4 x 3 / 4 − 3
= 3 x 12 = 36
Hence, the age of father is 36 years.
Son's age = 1 x
4(4 − 1) / 1 x 4 − 3 x 1

=
4 x 3 / 4 − 3
= 4 x 3 = 12
Hence, the age of son is 12 years.

18. Six years ago Jimmy was twice as old as Babbi. If the ratio of their present ages is 9 : 5 respectively, what is the difference between their present ages in years?
  A.  22 years
  B.  25 years
  C.  24 years
  D.  23 years
     
   
View Answer

Shortcut:
If the ratio of the ages of X and Y at present is a : b. 't' years earlier, the ratio was c : d. Then the present ages of X and Y are a x
t(c − d) / b x c − a x d
and b x
t(c − d) / b x c − a x d
years respectively.
t = 6, a = 9, b = 5, c = 2, d = 1
Using these values in the shortcut, we get:
Jimmy's age = 9 x
6(2 − 1) / 5 x 2 − 9 x 1

= 9 x
6 x 1 / 10 − 9
= 9 x 6 = 54
Hence, the age of Jimmy is 54 years.
t = 6, a = 9, b = 5, c = 2, d = 1
Babbi's age = 5 x
6(2 − 1) / 5 x 2 − 9 x 1

= 5 x
6 x 1 / 10 − 9
= 5 x 6 = 30
Hence, the age of Babbi is 30 years.
Difference between Jimmy & Babbi age = 54 − 30 = 24 years.

19. The ratio of the ages of the father and the son at present is 6:1. After 5 years the ratio will become 7 : 2. What is the sum of the present ages of the father and the son?
  A.  34 years
  B.  35 years
  C.  36 years
  D.  30 years
     
   
View Answer

Shortcut:
If the ratio of the ages of X and Y at present is a : b. After 't' years, the ratio will become c : d. Then the sum of present ages of X and Y is
[
t(c − d) / a x d − b x c
]
(a + b)
t = 5, a = 6, b = 1, c = 7, d = 2
Using these values in the shortcut, we get:
Jimmy's age =
[
5(7 − 2) / 6 x 2 − 1 x 7
]
(6 + 1)
=
5 x 5 / 12 − 7
x 7
=
5 x 5 / 5
x 7 = 5 x 7 = 35
Hence, the sum of their ages is 35 years.

20. If the product of the present ages of the father and his son is 900 years and the ratio of their present ages is 25 : 9. Find their present ages.
  A.  Father's age: 50 years; Son's age: 18 years
  B.  Father's age: 25 years; Son's age: 9 years
  C.  Father's age: 75 years; Son's age: 27 years
  D.  Father's age: 100 years; Son's age: 36 years
     
   
View Answer

Shortcut:
If the product of the present ages of X and Y is 'a' years and the raio of the present ages of X and Y is c : d. Then the present ages of X and Y are c x
√(
a / c x d
)
and d x
√(
a / c x d
)
years respectively.
a = 900, c = 25, d = 9
Using these values in the shortcut, we get:
Father's age = 25 x
√(
900 / 25 x 9
)
= 25 x
30 / 5 x 3
= 25 x 2 = 50
Hence, the age of father is 50 years.
a = 900, c = 25, d = 9
Son's age = 9 x
√(
900 / 25 x 9
)
= 9 x
30 / 5 x 3
= 9 x 2 = 18
Hence, the age of son is 18 years.

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