#1. Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit’s age. After further 8 years, how many times would he be of Ronit’s age?
A. 2.2 times
\[B.\;2\frac12Times\]
\[C.\;2\frac34Times\]
D. 3 times
Answer: Option A Solution: Let’s denote:
– \( S \) as Ronit’s current age. – \( F \) as the father’s current age.
We are given the following information:
1. **Father’s age is three times more than Ronit’s age.** This means: \[ F = 3S \]
2. **After 8 years, the father’s age would be two and a half times Ronit’s age.** In 8 years, Ronit’s age will be \( S + 8 \) and the father’s age will be \( F + 8 \). The relationship is: \[ F + 8 = 2.5 \times (S + 8) \]
Now we have two equations: 1. \( F = 3S \) 2. \( F + 8 = 2.5 \times (S + 8) \)
Step 1: Substitute the value of \( F \) from the first equation into the second equation:
\[ 3S + 8 = 2.5 \times (S + 8) \]
Step 2: Expand and simplify:
\[ 3S + 8 = 2.5S + 20 \]
Now, move all terms involving \( S \) to one side:
\[ 3S – 2.5S = 20 – 8 \] \[ 0.5S = 12 \]
Step 3: Solve for \( S \):
\[ S = \frac{12}{0.5} = 24 \]
So, Ronit’s current age \( S = 24 \) years.
Step 4: Find the father’s current age \( F \):
Since \( F = 3S \), we have:
\[ F = 3 \times 24 = 72 \]
Thus, the father’s current age is 72 years.
Step 5: Determine the ages after 16 years
After 8 more years (total of 16 years from the current time), Ronit’s age will be:
\[ S + 16 = 24 + 16 = 40 \quad \text{(Ronit’s age after 16 years)} \]
The father’s age will be:
\[ F + 16 = 72 + 16 = 88 \quad \text{(Father’s age after 16 years)} \]
Step 6: Find how many times the father’s age will be of Ronit’s age:
Now, we find how many times Ronit’s age will go into the father’s age:
\[ \frac{88}{40} = 2.2 \]
Final Answer: After 16 years, the father’s age will be 2.2 times Ronit’s age.
#2. The sum of ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?
A. 4 years
B. 8 years
C. 10 years
D. None of these
Answer: Option A Solution: Let’s denote the age of the youngest child as \( x \).
Since the children are born at intervals of 3 years, their ages will be: – Youngest child: \( x \) – Second youngest child: \( x + 3 \) – Third youngest child: \( x + 6 \) – Fourth youngest child: \( x + 9 \) – Fifth youngest child: \( x + 12 \)
The sum of their ages is given as 50 years. So, we can write the equation:
Final Answer: The age of the youngest child is 4 years.
#3. A father said to his son, “I was as old as you are at the present 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 years
B. 19 years
C. 33 years
D. 38 years
Answer: Option A Solution: Let’s define the following:
– Let the son’s current age be \( S \). – The father’s current age is given as 38 years.
According to the father’s statement, “I was as old as you are at the present at the time of your birth,” this means that the father’s age at the time of the son’s birth was equal to the son’s current age \( S \).
Step 1: Relationship between the father’s age and the son’s age
At the time of the son’s birth, the father’s age was \( 38 – S \), since the father is currently 38 years old. According to the statement, the father’s age at that time equals the son’s current age:
\[ 38 – S = S \]
Step 2: Solve for \( S \)
\[ 38 = 2S \] \[ S = \frac{38}{2} = 19 \]
Thus, the son’s current age is **19 years**.
Step 3: Find the son’s age five years ago
The son’s age five years ago was:
\[ 19 – 5 = 14 \]
Final Answer: The son’s age five years ago was 14 years.
#4. A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 27, the how old is B?
A. 7
B. 8
C. 9
D. 10
E. 11
Answer: Option D Solution: Let’s define the following:
– Let \( C \)’s age be \( x \). – Since B is twice as old as C, B’s age will be \( 2x \). – A is 2 years older than B, so A’s age will be \( 2x + 2 \).
Step 1: Set up the equation for the total of their ages
#5. Present ages of Sameer and Anand are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Anand’s present age in years?
A. 24
B. 27
C. 40
D. Cannot be determined
E. None of these
Answer: Option A Solution: Let’s denote the present ages of Sameer and Anand as \( 5x \) and \( 4x \), respectively, where \( x \) is a constant factor.
Step 1: Set up the ratio of their ages after 3 years
Three years later, Sameer’s age will be \( 5x + 3 \), and Anand’s age will be \( 4x + 3 \).
We are told that after 3 years, the ratio of their ages will be 11:9. This gives the equation:
\[ \frac{5x + 3}{4x + 3} = \frac{11}{9} \]
Step 2: Cross multiply and solve for \( x \)
Cross multiply the equation:
\[ 9(5x + 3) = 11(4x + 3) \]
Expanding both sides:
\[ 45x + 27 = 44x + 33 \]
Simplifying the equation:
\[ 45x – 44x = 33 – 27 \] \[ x = 6 \]
Step 3: Find Anand’s present age
Since Anand’s present age is \( 4x \), we substitute \( x = 6 \):
\[ \text{Anand’s age} = 4 \times 6 = 24 \]
Final Answer: Anand’s present age is 24 years.
#6. A man is 24 years older than his son. In two years, his age will be twice the age of his son. The present age of his son is:
A. 14 years
B. 18 years
C. 20 years
D. 22 years
E. None of these
Answer: Option D Solution: Let’s define:
– The son’s current age as \( x \). – The man’s current age will then be \( x + 24 \) because the man is 24 years older than his son.
Step 1: Set up the equation for their ages in 2 years
In two years: – The son’s age will be \( x + 2 \). – The man’s age will be \( (x + 24) + 2 = x + 26 \).
We are told that in two years, the man’s age will be twice the son’s age. So, we can set up the equation:
\[ x + 26 = 2 \times (x + 2) \]
Step 2: Solve for \( x \)
First, expand the right-hand side:
\[ x + 26 = 2x + 4 \]
Now, simplify the equation:
\[ x + 26 – x = 2x + 4 – x \] \[ 26 = x + 4 \] \[ x = 26 – 4 \] \[ x = 22 \]
Final Answer: The present age of the son is 22 years.
#7. Six years ago, the ratio of the ages of Kunal and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar’s age at present?
A. 16 years
B. 18 years
C. 20 years
D. Cannot be determined
E. None of these
Answer: Option A Solution: Let’s denote:
– Kunal’s present age as \( K \). – Sagar’s present age as \( S \).
Step 1: Set up the equations based on the given information
**Six years ago**, the ratio of their ages was 6:5. So:
\[ \frac{K – 6}{S – 6} = \frac{6}{5} \]
**Four years from now**, the ratio of their ages will be 11:10. So:
#8. The sum of the present ages of a father and his son is 60 years. Six years ago, father’s age was five times the age of the son. After 6 years, son’s age will be:
A. 12 years
B. 14 years
C. 18 years
D. 20 years
E. None of these
Answer: Option D Solution: Let’s denote the present age of the son as \( S \) and the present age of the father as \( F \).
Step 1: Set up the equations based on the given information
We are given: 1. The sum of their ages is 60 years: \[ F + S = 60 \]
2. Six years ago, the father’s age was five times the age of the son: \[ F – 6 = 5(S – 6) \]
Step 2: Solve the system of equations
First, simplify the second equation: \[ F – 6 = 5(S – 6) \] \[ F – 6 = 5S – 30 \] \[ F = 5S – 24 \quad \text{(Equation 2)} \]
Now, substitute Equation 2 into the first equation:
After 6 years, the son’s age will be: \[ 14 + 6 = 20 \]
Final Answer: The son’s age after 6 years will be 20 years.
#9. At present, the ratio between the ages of Arun and Deepak is 4 : 3. After 6 years, Arun’s age will be 26 years. What is the age of Deepak at present ?
A. 12 years
B. 15 years
C. 19 and half
D. 21 years
E. None of these
Answer: Option B Solution: Let’s denote:
– Arun’s present age as \( 4x \) (since the ratio of Arun’s age to Deepak’s age is 4:3). – Deepak’s present age as \( 3x \).
Step 1: Use the information about Arun’s age after 6 years
We are told that after 6 years, Arun’s age will be 26 years. So, we can write:
Since Deepak’s present age is \( 3x \), we substitute \( x = 5 \):
\[ \text{Deepak’s age} = 3 \times 5 = 15 \]
Final Answer: Deepak’s present age is 15 years.
#10. Sachin is younger than Rahul by 7 years. If their ages are in the respective ratio of 7 : 9, how old is Sachin?
A. 16 years
B. 18 years
C. 28 years
D. 24.5 years
E. None of these
Answer: Option D Solution: Let’s denote:
– Sachin’s age as \( S \) – Rahul’s age as \( R \)
We are given the following information:
1. Sachin is younger than Rahul by 7 years: \[ R = S + 7 \]
2. The ratio of their ages is 7:9: \[ \frac{S}{R} = \frac{7}{9} \]
Step 1: Use the ratio to express \( R \) in terms of \( S \)
From the ratio \( \frac{S}{R} = \frac{7}{9} \), we can write:
\[ S = \frac{7}{9}R \]
Step 2: Substitute \( R = S + 7 \) into the equation
Now, substitute \( R = S + 7 \) into the equation \( S = \frac{7}{9}R \):
\[ S = \frac{7}{9}(S + 7) \]
Step 3: Solve for \( S \)
Now, expand the right-hand side:
\[ S = \frac{7}{9}S + \frac{7}{9} \times 7 \] \[ S = \frac{7}{9}S + \frac{49}{9} \]
Multiply the entire equation by 9 to eliminate the fraction:
\[ 9S = 7S + 49 \]
Now, subtract \( 7S \) from both sides:
\[ 9S – 7S = 49 \] \[ 2S = 49 \]
Now, divide by 2:
\[ S = \frac{49}{2} = 24.5 \]
Final Answer: Sachin is 24.5 years old.
#11. The present ages of three persons in proportions 4 : 7 : 9. Eight years ago, the sum of their ages was 56. Find their present ages (in years).
A. 8, 20, 28
B. 16, 28, 36
C. 20, 35, 45
D. None of these
Answer: Option B Solution: Let’s denote the present ages of the three persons as:
– The first person’s age = \( 4x \) – The second person’s age = \( 7x \) – The third person’s age = \( 9x \)
Step 1: Use the information about their ages 8 years ago
Eight years ago, the sum of their ages was 56. So, we can write the equation for the sum of their ages 8 years ago:
Now that we have \( x = 4 \), we can find the present ages of the three persons:
– The first person’s age = \( 4x = 4 \times 4 = 16 \) – The second person’s age = \( 7x = 7 \times 4 = 28 \) – The third person’s age = \( 9x = 9 \times 4 = 36 \)
Final Answer: The present ages of the three persons are: – First person: 16 years – Second person: 28 years – Third person: 36 years
#12. Ayesha’s father was 38 years of age when she was born while her mother was 36 years old when her brother four years younger to her was born. What is the difference between the ages of her parents?
A. 2 years
B. 4 years
C. 6 years
D. 8 years
E. None of these
Answer: Option C Solution: Let’s define the following:
– Let Ayesha’s current age be \( A \). – Ayesha’s father’s current age is \( A + 38 \) because he was 38 years old when she was born. – Ayesha’s brother’s current age is \( A – 4 \), since her brother is four years younger than her. – Ayesha’s mother’s current age is \( (A – 4) + 36 \), because her mother was 36 years old when her brother was born.
Step 1: Find the difference in ages between Ayesha’s parents
The age difference between Ayesha’s father and mother is the difference between their ages:
Final Answer: The difference in ages between Ayesha’s parents is 6 years.
#13. 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. 32 years
B. 36 years
C. 40 years
D. 48 years
E. None of these
Answer: Option C Solution: Let’s denote:
– The person’s present age as \( P \). – The mother’s present age as \( M \).
Step 1: Set up the equations based on the given information
We are given:
1. The person’s present age is two-fifths of the age of his mother: \[ P = \frac{2}{5}M \]
2. After 8 years, the person’s age will be one-half of the mother’s age: \[ P + 8 = \frac{1}{2}(M + 8) \]
Step 2: Substitute \( P = \frac{2}{5}M \) into the second equation
Substitute \( P = \frac{2}{5}M \) into the second equation:
\[ \frac{2}{5}M + 8 = \frac{1}{2}(M + 8) \]
Step 3: Simplify the equation
First, clear the fraction by multiplying both sides of the equation by 10:
Final Answer: The mother’s present age is 40 years.
#14. Q is as much younger than R as he is older than T. If the sum of the ages of R and T is 50 years, what is definitely the difference between R and Q’s age?
A. 1 year
B. 2 years
C. 25 years
D. Data inadequate
E. None of these
Answer: Option D Solution: Let’s define:
– Q’s age as \( Q \) – R’s age as \( R \) – T’s age as \( T \)
Step 1: Translate the given information into equations
We are given the following:
1. Q is as much younger than R as he is older than T. This means: \[ R – Q = Q – T \] This can be rewritten as: \[ R = 2Q – T \quad \text{(Equation 1)} \]
2. The sum of the ages of R and T is 50 years: \[ R + T = 50 \quad \text{(Equation 2)} \]
Step 2: Solve the system of equations
From Equation 1, we know that \( R = 2Q – T \). Substitute this expression for \( R \) into Equation 2:
\[ (2Q – T) + T = 50 \]
Simplifying the equation:
\[ 2Q = 50 \]
Solving for \( Q \):
\[ Q = 25 \]
Step 3: Find the difference between R and Q’s age
We already know that \( R = 2Q – T \). Since \( Q = 25 \), substitute this value into the equation:
\[ R = 2(25) – T = 50 – T \]
Now, the difference between R and Q’s age is:
\[ R – Q = (50 – T) – 25 = 25 – T \]
Since we know \( R + T = 50 \), substituting \( R = 50 – T \) into this equation:
\[ (50 – T) + T = 50 \]
This confirms that the difference between R and Q’s age is:
\[ \boxed{25 \text{ years}} \]
#15. The age of father 10 years ago was thrice the age of his son. Ten years hence, father’s age will be twice that of his son. The ratio of their present ages is:
A. 5 : 2
B. 7 : 3
C. 9 : 2
D. 13 : 4
Answer: Option B Solution: Let’s denote:
– The father’s present age as \( F \). – The son’s present age as \( S \).
Step 1: Set up the equations based on the given information
We are given:
1. **Ten years ago**, the father’s age was three times the son’s age: \[ F – 10 = 3(S – 10) \]
2. **Ten years hence**, the father’s age will be twice the son’s age: \[ F + 10 = 2(S + 10) \]
Step 2: Simplify both equations
**Equation 1:** \[ F – 10 = 3(S – 10) \] \[ F – 10 = 3S – 30 \] \[ F = 3S – 20 \quad \text{(Equation 1)} \]
**Equation 2:** \[ F + 10 = 2(S + 10) \] \[ F + 10 = 2S + 20 \] \[ F = 2S + 10 \quad \text{(Equation 2)} \]
Step 3: Solve the system of equations
Now, we have the system of equations:
1. \( F = 3S – 20 \) 2. \( F = 2S + 10 \)
Since both expressions represent \( F \), we can set them equal to each other:
\[ 3S – 20 = 2S + 10 \]
Now, solve for \( S \):
\[ 3S – 2S = 10 + 20 \] \[ S = 30 \]
Step 4: Find the father’s present age
Now that we know \( S = 30 \), substitute this value into either equation for \( F \). Using Equation 2:
\[ F = 2S + 10 \] \[ F = 2(30) + 10 = 60 + 10 = 70 \]
Step 5: Find the ratio of their present ages
The ratio of their present ages is:
\[ \frac{F}{S} = \frac{70}{30} = \frac{7}{3} \]
Final Answer: The ratio of the father’s present age to the son’s present age is \( \boxed{7:3} \).
#16. The ages of Nitish and Vinnee are in the ratio 6 : 5 respectively. After 9 years the ratio of their ages will be 9 : 8. What is the difference in their ages now ?
B. 5 Years
C. 7 Years
D. 9 Years
E. None of these
Answer: Option A Solution: Let the present ages of Nitish and Vinnee be represented as:
– Nitish’s age = \( 6x \) – Vinnee’s age = \( 5x \)
Step 1: Set up the equation based on the ratio after 9 years
We are told that **after 9 years**, the ratio of their ages will be 9:8. So, we can write:
\[ \frac{6x + 9}{5x + 9} = \frac{9}{8} \]
Step 2: Cross-multiply to solve for \( x \)
Cross-multiply:
\[ 8(6x + 9) = 9(5x + 9) \]
Expand both sides:
\[ 48x + 72 = 45x + 81 \]
Step 3: Solve for \( x \)
Now, subtract \( 45x \) from both sides:
\[ 48x – 45x + 72 = 81 \] \[ 3x + 72 = 81 \]
Now, subtract 72 from both sides:
\[ 3x = 9 \]
Divide by 3:
\[ x = 3 \]
Step 4: Find the present ages of Nitish and Vinnee
Now that we know \( x = 3 \), we can find their present ages:
Final Answer: The difference in their ages now is 3 years.
#17. The ages of Shakti and Kanti are in the ratio of 8 : 7 respectively. After 10 years, the ratio of their ages will be 13 : 12. What is the difference between their ages ?
A. 2 Years
B. 4 Years
C. 8 Years
D. 6 Years
E. None of these
Answer: Option A Solution: Let the present ages of Shakti and Kanti be represented as:
– Shakti’s age = \( 8x \) – Kanti’s age = \( 7x \)
Step 1: Set up the equation based on the ratio after 10 years
We are told that **after 10 years**, the ratio of their ages will be 13:12. So, we can write the equation:
\[ \frac{8x + 10}{7x + 10} = \frac{13}{12} \]
Step 2: Cross-multiply to solve for \( x \)
Cross-multiply:
\[ 12(8x + 10) = 13(7x + 10) \]
Expand both sides:
\[ 96x + 120 = 91x + 130 \]
Step 3: Solve for \( x \)
Now, subtract \( 91x \) from both sides:
\[ 96x – 91x + 120 = 130 \] \[ 5x + 120 = 130 \]
Now, subtract 120 from both sides:
\[ 5x = 10 \]
Divide by 5:
\[ x = 2 \]
Step 4: Find the present ages of Shakti and Kanti
Now that we know \( x = 2 \), we can find their present ages:
Final Answer: The difference in their ages is 2 years.
#18. The ages of A and B are in the ratio 6 : 5 and the sum of their ages is 44 years. What will be the ratio of their ages after 8 years ?
A. 7 : 6
B. 8 : 7
C. 9 : 8
D. 3 : 4
Answer: Option B Solution: Let’s denote the present ages of A and B as:
– A’s age = \( 6x \) – B’s age = \( 5x \)
Step 1: Use the sum of their ages to find \( x \)
We are given that the sum of their ages is 44 years:
\[ 6x + 5x = 44 \]
Simplify the equation:
\[ 11x = 44 \]
Solve for \( x \):
\[ x = \frac{44}{11} = 4 \]
Step 2: Find their present ages
Now that we know \( x = 4 \), we can find their present ages:
– A’s age after 8 years = \( 24 + 8 = 32 \) – B’s age after 8 years = \( 20 + 8 = 28 \)
Step 4: Find the ratio of their ages after 8 years
The ratio of their ages after 8 years is:
\[ \frac{A’s \, age \, after \, 8 \, years}{B’s \, age \, after \, 8 \, years} = \frac{32}{28} = \frac{8}{7} \]
Final Answer: The ratio of their ages after 8 years will be \( \boxed{8:7} \).
19. Farah got married 8 years ago, Today her age is\[1\frac27\] times her age at the time of her marriage. At present her daughter’s age is one-sixth of her age. What was her daughter’s age 3 years ago ?
A. 4 Years
B. 3 Years
C. 6 Years
D. Cannot be determind
E. None of these
Answer: Option B Solution: Let’s break down the information and solve the problem step by step.
Step 1: Define Variables
Let:
– Farah’s age at the time of her marriage = \( x \) – Farah’s present age = \( 1 \frac{2}{7} \times x = \frac{9}{7} \times x \) – Farah’s daughter’s present age = \( y \)
We are also given that:
– Farah’s daughter’s present age is one-sixth of Farah’s present age: \[ y = \frac{1}{6} \times \left( \frac{9}{7} \times x \right) \]
Step 2: Express Farah’s Age 8 Years Ago
Since Farah got married 8 years ago, her age at the time of her marriage was \( x \), and her present age is \( \frac{9}{7} \times x \). Therefore, her age now is \( \frac{9}{7} \times x \), and we know this is 8 years after her marriage:
\[ \frac{9}{7} \times x = x + 8 \]
Step 3: Solve for \( x \)
Now, solve the equation for \( x \):
\[ \frac{9}{7}x = x + 8 \]
Multiply both sides by 7 to eliminate the fraction:
\[ 9x = 7x + 56 \]
Now, subtract \( 7x \) from both sides:
\[ 2x = 56 \]
Solve for \( x \):
\[ x = 28 \]
Step 4: Find Farah’s Present Age
Now that we know \( x = 28 \), we can find Farah’s present age:
Now, using the equation \( y = \frac{1}{6} \times \text{Farah’s present age} \):
\[ y = \frac{1}{6} \times 36 = 6 \]
So, Farah’s daughter is 6 years old now.
Step 6: Find the Daughter’s Age 3 Years Ago
To find the daughter’s age 3 years ago, subtract 3 from her current age:
\[ 6 – 3 = 3 \]
Final Answer:
Farah’s daughter was 3 years old three years ago.
#20. The age of a mother today is thrice that of her daughter. After 12 years , the age of the mother will be twice that of her daughter. The present age of the daughter is = ?
A. 12 Years
B. 14 Years
C. 16 Years
D. 18 Years
Answer: Option A Solution: Let’s define the following:
– The mother’s present age = \( M \) – The daughter’s present age = \( D \)
Step 1: Set up the equations based on the given information
We are given the following:
1. The mother’s present age is three times the daughter’s present age: \[ M = 3D \quad \text{(Equation 1)} \]
2. After 12 years, the mother’s age will be twice the daughter’s age: \[ M + 12 = 2(D + 12) \quad \text{(Equation 2)} \]
Step 2: Solve the system of equations
Substitute \( M = 3D \) from Equation 1 into Equation 2:
\[ 3D + 12 = 2(D + 12) \]
Simplify the equation:
\[ 3D + 12 = 2D + 24 \]
Now, subtract \( 2D \) from both sides:
\[ D + 12 = 24 \]
Subtract 12 from both sides:
\[ D = 12 \]
Step 3: Find the mother’s present age
Now that we know \( D = 12 \), substitute this value into Equation 1:
\[ M = 3D = 3 \times 12 = 36 \]
Final Answer: The present age of the daughter is 12 years.
#21. The present age of Mr. Sanyal is three times the age of his son. Six years hence , the ratio of their ages will be 5 : 2. What is the present age of Mr. Sanyal ?
A. 48 years
B. 50 years
C. 54 years
D. 60 years
E. None of these
Answer: Option C Solution: Let’s denote the present age of Mr. Sanyal as \( S \) and the present age of his son as \( s \).
Step 1: Set up the equations based on the given information
We are told the following:
1. The present age of Mr. Sanyal is three times the age of his son: \[ S = 3s \quad \text{(Equation 1)} \]
2. Six years hence, the ratio of their ages will be 5:2: \[ \frac{S + 6}{s + 6} = \frac{5}{2} \quad \text{(Equation 2)} \]
Step 2: Solve the system of equations
Substitute \( S = 3s \) from Equation 1 into Equation 2:
\[ \frac{3s + 6}{s + 6} = \frac{5}{2} \]
Step 3: Cross-multiply to solve for \( s \)
Cross-multiply:
\[ 2(3s + 6) = 5(s + 6) \]
Simplify both sides:
\[ 6s + 12 = 5s + 30 \]
Step 4: Solve for \( s \)
Now, subtract \( 5s \) from both sides:
\[ 6s – 5s + 12 = 30 \] \[ s + 12 = 30 \]
Subtract 12 from both sides:
\[ s = 18 \]
Step 5: Find Mr. Sanyal’s present age
Now that we know \( s = 18 \), substitute this into Equation 1 to find \( S \):
\[ S = 3s = 3 \times 18 = 54 \]
Final Answer: The present age of Mr. Sanyal is 54 years.
#22. The average of the ages of a man and his daughter is 34 years. If the respective ratio of their ages four years from now is 14 : 5,what is daughter’s present age ?
A. 10 years
B. 12 years
C. 18 years
D. Cannot be determined
E. None of these
Answer: Option E Solution: Let the present age of the man be \( M \) and the present age of the daughter be \( D \).
Step 1: Use the average age condition
The average age of the man and his daughter is given as 34 years. Therefore, we can write the equation:
\[ \frac{M + D}{2} = 34 \]
Multiply both sides by 2 to eliminate the fraction:
\[ M + D = 68 \quad \text{(Equation 1)} \]
Step 2: Use the ratio of their ages four years from now
The ratio of their ages 4 years from now is given as 14:5. So, we can write the equation:
\[ \frac{M + 4}{D + 4} = \frac{14}{5} \]
Cross-multiply:
\[ 5(M + 4) = 14(D + 4) \]
Simplify the equation:
\[ 5M + 20 = 14D + 56 \]
Now, subtract 20 from both sides:
\[ 5M = 14D + 36 \quad \text{(Equation 2)} \]
Step 3: Solve the system of equations
We now have the system of equations:
1. \( M + D = 68 \) 2. \( 5M = 14D + 36 \)
Solve for \( M \) from Equation 1:
From Equation 1, solve for \( M \):
\[ M = 68 – D \]
Substitute this into Equation 2:
Substitute \( M = 68 – D \) into Equation 2:
\[ 5(68 – D) = 14D + 36 \]
Simplify the equation:
\[ 340 – 5D = 14D + 36 \]
Now, subtract 36 from both sides:
\[ 304 – 5D = 14D \]
Now, add \( 5D \) to both sides:
\[ 304 = 19D \]
Solve for \( D \):
\[ D = \frac{304}{19} = 16 \]
Final Answer: The daughter’s present age is 16 years.
#23. The age of a father 10 years ago was thrice the age of his son. 10 years hence , the father’s age will be twice that of his son. The ratio of their present age is = ?
A. 8 : 5
B. 7 : 3
C. 9 : 5
D. 5 : 2
Answer: Option B Solution: Let the present age of the father be \( F \) and the present age of the son be \( S \).
Step 1: Use the information from 10 years ago
We are told that **10 years ago**, the father’s age was three times the son’s age:
From Equation 1: \[ F – 10 = 3(S – 10) \] \[ F – 10 = 3S – 30 \] \[ F = 3S – 20 \quad \text{(Equation 3)} \]
From Equation 2: \[ F + 10 = 2(S + 10) \] \[ F + 10 = 2S + 20 \] \[ F = 2S + 10 \quad \text{(Equation 4)} \]
Solve for \( F \) and \( S \):
Now, set the two expressions for \( F \) equal to each other:
\[ 3S – 20 = 2S + 10 \]
Subtract \( 2S \) from both sides:
\[ S – 20 = 10 \]
Add 20 to both sides:
\[ S = 30 \]
Now substitute \( S = 30 \) into either Equation 3 or Equation 4 to find \( F \). Using Equation 4:
\[ F = 2(30) + 10 = 60 + 10 = 70 \]
Step 4: Find the ratio of their present ages
The present age ratio of the father and the son is:
\[ \frac{F}{S} = \frac{70}{30} = \frac{7}{3} \]
Final Answer: The ratio of their present ages is 7:3.
#24. At present Suresh’s age is twice the age of his daughter. After 6 year from now, the ratio of the ages of Suresh and his daughter will be 23 : 13. What is the present age of Suresh ?
A. 36 years
B. 40 years
C. 46 years
D. Cannot be determined
E. None of these
Answer: Option B Solution: Let the present age of Suresh be \( S \) and the present age of his daughter be \( D \).
Step 1: Use the given relationship between their current ages
We are told that **Suresh’s age is twice the age of his daughter**:
\[ S = 2D \quad \text{(Equation 1)} \]
Step 2: Use the information after 6 years
After 6 years, the ratio of their ages will be 23:13:
\[ \frac{S + 6}{D + 6} = \frac{23}{13} \]
Step 3: Cross-multiply to solve for \( S \) and \( D \)
Cross-multiply:
\[ 13(S + 6) = 23(D + 6) \]
Expand both sides:
\[ 13S + 78 = 23D + 138 \]
Step 4: Substitute \( S = 2D \) from Equation 1 into the equation
Substitute \( S = 2D \) into the above equation:
\[ 13(2D) + 78 = 23D + 138 \]
Simplify the equation:
\[ 26D + 78 = 23D + 138 \]
Step 5: Solve for \( D \)
Now, subtract \( 23D \) from both sides:
\[ 3D + 78 = 138 \]
Subtract 78 from both sides:
\[ 3D = 60 \]
Divide by 3:
\[ D = 20 \]
Step 6: Find Suresh’s present age
Now that we know \( D = 20 \), substitute this into Equation 1 to find \( S \):
\[ S = 2D = 2 \times 20 = 40 \]
Final Answer: The present age of Suresh is 40 years.
#25. Ten years ago, a man was seven times as old as his son. Two years hence, twice his age will be equal to five times the age of his son. What is the present age of the son ?
A. 12 years
B. 13 years
C. 14 years
D. 15 years
Answer: Option C Solution: Let the present age of the man be \( M \) and the present age of the son be \( S \).
Step 1: Use the information from 10 years ago
We are told that **10 years ago**, the man was seven times as old as his son:
\[ M – 10 = 7(S – 10) \quad \text{(Equation 1)} \] Step 2: Use the information from 2 years hence
We are also told that **2 years hence**, twice the man’s age will be equal to five times the son’s age:
Final Answer: The present age of the son is 14 years.
#26. The ages of Samina and Suhana are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their ages?
A. 6 years
B. 8 years
C. 10 years
D. 12 years
E. None of these
Answer: Option B Solution: Let the present age of Samina be \( S \) and the present age of Suhana be \( H \).
Step 1: Use the given ratio of their present ages
We are told that the present ratio of their ages is 7:3, so:
\[ \frac{S}{H} = \frac{7}{3} \]
This implies:
\[ S = \frac{7}{3}H \quad \text{(Equation 1)} \]
Step 2: Use the ratio of their ages after 6 years
We are also told that **after 6 years**, the ratio of their ages will be 5:3, so:
\[ \frac{S + 6}{H + 6} = \frac{5}{3} \]
Step 3: Cross-multiply and simplify
Cross-multiply:
\[ 3(S + 6) = 5(H + 6) \]
Expand both sides:
\[ 3S + 18 = 5H + 30 \]
Now, subtract 18 from both sides:
\[ 3S = 5H + 12 \quad \text{(Equation 2)} \]
Step 4: Substitute \( S = \frac{7}{3}H \) from Equation 1 into Equation 2
Substitute \( S = \frac{7}{3}H \) into Equation 2:
\[ 3 \times \frac{7}{3}H = 5H + 12 \]
Simplify:
\[ 7H = 5H + 12 \]
Now, subtract \( 5H \) from both sides:
\[ 2H = 12 \]
Divide by 2:
\[ H = 6 \]
Step 5: Find Samina’s age
Now that we know \( H = 6 \), substitute this value into Equation 1 to find \( S \):
\[ S = \frac{7}{3} \times 6 = 14 \]
Step 6: Find the difference in their ages
The difference in their ages is:
\[ S – H = 14 – 6 = 8 \]
Final Answer: The difference in their ages is 8 years.
#27. The ages of A and B are presently in the ratio of 5 : 6 respectively. Six years hence, this ratio became 6 : 7 respectively. What was B’s age 5 years ago ?
A. 25 years
B. 30 years
C. 31 years
D. 36 years
E. None of these
Answer: Option C Solution: Let the present age of A be \( A \) and the present age of B be \( B \).
Step 1: Use the given ratio of their present ages
We are told that the present ratio of their ages is 5:6, so:
\[ \frac{A}{B} = \frac{5}{6} \]
This implies:
\[ A = \frac{5}{6}B \quad \text{(Equation 1)} \]
Step 2: Use the ratio of their ages after 6 years
We are also told that **six years hence**, the ratio of their ages will be 6:7:
\[ \frac{A + 6}{B + 6} = \frac{6}{7} \]
Step 3: Cross-multiply and simplify
Cross-multiply:
\[ 7(A + 6) = 6(B + 6) \]
Expand both sides:
\[ 7A + 42 = 6B + 36 \]
Now, subtract 36 from both sides:
\[ 7A + 6 = 6B \]
Step 4: Substitute \( A = \frac{5}{6}B \) from Equation 1 into the equation
Substitute \( A = \frac{5}{6}B \) into the equation:
\[ 7 \times \frac{5}{6}B + 6 = 6B \]
Simplify:
\[ \frac{35}{6}B + 6 = 6B \]
Multiply the entire equation by 6 to eliminate the fraction:
\[ 35B + 36 = 36B \]
Now, subtract 35B from both sides:
\[ 36 = B \]
Step 5: Find B’s age 5 years ago
We have \( B = 36 \), so the present age of B is 36 years. To find B’s age 5 years ago:
\[ B – 5 = 36 – 5 = 31 \]
Final Answer: B’s age 5 years ago was 31 years.
#28. The sum of the ages of a daughter and her mother is 56 years. After 4 years, the age of the mother will be three times that of the daughter. At present their ages are ?
A. 10 years, 46 years
B. 12 years, 44 years
C. 11 years, 45 years
D. 13 years, 43 years
Answer: Option B Solution: Let the present age of the daughter be \( D \) and the present age of the mother be \( M \).
Step 1: Use the given sum of their ages
We are told that the sum of their ages is 56 years:
\[ D + M = 56 \quad \text{(Equation 1)} \]
Step 2: Use the given condition after 4 years
After 4 years, the mother’s age will be three times the daughter’s age:
\[ M + 4 = 3(D + 4) \quad \text{(Equation 2)} \]
Step 3: Solve the system of equations
Expand Equation 2:
\[ M + 4 = 3(D + 4) \] \[ M + 4 = 3D + 12 \] \[ M = 3D + 8 \quad \text{(Equation 3)} \]
Substitute Equation 3 into Equation 1:
Substitute \( M = 3D + 8 \) into \( D + M = 56 \):
\[ D + (3D + 8) = 56 \]
Simplify:
\[ 4D + 8 = 56 \]
Subtract 8 from both sides:
\[ 4D = 48 \]
Now divide by 4:
\[ D = 12 \]
Step 4: Find the mother’s age
Now that we know \( D = 12 \), substitute this value into Equation 1 to find \( M \):
\[ M + 12 = 56 \] \[ M = 44 \]
Final Answer: The present age of the daughter is 12 years, and the present age of the mother is 44 years.
#29. The present age of son is half of the present age of his mother. Ten years ago, his mother’s age was thrice the age of her son. What is the present age of the son ?
A. 20 years
B. 25 years
C. 30 years
D. 40 years
Answer: Option A Solution: Let the present age of the son be \( S \) and the present age of the mother be \( M \).
Step 1: Use the relationship between their current ages
We are told that the **son’s present age is half of his mother’s present age**, so:
\[ S = \frac{1}{2}M \quad \text{(Equation 1)} \]
Step 2: Use the condition from 10 years ago
We are also told that **10 years ago**, the mother’s age was three times the son’s age. So, 10 years ago, the mother’s age was \( M – 10 \) and the son’s age was \( S – 10 \), and the relationship was:
Final Answer: The present age of the son is 20 years.
#30. Rajan got married 8 years ago. His present age is \[\frac65\] times his age at the time of his marriage. Rajan’s sister was 10 years younger to him at the time of his marrige. The age of Rajan’s sister is = ?
A. 32 years
B. 36 years
C. 38 years
D. 40 years
Answer: Option C Solution: Let’s denote Rajan’s present age by \( R \) and his age at the time of marriage by \( M \).
Step 1: Use the given ratio of ages
We are told that Rajan’s present age is \( \frac{6}{5} \) times his age at the time of marriage. This gives us the equation:
\[ R = \frac{6}{5} M \quad \text{(Equation 1)} \]
Also, Rajan got married 8 years ago, so:
\[ M = R – 8 \quad \text{(Equation 2)} \]
Step 2: Use the information about Rajan’s sister
We are told that Rajan’s sister was 10 years younger than him at the time of his marriage. So, her age at the time of his marriage was:
From Equation 2, \( M = R – 8 \), substitute this into Equation 1:
\[ R = \frac{6}{5} (R – 8) \]
Now, solve for \( R \):
\[ R = \frac{6}{5}R – \frac{6}{5} \times 8 \] \[ R = \frac{6}{5}R – \frac{48}{5} \]
Multiply both sides by 5 to eliminate the fraction:
\[ 5R = 6R – 48 \]
Now, subtract \( 5R \) from both sides:
\[ 0 = R – 48 \]
So, \( R = 48 \).
Step 4: Find Rajan’s sister’s age
Now that we know \( R = 48 \), substitute this value into \( M = R – 8 \) to find Rajan’s age at the time of marriage:
\[ M = 48 – 8 = 40 \]
So, Rajan’s sister’s age at the time of his marriage was:
\[ M – 10 = 40 – 10 = 30 \]
Thus, her present age is:
\[ 30 + 8 = 38 \]
Final Answer: Rajan’s sister’s present age is 38 years.
#31. A couple has a son and a daughter. The age of the father is four times that of the son and the age of the daughter is one-third of that of her mother. The wife is 6 years younger to her husband and the sister is 3 years older then her brother. The mother’s age is = ?
A. 42 years
B. 48 years
C. 54 years
D. 63 years
Answer: Option C Solution: Let’s denote the ages as follows:
– The son’s age = \( S \) – The father’s age = \( F \) – The daughter’s age = \( D \) – The mother’s age = \( M \)
Step 1: Use the relationship between the father’s and son’s ages We are told that the father’s age is four times that of the son:
\[ F = 4S \quad \text{(Equation 1)} \]
Step 2: Use the relationship between the mother’s and daughter’s ages We are also told that the daughter’s age is one-third of the mother’s age:
\[ D = \frac{1}{3}M \quad \text{(Equation 2)} \]
Step 3: Use the relationship between the wife and husband’s ages We are told that the wife is 6 years younger than her husband:
\[ M = F – 6 \quad \text{(Equation 3)} \]
Step 4: Use the relationship between the sister and brother’s ages We are told that the sister (daughter) is 3 years older than her brother (son):
\[ D = S + 3 \quad \text{(Equation 4)} \]
Step 5: Solve the system of equations
Step 5a: Substitute Equation 1 into Equation 3
From Equation 1, \( F = 4S \), substitute this into Equation 3:
\[ M = 4S – 6 \quad \text{(Equation 5)} \]
Step 5b: Substitute Equation 5 into Equation 2
From Equation 5, \( M = 4S – 6 \), substitute this into Equation 2:
\[ D = \frac{1}{3}(4S – 6) \]
Simplify:
\[ D = \frac{4S – 6}{3} \quad \text{(Equation 6)} \]
Step 5c: Substitute Equation 4 into Equation 6
From Equation 4, \( D = S + 3 \), substitute this into Equation 6:
\[ S + 3 = \frac{4S – 6}{3} \]
Multiply both sides by 3 to eliminate the fraction:
\[ 3(S + 3) = 4S – 6 \]
Expand both sides:
\[ 3S + 9 = 4S – 6 \]
Now, subtract \( 3S \) from both sides:
\[ 9 = S – 6 \]
Add 6 to both sides:
\[ S = 15 \]
Step 6: Find the mother’s age
Now that we know \( S = 15 \), substitute this into Equation 5 to find \( M \):
\[ M = 4S – 6 = 4(15) – 6 = 60 – 6 = 54 \]
Final Answer: The mother’s age is 54 years.
#32. Reenu’s father was 38 years of age when she was born while her mother was 36 years old when her brother 4 years younger to her was born. What is the difference between the ages of her parents ?
A. 2 years
B. 4 years
C. 6 years
D. 8 years
E. None of these
Answer: Option C Solution: Let’s break down the information given and solve the problem step by step.
Step 1: Define the variables – Let Reenu’s current age be \( R \). – Let Reenu’s father’s current age be \( F \). – Let Reenu’s mother’s current age be \( M \). – Reenu’s brother is 4 years younger than her, so his age is \( R – 4 \).
Step 2: Use the information about the father’s age We are told that Reenu’s father was 38 years old when she was born. This means the difference in their ages is 38 years. Thus:
\[ F = R + 38 \quad \text{(Equation 1)} \]
Step 3: Use the information about the mother’s age We are told that Reenu’s mother was 36 years old when Reenu’s brother was born. Since Reenu’s brother is 4 years younger than Reenu, Reenu’s mother was 36 years old \( R – 4 \) years ago. This means:
\[ M = (R – 4) + 36 \quad \text{(Equation 2)} \] \[ M = R + 32 \quad \text{(Equation 3)} \]
Step 4: Find the difference between the parents’ ages Now we need to find the difference between Reenu’s mother’s and father’s ages. Using Equations 1 and 3, we can compute:
\[ F – M = (R + 38) – (R + 32) \] \[ F – M = 38 – 32 \] \[ F – M = 6 \]
Final Answer: The difference between the ages of Reenu’s parents is 6 years.
#33. A man was asked to state his age in years. His reply was, “Take my age 3 years hence, multiply it by 3 and then subtract 3 times my age 3 years ago and you will know how old I am”. What is the age of the man ?
A. 18 years
B. 20 years
C. 24 years
D. 32 years
Answer: Option A Solution: Let the man’s present age be \( x \) years.
Step 1: Set up the equation based on the given information
– The man’s age **3 years hence** will be \( x + 3 \). – The man’s age **3 years ago** was \( x – 3 \).
Now, according to the problem, we are told that:
– Multiply the man’s age 3 years hence by 3: \( 3(x + 3) \). – Subtract 3 times the man’s age 3 years ago: \( 3(x – 3) \).
We are given that this entire expression equals the man’s present age. So, we can write the equation:
\[ 3(x + 3) – 3(x – 3) = x \]
Step 2: Simplify the equation
Expand both terms:
\[ 3(x + 3) = 3x + 9 \] \[ 3(x – 3) = 3x – 9 \]
Now, substitute these into the equation:
\[ (3x + 9) – (3x – 9) = x \]
Simplify the left side:
\[ 3x + 9 – 3x + 9 = x \] \[ 18 = x \]
Final Answer: The man’s present age is 18 years.
#34. The ratio of a man’s age and his son’s age is 7 : 3 and the product of their ages is 756. The ratio of their ages after 6 years will be ?
A. 5 : 2
B. 2 : 1
C. 11 : 7
D. 13 : 9
E. None of these
Answer: Option B Solution: Let the present age of the man be \( 7x \) and the present age of his son be \( 3x \), where \( x \) is a constant.
Step 1: Use the information about the product of their ages We are told that the product of their ages is 756. So, we can write the equation:
\[ (7x) \times (3x) = 756 \]
Simplifying the left side:
\[ 21x^2 = 756 \]
Now, divide both sides by 21:
\[ x^2 = \frac{756}{21} = 36 \]
Taking the square root of both sides:
\[ x = 6 \]
Step 2: Find the present ages of the man and his son Now that we know \( x = 6 \), we can find the present ages:
– The man’s present age is \( 7x = 7 \times 6 = 42 \) years. – The son’s present age is \( 3x = 3 \times 6 = 18 \) years.
Step 3: Find the ratio of their ages after 6 years After 6 years:
– The man’s age will be \( 42 + 6 = 48 \) years. – The son’s age will be \( 18 + 6 = 24 \) years.
The ratio of their ages after 6 years will be:
\[ \frac{48}{24} = 2 \]
Final Answer: The ratio of their ages after 6 years will be 2 : 1.
#35. The ratio between the parents ages of A and B is 5 : 3 respectively. The ratio between A’s age 4 years ago and B’s 4 years hence is 1 : 1. What is the ratio between A’s age 4 years hence and B’s age 4 years ago?
A. 1 : 3
B. 3 : 1
C. 2 : 1
D. 4 : 1
E. None of these
Answer: Option B Solution: Let’s define the present ages of A and B as follows:
– Let A’s present age be \( A \). – Let B’s present age be \( B \).
Step 1: Use the given ratio of parents’ ages We are told that the ratio between the parents’ ages of A and B is 5:3. This means:
\[ \frac{A}{B} = \frac{5}{3} \]
So, we can express A’s age in terms of B’s age:
\[ A = \frac{5}{3}B \quad \text{(Equation 1)} \]
Step 2: Use the ratio of A’s age 4 years ago and B’s age 4 years hence We are told that the ratio between A’s age 4 years ago and B’s age 4 years hence is 1:1. This means:
\[ \frac{A – 4}{B + 4} = 1 \]
Simplifying the equation:
\[ A – 4 = B + 4 \]
\[ A = B + 8 \quad \text{(Equation 2)} \]
Step 3: Solve the system of equations Now we have two equations:
1. \( A = \frac{5}{3}B \) 2. \( A = B + 8 \)
We can set these two expressions for \( A \) equal to each other:
\[ \frac{5}{3}B = B + 8 \]
Multiply both sides by 3 to eliminate the fraction:
\[ 5B = 3B + 24 \]
Now, subtract \( 3B \) from both sides:
\[ 2B = 24 \]
So,
\[ B = 12 \]
Step 4: Find A’s present age Substitute \( B = 12 \) into Equation 1:
\[ A = \frac{5}{3} \times 12 = 20 \]
So, A’s present age is 20 years and B’s present age is 12 years.
Step 5: Find the ratio between A’s age 4 years hence and B’s age 4 years ago – A’s age 4 years hence will be \( 20 + 4 = 24 \). – B’s age 4 years ago was \( 12 – 4 = 8 \).
The ratio of A’s age 4 years hence to B’s age 4 years ago is:
\[ \frac{24}{8} = 3 \]
Final Answer: The ratio between A’s age 4 years hence and B’s age 4 years ago is 3 : 1.
#36. The ratio between the ages of Neelam and Shiny is 5 : 6 respectively. If the ratio between the one-third age of Neelam and half of Shiny’s age is 5 : 9, then what is Shiny’s age = ?
A. 25 years
B. 30 years
C. 36 years
D. Cannot be determined
E. None of these
Answer: Option D Solution: Let Neelam’s present age be \( N \) and Shiny’s present age be \( S \).
Step 1: Use the first ratio of their ages We are given that the ratio of their ages is 5:6. This means:
\[ \frac{N}{S} = \frac{5}{6} \]
So, we can express Neelam’s age in terms of Shiny’s age:
\[ N = \frac{5}{6}S \quad \text{(Equation 1)} \]
Step 2: Use the second ratio involving their ages We are told that the ratio of one-third of Neelam’s age to half of Shiny’s age is 5:9. This means:
Step 3: Solve for \( N \) and \( S \) We now have the equation:
\[ \frac{2N}{3S} = \frac{5}{9} \]
Cross-multiply to eliminate the fractions:
\[ 2N \times 9 = 5 \times 3S \]
\[ 18N = 15S \]
Now, substitute \( N = \frac{5}{6}S \) from Equation 1 into this equation:
\[ 18 \times \frac{5}{6}S = 15S \]
Simplify the left side:
\[ 15S = 15S \]
This confirms the equation is correct.
Step 4: Find Shiny’s age The ratio holds true, and it does not provide any contradictions, so we can conclude that the relationship is valid for any value satisfying the given ratio. However, this problem does not give a specific value for \( S \). Based on the ratios provided, Shiny’s age can be any value that satisfies the given proportions.
Given this, if you’d like a more specific calculation or more details, let me know.
#37. 18 years ago, a man was three times as old as his son. Now, the man is twice as old as his son. The sum of the present ages of the man and his son is = ?
A. 54 years
B. 72 years
C. 105 years
D. 108 years
Answer: Option D Solution: Let’s define the present ages of the man and his son as follows:
– Let the present age of the man be \( M \). – Let the present age of the son be \( S \).
Step 1: Use the information from 18 years ago We are told that 18 years ago, the man was three times as old as his son. This means:
\[ M – 18 = 3(S – 18) \]
Step 2: Use the information from the present day We are also told that currently, the man is twice as old as his son. This gives the equation:
\[ M = 2S \]
Step 3: Solve the system of equations Now we have the system of equations:
1. \( M – 18 = 3(S – 18) \) 2. \( M = 2S \)
Substitute \( M = 2S \) into the first equation:
\[ 2S – 18 = 3(S – 18) \]
Expand both sides:
\[ 2S – 18 = 3S – 54 \]
Now, subtract \( 2S \) from both sides:
\[ -18 = S – 54 \]
Add 54 to both sides:
\[ S = 36 \]
Step 4: Find the man’s age Now that we know \( S = 36 \), substitute this value into \( M = 2S \):
\[ M = 2 \times 36 = 72 \]
Step 5: Find the sum of their present ages The sum of the present ages of the man and his son is:
\[ M + S = 72 + 36 = 108 \]
Final Answer: The sum of the present ages of the man and his son is 108 years.
#38. The age of a man 10 years ago was thrice the age of his son. 10 years hence, the man’s age will be twice the age of his son. The ratio of their present ages is = ?
A. 5 : 2
B. 7 : 3
C. 9 : 2
D. 13 : 4
Answer: Option B Solution: Let’s define the present ages of the man and his son as follows:
– Let the present age of the man be \( M \). – Let the present age of the son be \( S \).
Step 1: Use the information from 10 years ago We are told that 10 years ago, the man’s age was three times the age of his son. This means:
\[ M – 10 = 3(S – 10) \]
Step 2: Use the information from 10 years hence We are also told that in 10 years, the man’s age will be twice the age of his son. This gives the equation:
\[ M + 10 = 2(S + 10) \]
Step 3: Solve the system of equations Now we have the system of equations:
1. \( M – 10 = 3(S – 10) \) 2. \( M + 10 = 2(S + 10) \)
Expand and simplify both equations:
From the first equation:
\[ M – 10 = 3(S – 10) \] \[ M – 10 = 3S – 30 \] \[ M = 3S – 20 \quad \text{(Equation 1)} \]
From the second equation:
\[ M + 10 = 2(S + 10) \] \[ M + 10 = 2S + 20 \] \[ M = 2S + 10 \quad \text{(Equation 2)} \]
Step 4: Solve for \( M \) and \( S \) Now, equate the two expressions for \( M \):
\[ 3S – 20 = 2S + 10 \]
Subtract \( 2S \) from both sides:
\[ S – 20 = 10 \]
Add 20 to both sides:
\[ S = 30 \]
Step 5: Find the man’s present age Now that we know \( S = 30 \), substitute this value into \( M = 2S + 10 \) (Equation 2):
\[ M = 2 \times 30 + 10 = 60 + 10 = 70 \]
Step 6: Find the ratio of their present ages The ratio of their present ages is:
\[ \frac{M}{S} = \frac{70}{30} = \frac{7}{3} \]
Final Answer: The ratio of their present ages is 7 : 3.
#39. Tanya’s grandfather was 8 times older to her 16 years ago. He would be 3 times of her age 8 years from now. 8 years ago, what was the ratio of Tanya’s age to that of her grandfather ?
A. 1 : 2
B. 1 : 5
C. 3 : 8
D. None of these
Answer: Option D Solution: Let’s define the present ages of Tanya and her grandfather as follows:
– Let Tanya’s present age be \( T \). – Let her grandfather’s present age be \( G \).
Step 1: Use the information from 16 years ago We are told that 16 years ago, Tanya’s grandfather was 8 times older than her. This means:
\[ G – 16 = 8(T – 16) \]
Step 2: Use the information from 8 years from now We are also told that 8 years from now, Tanya’s grandfather will be 3 times her age. This gives the equation:
\[ G + 8 = 3(T + 8) \]
Step 3: Solve the system of equations Now we have the system of equations:
1. \( G – 16 = 8(T – 16) \) 2. \( G + 8 = 3(T + 8) \)
Expand and simplify both equations:
From the first equation:
\[ G – 16 = 8(T – 16) \] \[ G – 16 = 8T – 128 \] \[ G = 8T – 112 \quad \text{(Equation 1)} \]
From the second equation:
\[ G + 8 = 3(T + 8) \] \[ G + 8 = 3T + 24 \] \[ G = 3T + 16 \quad \text{(Equation 2)} \]
Step 4: Solve for \( G \) and \( T \) Now, equate the two expressions for \( G \):
\[ 8T – 112 = 3T + 16 \]
Subtract \( 3T \) from both sides:
\[ 5T – 112 = 16 \]
Add 112 to both sides:
\[ 5T = 128 \]
Now, divide both sides by 5:
\[ T = \frac{128}{5} = 25.6 \]
Step 5: Find the grandfather’s present age Now that we know \( T = 25.6 \), substitute this value into \( G = 3T + 16 \) (Equation 2):
\[ G = 3 \times 25.6 + 16 = 76.8 + 16 = 92.8 \]
Step 6: Find the ratio of their ages 8 years ago Eight years ago:
– Tanya’s age was \( 25.6 – 8 = 17.6 \). – Her grandfather’s age was \( 92.8 – 8 = 84.8 \).
The ratio of their ages 8 years ago is:
\[ \frac{17.6}{84.8} = \frac{2}{9} \]
Final Answer: The ratio of Tanya’s age to her grandfather’s age 8 years ago was 2 : 9.
#40. 6 years ago , the ratio of the ages of Kunal and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar’s age at present ?
A. 16 years
B. 18 years
C. 20 years
D. Cannot be determined
E. None of these
Answer: Option A Solution: Let’s use algebra to solve the problem.
Let Kunal’s age 6 years ago be \( 6x \), and Sagar’s age 6 years ago be \( 5x \), where \( x \) is a constant multiplier.
Now, we know that the ratio of their ages 6 years ago was 6:5, and the current ages of Kunal and Sagar will be:
– Kunal’s current age = \( 6x + 6 \) – Sagar’s current age = \( 5x + 6 \)
According to the problem, 4 years from now, the ratio of their ages will be 11:10. So:
– Kunal’s age in 4 years = \( 6x + 6 + 4 = 6x + 10 \) – Sagar’s age in 4 years = \( 5x + 6 + 4 = 5x + 10 \)
We are given that the ratio of their ages in 4 years is 11:10. Therefore, we have the equation:
\[ \frac{6x + 10}{5x + 10} = \frac{11}{10} \]
Now let’s solve this equation to find \( x \).
Cross-multiply:
\[ 10(6x + 10) = 11(5x + 10) \]
Simplify both sides:
\[ 60x + 100 = 55x + 110 \]
Subtract \( 55x \) from both sides:
\[ 5x + 100 = 110 \]
Subtract 100 from both sides:
\[ 5x = 10 \]
Solve for \( x \):
\[ x = 2 \]
Now that we know \( x = 2 \), we can find Sagar’s current age:
#41. Sneh’s age is \[\frac16\] th of her father’s age. Sneh’s father’s age will be twice of Vimal’s age after 10 years. If Vimal’s 8th birthday was celebrated 2 years ago,then what is Sneh’s present age ?
\[A.\;\frac23years\]
B. 10 years
C. 12 years
D. 15 years
E. None of these
Answer: Option A Solution: Let’s break this problem down step by step using algebra.
Step 1: Define variables – Let Sneh’s present age be \( S \). – Let her father’s present age be \( F \). – Let Vimal’s present age be \( V \).
Step 2: Express relations based on the problem – According to the first part, Sneh’s age is \(\frac{1}{6}\) of her father’s age: \[ S = \frac{1}{6} F \quad \text{(Equation 1)} \]
– The second part tells us that Sneh’s father’s age will be twice Vimal’s age after 10 years: \[ F + 10 = 2(V + 10) \quad \text{(Equation 2)} \]
– Vimal’s 8th birthday was celebrated 2 years ago, so Vimal’s present age is: \[ V = 8 + 2 = 10 \quad \text{(Equation 3)} \]
Step 3: Solve the equations Substitute \( V = 10 \) from Equation 3 into Equation 2:
\[ F + 10 = 2(10 + 10) \] \[ F + 10 = 2 \times 20 \] \[ F + 10 = 40 \] \[ F = 40 – 10 = 30 \]
Now, substitute \( F = 30 \) into Equation 1 to find Sneh’s age:
\[ S = \frac{1}{6} \times 30 = 5 \]
Conclusion Sneh’s present age is 5 years.
#42. The ages of Sulekha and Arunima are in the ratio 9 : 8 respectively. After 5 years, the ratio of their ages will be 10 : 9. What is the difference in their ages ?
A. 4 years
B. 5 years
C. 6 years
D. 7 years
E. None of these
Answer: Option B Solution: Let’s use algebra to solve this problem.
Step 1: Define variables – Let the present age of Sulekha be \( 9x \). – Let the present age of Arunima be \( 8x \), where \( x \) is a constant multiplier.
Step 2: Set up the equation for the ages after 5 years After 5 years, their ages will be:
– Sulekha’s age after 5 years = \( 9x + 5 \) – Arunima’s age after 5 years = \( 8x + 5 \)
We are given that the ratio of their ages after 5 years will be 10:9. So, we can write the equation:
\[ \frac{9x + 5}{8x + 5} = \frac{10}{9} \]
Step 3: Solve the equation Now, let’s cross-multiply to solve for \( x \):
\[ 9(9x + 5) = 10(8x + 5) \]
Expand both sides:
\[ 81x + 45 = 80x + 50 \]
Subtract \( 80x \) from both sides:
\[ x + 45 = 50 \]
Subtract 45 from both sides:
\[ x = 5 \]
Step 4: Calculate the difference in their ages The present age of Sulekha is \( 9x = 9 \times 5 = 45 \) years.
The present age of Arunima is \( 8x = 8 \times 5 = 40 \) years.
The difference in their ages is:
\[ 45 – 40 = 5 \text{ years} \]
Conclusion The difference in their ages is 5 years.
#43. X’s age 3 years ago was three times the present age of Y. At present Z’s age is twice the age of Y. Also Z is 12 years younger than X. What is the present age of Z ?
A. 15 years
B. 24 years
C. 12 years
D. 18 years
E. 6 years
Answer: Option D Solution: Let’s break down the problem and use algebra to find the present age of Z.
Step 1: Define variables – Let the present age of \( X \) be \( X \). – Let the present age of \( Y \) be \( Y \). – Let the present age of \( Z \) be \( Z \).
Step 2: Translate the given information into equations
1. **X’s age 3 years ago was three times the present age of Y:**
\[ X – 3 = 3Y \quad \text{(Equation 1)} \]
2. **Z’s age is twice the age of Y:**
\[ Z = 2Y \quad \text{(Equation 2)} \]
3. **Z is 12 years younger than X:**
\[ Z = X – 12 \quad \text{(Equation 3)} \]
Step 3: Solve the system of equations
From Equation 2, we know that:
\[ Z = 2Y \]
Substitute this expression for \( Z \) into Equation 3:
\[ 2Y = X – 12 \]
Now, solve for \( X \):
\[ X = 2Y + 12 \quad \text{(Equation 4)} \]
Next, substitute Equation 4 into Equation 1:
\[ (2Y + 12) – 3 = 3Y \]
Simplify the equation:
\[ 2Y + 9 = 3Y \]
Subtract \( 2Y \) from both sides:
\[ 9 = Y \]
So, \( Y = 9 \).
Step 4: Find Z’s age
Now that we know \( Y = 9 \), substitute this value into Equation 2:
\[ Z = 2Y = 2 \times 9 = 18 \]
Conclusion
The present age of \( Z \) is 18 years.
#44. Eight year ago, Poorvi’s age was equal to the sum of the present ages of her one son and one daughter. Five years hence, the respective ratio between the ages of her daughter and her son that time will be 7 : 6. If Poorvi’s husband is 7 years elder to her and his present age is three times the present age of their son, what is the present age of the daughter ? (in year)
A. 15 years
B. 23 years
C. 19 years
D. 27 years
E. 13 years
Answer: Option B Solution: Let’s define the variables for this problem:
– Let **Poorvi’s present age** be \( P \). – Let **her son’s present age** be \( S \). – Let **her daughter’s present age** be \( D \). – **Poorvi’s husband’s present age** is \( P_H \).
Step 1: Translate the given information into equations
1. **Eight years ago, Poorvi’s age was equal to the sum of the present ages of her son and daughter:**
\[ P – 8 = S + D \quad \text{(Equation 1)} \]
2. **Five years from now, the ratio of her daughter’s age to her son’s age will be 7:6:**
From Equation 3 and Equation 4, we can equate the expressions for \( P_H \):
\[ P + 7 = 3S \]
Solve for \( P \):
\[ P = 3S – 7 \quad \text{(Equation 5)} \]
Substitute Equation 5 into Equation 1:
\[ (3S – 7) – 8 = S + D \]
Simplify:
\[ 3S – 15 = S + D \]
Subtract \( S \) from both sides:
\[ 2S – 15 = D \quad \text{(Equation 6)} \]
Now, substitute \( D = 2S – 15 \) from Equation 6 into Equation 2:
\[ \frac{(2S – 15) + 5}{S + 5} = \frac{7}{6} \]
Simplify the numerator:
\[ \frac{2S – 10}{S + 5} = \frac{7}{6} \]
Cross-multiply:
\[ 6(2S – 10) = 7(S + 5) \]
Expand both sides:
\[ 12S – 60 = 7S + 35 \]
Subtract \( 7S \) from both sides:
\[ 5S – 60 = 35 \]
Add 60 to both sides:
\[ 5S = 95 \]
Solve for \( S \):
\[ S = 19 \]
Step 3: Find the daughter’s age
Now that we know \( S = 19 \), substitute it into Equation 6 to find \( D \):
\[ D = 2(19) – 15 = 38 – 15 = 23 \]
Conclusion
The present age of Poorvi’s daughter is 23 years.
#45. The sum of present ages of a father and his son is 8 years more than the present age of the mother. The mother is 22 years older than the son. What will be the age of the father after 4 years?
A. 34 years
B. 36 years
C. 40 years
D. 38 years
E. 28 years
Answer: Option A Solution: Let’s define the variables:
– Let the **father’s present age** be \( F \). – Let the **son’s present age** be \( S \). – Let the **mother’s present age** be \( M \).
Step 1: Translate the given information into equations
1. **The sum of the present ages of the father and the son is 8 years more than the present age of the mother:**
\[ F + S = M + 8 \quad \text{(Equation 1)} \]
2. **The mother is 22 years older than the son:**
\[ M = S + 22 \quad \text{(Equation 2)} \]
Step 2: Solve the equations
Substitute Equation 2 into Equation 1:
\[ F + S = (S + 22) + 8 \]
Simplify:
\[ F + S = S + 30 \]
Subtract \( S \) from both sides:
\[ F = 30 \]
Step 3: Find the father’s age after 4 years
Since the father’s present age is \( F = 30 \), after 4 years, his age will be:
\[ F + 4 = 30 + 4 = 34 \]
Conclusion
The father’s age after 4 years will be 34 years.
#46. Rahul is as much younger than Sagar as he is older than Purav. If the sum of the ages of Purav and Sagar is 66 years, and Sagar’s age is 48 years, then what is the difference between Rahul and Purav’s age ? ( in years)
A. 18
B. 15
C. 16
D. 20
E. Cannot be determined
Answer: Option B Solution: Let’s break down the problem step by step using algebra.
Step 1: Define the variables – Let **Rahul’s age** be \( R \). – Let **Sagar’s age** be \( S \). – Let **Purav’s age** be \( P \).
From the problem, we know:
– Sagar’s age is 48 years: \( S = 48 \). – The sum of Purav and Sagar’s ages is 66 years: \( P + S = 66 \). – Rahul is as much younger than Sagar as he is older than Purav. This means the difference between Rahul and Sagar’s age is the same as the difference between Rahul and Purav’s age.
Step 2: Set up the equation based on the given condition Since Rahul is as much younger than Sagar as he is older than Purav, we can write:
\[ S – R = R – P \quad \text{(Equation 1)} \]
This means the difference between Sagar’s and Rahul’s ages is equal to the difference between Rahul’s and Purav’s ages.
Step 3: Use the sum of Purav’s and Sagar’s ages From the sum of Purav’s and Sagar’s ages, we know:
\[ P + S = 66 \]
Substitute \( S = 48 \) into this equation:
\[ P + 48 = 66 \]
Solve for \( P \):
\[ P = 66 – 48 = 18 \]
Step 4: Solve for Rahul’s age Now, substitute \( P = 18 \) and \( S = 48 \) into Equation 1:
\[ 48 – R = R – 18 \]
Simplify this equation:
\[ 48 + 18 = 2R \]
\[ 66 = 2R \]
Solve for \( R \):
\[ R = \frac{66}{2} = 33 \]
Step 5: Find the difference between Rahul and Purav’s ages Now that we know \( R = 33 \) and \( P = 18 \), the difference between Rahul’s and Purav’s ages is:
\[ R – P = 33 – 18 = 15 \]
Conclusion The difference between Rahul and Purav’s ages is 15 years.
#48. Ten years hence, the respective ratio between Simmi’s age ans Niti’s age will be 7 : 9. Two years ago, the respective ratio between Simmi’s and Niti’s age was 1 : 3. If Abhay is 4 years older then Niti, what is Abhay’s present age ? (in years)
A. 8
B. 4
C. 16
D. 12
E. 20
Answer: Option D Solution: Let’s break down the problem step by step.
Step 1: Define the variables – Let **Simmi’s present age** be \( S \). – Let **Niti’s present age** be \( N \). – Let **Abhay’s present age** be \( A \).
Step 2: Translate the given information into equations
1. **Ten years hence, the ratio between Simmi’s age and Niti’s age will be 7:9:** \[ \frac{S + 10}{N + 10} = \frac{7}{9} \quad \text{(Equation 1)} \]
2. **Two years ago, the ratio between Simmi’s and Niti’s age was 1:3:** \[ \frac{S – 2}{N – 2} = \frac{1}{3} \quad \text{(Equation 2)} \]
3. **Abhay is 4 years older than Niti:** \[ A = N + 4 \quad \text{(Equation 3)} \]
Step 4: Solve the system of equations (Equations 4 and 5)
We now have the system of equations: 1. \( 3S – N = 4 \) (Equation 4) 2. \( 9S – 7N = -20 \) (Equation 5)
Solve Equation 4 for \( N \):
\[ N = 3S – 4 \]
Substitute this into Equation 5:
\[ 9S – 7(3S – 4) = -20 \] Expand:
\[ 9S – 21S + 28 = -20 \] Simplify:
\[ -12S + 28 = -20 \] Subtract 28 from both sides:
\[ -12S = -48 \] Solve for \( S \):
\[ S = 4 \]
Find \( N \): Now that we know \( S = 4 \), substitute this into \( N = 3S – 4 \):
\[ N = 3(4) – 4 = 12 – 4 = 8 \]
Step 5: Find Abhay’s age From Equation 3, we know:
\[ A = N + 4 \] Substitute \( N = 8 \):
\[ A = 8 + 4 = 12 \]
Conclusion Abhay’s present age is 12 years.
#49. Eight years ago, the ratio of ages of A and B was 5 : 4. The ratio of their present ages is 6 : 5. What will be the sum (in years) of the ages of A and B after 7 years from now?
A. 90
B. 80
C. 102
D. 112
Answer: Option C Solution: Let’s define the variables:
– Let **A’s present age** be \( A \). – Let **B’s present age** be \( B \).
Step 1: Translate the given information into equations
1. **Eight years ago, the ratio of ages of A and B was 5:4:**
Now that we know \( B = 40 \), substitute this value into Equation 3:
\[ A = \frac{6}{5} \times 40 = 48 \]
Step 3: Find the sum of their ages after 7 years
In 7 years, A’s age will be:
\[ A + 7 = 48 + 7 = 55 \]
In 7 years, B’s age will be:
\[ B + 7 = 40 + 7 = 47 \]
So, the sum of their ages after 7 years will be:
\[ 55 + 47 = 102 \]
Conclusion
The sum of the ages of A and B after 7 years will be 102 years.
#50. The ratio of a man’s age to his father’s age is 4 : 5, and the ratio of his age to his son’s age is 6 : 1. Four years ago these ratios were 11 : 14 and 11 : 1, respectively. The ratio of the age of the grandfather to that of the grandson 12 years from now will be:
A. 12 : 5
B. 14 : 3
C. 18 : 5
D. 18 : 7
Answer: Option C Solution: Let’s define the variables:
– Let **the man’s present age** be \( M \). – Let **the father’s present age** be \( F \). – Let **the son’s present age** be \( S \). – **The grandfather’s present age** is \( G \), which is the father’s age (since he is the grandfather).
Step 1: Set up the equations based on the given ratios
1. **The ratio of the man’s age to his father’s age is 4:5**: \[ \frac{M}{F} = \frac{4}{5} \quad \text{(Equation 1)} \] This implies: \[ M = \frac{4}{5}F \quad \text{(Equation 2)} \]
2. **The ratio of the man’s age to his son’s age is 6:1**: \[ \frac{M}{S} = \frac{6}{1} \quad \text{(Equation 3)} \] This implies: \[ M = 6S \quad \text{(Equation 4)} \]
3. **Four years ago, the ratio of the man’s age to his father’s age was 11:14**: \[ \frac{M – 4}{F – 4} = \frac{11}{14} \quad \text{(Equation 5)} \]
4. **Four years ago, the ratio of the man’s age to his son’s age was 11:1**: \[ \frac{M – 4}{S – 4} = \frac{11}{1} \quad \text{(Equation 6)} \] This simplifies to: \[ M – 4 = 11(S – 4) \quad \text{(Equation 7)} \]
Step 2: Solve the system of equations
Step 2.1: Use Equation 4 to substitute \( M = 6S \) into Equation 7: Substitute \( M = 6S \) into \( M – 4 = 11(S – 4) \):
\[ 6S – 4 = 11(S – 4) \] Expand the equation:
\[ 6S – 4 = 11S – 44 \] Simplify:
\[ 6S – 11S = -44 + 4 \] \[ -5S = -40 \] Solve for \( S \):
\[ S = 8 \]
Step 2.2: Find the man’s present age \( M \): Substitute \( S = 8 \) into \( M = 6S \):
\[ M = 6 \times 8 = 48 \]
Step 2.3: Find the father’s present age \( F \): Substitute \( M = 48 \) into \( M = \frac{4}{5}F \):
\[ 48 = \frac{4}{5}F \] Multiply both sides by 5:
\[ 240 = 4F \] Solve for \( F \):
\[ F = \frac{240}{4} = 60 \]
Step 2.4: Find the grandfather’s present age \( G \): Since the grandfather is the father, \( G = F \). So, the grandfather’s present age is \( G = 60 \).
Step 3: Find the ratio of the grandfather’s age to the grandson’s age 12 years from now
In 12 years: – The grandfather’s age will be \( G + 12 = 60 + 12 = 72 \). – The grandson’s age will be \( S + 12 = 8 + 12 = 20 \).
The ratio of the grandfather’s age to the grandson’s age will be:
