Problem 1 :
If A is older than B, C is older than B but younger than A. D is younger than E and B. B is older than E. Then identify who is the yongest?
(A) A (B) B (C) C (D) D
Solution :
Given : A is older than B, C is older than B but younger than A.
A > C > B ----(1)
Given : D is younger than E and B. B is older than E.
B > E > D ----(2)
From (1) and (2), we have
A > C > B > E > D
Clearly, D is the youngest.
The correct answer choice is option (D).
Problem 2 :
The present age of A and B are in the ratio 4 : 5 and after five years they will be in the ratio 5 : 6, then the sum of their present ages is
(A) 55 years (B) 45 years (C) 35 years (D) 25 years
Solution :
Given : The ratio between the present ages of A and B is
= 4 : 5
From the above ratio,
present age of A = 4x
present age of B = 5x
After five years,
age of A = 4x + 5
age of B = 5x + 5
After five years, the ratio between their ages will be
= (4x + 5) : (5x + 5) ----(1)
Given : After five years, the ratio between their ages will be
= 5 : 6 ----(2)
From (1) and (2), we have
(4x + 5) : (5x + 5) = 5 : 6
6(4x + 5) = 5(5x + 5)
24x + 30 = 25x + 25
-x = -5
x = 5
present age of A = 4x = 4(5) = 20 years
present age of B = 5x = 5(5) = 25 years
Sum of the present ages of A and B :
= 20 + 25
= 45 years
The correct answer choice is option (B).
Problem 3 :
Tharani's age is 3 less than that of Rani. If Rani's age is 18, find the ratio between the ages of Tharani and Rani.
(A) 5 : 6 (B) 6 : 5 (C) 3 : 6 (D) 6 : 3
Solution :
Given : Tharani's age is 3 less than that of Rani and Rani's age is 18.
Tharani's age = 18 - 3 = 15
Ratio between the ages of Tharani and Rani :
= 15 : 18
= 5 : 6
The correct answer choice is option (A).
Problem 4 :
The sum of the ages of Rani and Mary is 14 years more than the sum of the ages of Mary and Nancy. How many years is Nancy younger than Rani?
(A) 12 (B) 16 (C) 14 (D) 28
Solution :
Let r, m and n be the ages of Rani, Mary and Nancy.
Given : The sum of the ages of Rani and Mary is 14 years more than the sum of the ages of Mary and Nancy.
r + m = m + n + 14
r = n + 14
n = r - 14
Nancy is 14 years younger than Rani.
The correct answer choice is option (C).
Problem 5 :
Arun is now half as old as his father. Twelve years ago, the father's age was three times as old as Arun. Find the present age of Arun.
(A) 24 (B) 48 (C) 36 (D) 12
Solution :
Let a and f be the present ages of Arun and his father.
Given : Arun is now half as old as his father.
a = ½f
f = 2a
Given : Twelve years ago, the father's age was three times as old as Arun.
f - 12 = 3(a - 12)
Substitute f = 2a.
2a - 12 = 3(a - 12)
2a - 12 = 3a - 36
-a = -24
a = 24
The present age of Arun is 24 years.
The correct answer choice is option (A).
Problem 6 :
Ten years ago, the ages of X and Y were in trhe ratio 3 : 1. The ratio of their present ages is 2 : 1. Their present ages are
(A) 40, 20 (B) 20, 10 (C) 30, 15 (D) 60, 30
Solution :
Given : The ratio of the present ages of X and Y is
2 : 1
From the above ratio,
present age of X = 2k
present age of of Y = k
Ten years ago,
age of X = 2k - 10
age of of Y = k - 10
Ten years ago, the ratio of the ages of X and Y was
= (2k - 10) : (k - 10) ----(1)
Given : Ten years ago, the ratio of the ages of X and Y was
= 3 : 1 ----(2)
From (1) and (2),
(2k - 10) : (k - 10) = 3 : 1
2k - 10 = 3(k - 10)
2k - 10 = 3k - 30
-k = -20
k = 20
present age of X = 2k = 2(20) = 40
present age of of Y = k = 20
The present ages of X and Y are 40 and 20.
The correct answer choice is option (A).
Problem 7 :
A mother is 20 years older than her daughter. Four years ago, the age of the mother was five times the age of her daughter. Find the present age of daughter.
(A) 9 (B) 12 (C) 18 (D) 16
Solution :
Let m and d be the present ages of mother and daughter.
Given : The mother is 20 years older than her daughter.
m = d + 20
Given : Four years ago, the age of the mother was five times the age of her daughter.
m - 4 = 5(d - 4)
Substitute m = d + 20.
d + 20 - 4 = 5(d - 4)
d + 16 = 5d - 20
-4d = -36
d = 9
The present age of the daughter is 9 years.
The correct answer choice is option (A).
Problem 8 :
A person's present age is two-fifth of the age of his mother. After 8 years, he will be one-half of the age of his mother. How old is the mother at present?
(A) 42 (B) 40 (C) 45 (D) 48
Solution :
Let p and m be the present ages of the person and his mother.
Given : The person's present age is two-fifth of the age of his mother.
Given : After 8 years, the person's age will be one-half of the age of his mother.
The least common multiple of the denominators 5 and 2 is 10. Multiply both sides by 10 to get rid of the denominators and solve for m.
2(2m) + 80 = 5(m + 8)
4m + 80 = 5m + 40
-m = -40
m = 40
Mother is 40 years old at present.
The correct answer choice is option (B).
Problem 9 :
Examine the following statements :
I. Either A and B are of the same age or A is older than B.
II. Either C and D are of the same age or D is older than C.
III. B is older than C.
Which one of the following conclusions can be drawn from the above statements?
(A) A is older than C
(B) D is older than C
(C) A is older than B
(D) B and D are of the same age
Solution :
According to the given statements, we have
A = B or A > B
C = D or D > C
B > C
Combining the above staements together,
A ≥ B > C ≤ D
Clearly A is older than C.
The correct answer choice is option (A).
Problem 10 :
A father said to his son, "Your age now is the same as my age at the time of your birth". If the father's age is 38 years now, the son's age five years back was
(A) 14 (B) 19 (C) 24 (D) 38
Solution :
Let x be the present age of son.
So, the son was born x years ago.
At present, father's age is 38 years. Then, the father's age at the time of his son's birth is (38 - x) years.
Given : Son's present age is the same as father's age at the time of son's birth.
x = 38 - x
2x = 38
x = 19
The present age of son is 19 years.
Then the age of the son 5 years back was
= 19 - 5
= 14 years
The correct answer choice is option (A).
Problem 11 :
The ratio between the present ages of P and Q is 2 : 3. If the difference between their ages is 8 years, find P's present age.
(A) 16 (B) 24 (C) 12 (D) 30
Solution :
Given : The ratio between the present ages of P and Q is
2 : 3
From the above ratio,
present age of P = 2x
Present age of Q = 3x
Given : The difference between their ages is 8 years
3x - 2x = 8
x = 8
present age of P = 2(8) = 16 years
The correct answer choice is option (A).
Problem 12 :
The sum of the ages of a father and a son is 50 years. Six years ago, the product of their ages was twice the father's age at that time. The present ages of father and son respectively are
(A) 40 years, 10 years
(B) 41 years, 9 years
(C) 38 years, 12 years
(D) 42 years, 8 years
Solution :
Let f and s be the present ages of father and son.
Given : The sum of the ages of a father and a son is 50 years.
f + s = 50
f = 50 - s ----(1)
Given : Six years ago, the product of their ages was twice the father's age at that time.
(f - 6)(s - 6) = 2(f - 6)
Dividce both sides by (f - 6).
s - 6 = 2
s = 8
Substitute s = 8 into (1).
f = 50 - 8
f = 42
Present age of father = 42 years
Present age of son = 8 years
The correct answer choice is option (D).
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