Problems on Ages Questions in English

(B) Let the ages of $Ram$ and $Rahim$ $10$ years ago be $x$ and $3x$ respectively.
Then,the present age of $Ram = (x + 10)$ and the present age of $Rahim = (3x + 10)$.
According to the question,the ratio of their ages $5$ years hence will be $2:3$:
$\frac{(x + 10 + 5)}{(3x + 10 + 5)} = \frac{2}{3}$
$\frac{x + 15}{3x + 15} = \frac{2}{3}$
$3(x + 15) = 2(3x + 15)$
$3x + 45 = 6x + 30$
$3x = 15$
$x = 5$
Now,calculate the present ages:
Present age of $Ram = 5 + 10 = 15$
Present age of $Rahim = 3(5) + 10 = 25$
Required ratio of present ages = $15:25 = 3:5$.

(C) Let the sum of the ages of the original $11$ players be $S$.
The average age of the original team is $S/11$.
When two players aged $18$ and $20$ are replaced by two new players with ages $x$ and $y$,the new sum of ages becomes $S - 18 - 20 + x + y = S + (x + y - 38)$.
The new average age is $(S + x + y - 38) / 11$.
According to the problem,the average age increases by $2$ months ($1/6$ of a year):
$(S + x + y - 38) / 11 = S/11 + 2/12$
Multiplying by $11$:
$S + x + y - 38 = S + 22/12$
$x + y - 38 = 22 \text{ months}$
$x + y = 38 \text{ years } 22 \text{ months}$.
The average age of the two new players is $(x + y) / 2 = (38 \text{ years } 22 \text{ months}) / 2 = 19 \text{ years } 11 \text{ months}$.

(A) Shan's age $= 55$ years.
Sathian is $5$ years junior to Shan,so Sathian's age $= 55 - 5 = 50$ years.
Sathian is $6$ years senior to Balan,so Balan's age $= 50 - 6 = 44$ years.
Devan is $7$ years junior to Balan,so Devan's age $= 44 - 7 = 37$ years.
The age difference between Shan and Devan $= 55 - 37 = 18$ years.

(C) Let the son's present age be $x$ years. Then the father's present age is $(x+30)$ years.
Father's age after $10$ years $= (x+30) + 10 = (x+40)$ years.
Son's age after $10$ years $= (x+10)$ years.
According to the problem:
$(x+40) = 3(x+10)$
Expanding the equation:
$x + 40 = 3x + 30$
Rearranging the terms:
$40 - 30 = 3x - x$
$10 = 2x$
Solving for $x$:
$x = 5$
Therefore,the son's present age is $5$ years.

(A) $1$. Rehana's age after $7 \text{ years}$ is $85 \text{ years}$,so her present age is $85 - 7 = 78 \text{ years}$.
$2$. Wasim is $12 \text{ years}$ younger than Rehana,so Wasim's present age is $78 - 12 = 66 \text{ years}$.
$3$. The ratio of Manoj's age to Wasim's age is $3:11$. Let Manoj's age be $M$. Then $M / 66 = 3 / 11$,which gives $M = (3 \times 66) / 11 = 18 \text{ years}$.
$4$. Manoj's father is $25 \text{ years}$ older than Manoj,so his father's present age is $18 + 25 = 43 \text{ years}$.

(C) Let Raman's present age be $x$ years.
$\therefore$ His daughter's present age $= x/3$ years.
His mother's present age $= 13x/9$ years.
According to the question, the sum of their ages is $125$ years:
$x + x/3 + 13x/9 = 125$
Multiplying by $9$ to clear the denominator:
$9x + 3x + 13x = 125 \times 9$
$25x = 1125$
$x = 1125 / 25 = 45$ years.
Now, Raman's daughter's age $= 45 / 3 = 15$ years.
Raman's mother's age $= (13 \times 45) / 9 = 13 \times 5 = 65$ years.
The difference between the ages of Raman's mother and daughter $= 65 - 15 = 50$ years.

(C) Let the present age of $Ram$ be $6x$ years and $Rakesh$ be $11x$ years.
According to the problem,$4$ years ago,their ages were $(6x - 4)$ and $(11x - 4)$ respectively.
The ratio of their ages $4$ years ago was $1:2$,so:
$\frac{6x - 4}{11x - 4} = \frac{1}{2}$
By cross-multiplying,we get:
$2(6x - 4) = 1(11x - 4)$
$12x - 8 = 11x - 4$
$12x - 11x = 8 - 4$
$x = 4$
Therefore,the present age of $Rakesh$ is $11 \times 4 = 44$ years.
After $5$ years,$Rakesh$'s age will be $44 + 5 = 49$ years.

(D) Let the present ages of the father and the son be $5x$ and $2x$ years,respectively.
After $4$ years,the age of the son will be $(2x + 4)$ years.
According to the problem,after $4$ years,the ratio of the son's age to the mother's age will be $1:2$.
Let the mother's age after $4$ years be $M_{future}$.
Then,$\frac{2x + 4}{M_{future}} = \frac{1}{2}$,which implies $M_{future} = 2(2x + 4) = 4x + 8$.
The present age of the mother is $M_{present} = M_{future} - 4 = (4x + 8) - 4 = 4x + 4$.
The ratio of the father's present age to the mother's present age is $\frac{5x}{4x + 4}$.
Since the value of $x$ is not provided and cannot be derived from the given information,the ratio depends on $x$.
Therefore,the ratio cannot be determined.

(D) Let Radha's present age be $x$ years.
According to the problem,her age $12$ years ago was $(x - 12)$ years.
The condition states: $x = 2(x - 12) - 3$.
Solving for $x$:
$x = 2x - 24 - 3$
$x = 2x - 27$
$x = 27$ years.
So,Radha's present age is $27$ years.
The ratio of Raj's present age to Radha's present age is $4:9$.
Let Raj's present age be $R$.
$R / 27 = 4 / 9$
$R = (4 / 9) \times 27 = 12$ years.
Raj's age after $5$ years will be $12 + 5 = 17$ years.

(B) Let the present ages of Swati $(S)$ and Trupti $(T)$ be $4x$ and $5x$ respectively.
According to the problem,after $6 \text{ years}$,the ratio of their ages will be $6:7$.
So,$\frac{4x + 6}{5x + 6} = \frac{6}{7}$.
Cross-multiplying,we get: $7(4x + 6) = 6(5x + 6)$.
$28x + 42 = 30x + 36$.
$42 - 36 = 30x - 28x$.
$6 = 2x$,which implies $x = 3$.
The present age of Swati is $4 \times 3 = 12 \text{ years}$.
The present age of Trupti is $5 \times 3 = 15 \text{ years}$.
The difference between their ages is $15 - 12 = 3 \text{ years}$.

(D) Let the present ages of Anju and Sandhya be $13x$ and $17x$ respectively.
According to the problem,$4$ years ago,their ages were $(13x - 4)$ and $(17x - 4)$.
The ratio was given as $11: 15$,so:
$\frac{13x - 4}{17x - 4} = \frac{11}{15}$
Cross-multiplying gives:
$15(13x - 4) = 11(17x - 4)$
$195x - 60 = 187x - 44$
$195x - 187x = 60 - 44$
$8x = 16$
$x = 2$
So,their present ages are $13 \times 2 = 26$ years and $17 \times 2 = 34$ years.
After $6$ years,their ages will be $(26 + 6) = 32$ years and $(34 + 6) = 40$ years.
The ratio of their ages after $6$ years will be $\frac{32}{40} = \frac{4}{5}$,which is $4:5$.

(C) Let the present ages of Vishal and Shekhar be $14x$ and $17x$ respectively.
According to the problem,after $6 \, \text{years}$,the ratio of their ages will be $17: 20$.
So,$\frac{14x + 6}{17x + 6} = \frac{17}{20}$.
Cross-multiplying,we get: $20(14x + 6) = 17(17x + 6)$.
$280x + 120 = 289x + 102$.
Rearranging the terms: $289x - 280x = 120 - 102$.
$9x = 18$.
$x = 2$.
Shekhar's present age is $17x = 17 \times 2 = 34 \, \text{years}$.

(D) Let Ram's present age be $R$, his son's age be $S$, and his father's age be $F$.
According to the problem, $R = 3S$, which implies $S = \frac{R}{3}$.
Also, $R = \frac{2}{5}F$, which implies $F = \frac{5R}{2}$.
The average of their ages is given as $46 \, \text{years}$, so $\frac{R + S + F}{3} = 46$.
This gives $R + S + F = 138$.
Substituting the values of $S$ and $F$ in terms of $R$: $R + \frac{R}{3} + \frac{5R}{2} = 138$.
Multiplying by $6$ to clear the denominators: $6R + 2R + 15R = 138 \times 6$.
$23R = 828$, so $R = \frac{828}{23} = 36$.
Now, $S = \frac{36}{3} = 12 \, \text{years}$ and $F = \frac{5 \times 36}{2} = 90 \, \text{years}$.
The difference between the son's age and the father's age is $F - S = 90 - 12 = 78 \, \text{years}$.

(D) Let the present age of Anubha be $x$ and her mother be $2x$.
According to the question,after $6$ years,the ratio of their ages will be $11:20$:
$\frac{x+6}{2x+6} = \frac{11}{20}$
Cross-multiplying,we get:
$20(x+6) = 11(2x+6)$
$20x + 120 = 22x + 66$
Rearranging the terms to solve for $x$:
$22x - 20x = 120 - 66$
$2x = 54$
$x = 27$
So,the present age of Anubha is $27$ years and her mother is $2 \times 27 = 54$ years.
To find the ratio of their ages $9$ years ago:
Anubha's age $9$ years ago $= 27 - 9 = 18$ years.
Mother's age $9$ years ago $= 54 - 9 = 45$ years.
Ratio $= \frac{18}{45} = \frac{2}{5} = 2:5$.

(B) Let the present age of Tina be $9x$ and the present age of Rakesh be $10x$.
According to the problem,$10 \text{ years}$ ago,their ages were $(9x - 10)$ and $(10x - 10)$ respectively.
The ratio of their ages $10 \text{ years}$ ago was $4:5$,so we can write:
$\frac{9x - 10}{10x - 10} = \frac{4}{5}$
Cross-multiplying the terms:
$5(9x - 10) = 4(10x - 10)$
$45x - 50 = 40x - 40$
Rearranging the terms to solve for $x$:
$45x - 40x = 50 - 40$
$5x = 10$
$x = 2$
Therefore,the present age of Rakesh is $10x = 10 \times 2 = 20 \text{ years}$.

(C) Let the present age of the son be $x$ years.
According to the problem, the son will be $15$ years old after $3$ years, so $x + 3 = 15$.
Solving for $x$, we get $x = 15 - 3 = 12$ years.
The father's present age is four times the son's age, so Father's age $= 4x = 4 \times 12 = 48$ years.
The man is $3$ years older than his wife, which means Wife's age $= \text{Father's age} - 3$.
Therefore, Wife's age $= 48 - 3 = 45$ years.

(D) Let the present age of $A$ be $x$ years and the present age of $B$ be $y$ years.
According to the problem,$7$ years ago:
$\frac{x-7}{y-7} = \frac{4}{5}$
$5(x-7) = 4(y-7)$
$5x - 35 = 4y - 28$
$5x - 4y = 7$ ....$(1)$
$7$ years hence:
$\frac{x+7}{y+7} = \frac{5}{6}$
$6(x+7) = 5(y+7)$
$6x + 42 = 5y + 35$
$6x - 5y = -7$ ....$(2)$
To solve for $y$,multiply equation $(1)$ by $6$ and equation $(2)$ by $5$:
$30x - 24y = 42$
$30x - 25y = -35$
Subtracting the second from the first:
$(-24y) - (-25y) = 42 - (-35)$
$y = 77$
Thus,the present age of $B$ is $77$ years.

(D) Let the present ages of Raj and Radha be $4x$ and $9x$ respectively.
According to the problem,Radha's present age is $3 \text{ years}$ less than twice her age $12 \text{ years}$ ago.
So,$9x = 2(9x - 12) - 3$.
Expanding the equation: $9x = 18x - 24 - 3$.
$9x = 18x - 27$.
$27 = 9x$,which gives $x = 3$.
Raj's present age is $4x = 4 \times 3 = 12 \text{ years}$.
After $5 \text{ years}$,Raj's age will be $12 + 5 = 17 \text{ years}$.

(D) Let the present age of the father be $5x$ and the son be $2x$.
After $4$ years,the age of the son will be $(2x + 4)$.
According to the problem,the ratio of the son's age to the mother's age after $4$ years is $1:2$.
So,$\frac{2x + 4}{M + 4} = \frac{1}{2}$,where $M$ is the mother's present age.
$2(2x + 4) = M + 4$
$4x + 8 = M + 4$
$M = 4x + 4$.
The ratio of the father's age to the mother's age is $\frac{F}{M} = \frac{5x}{4x + 4}$.
Since the value of $x$ is not provided,the ratio depends on $x$ and cannot be determined.

(B) Let the present ages of Tania and Rakesh be $9x$ and $10x$ respectively.
According to the problem,$10 \text{ years}$ ago,their ages were $(9x - 10)$ and $(10x - 10)$.
The ratio of their ages $10 \text{ years}$ ago was $4:5$,so:
$\frac{9x - 10}{10x - 10} = \frac{4}{5}$
Cross-multiplying the terms:
$5(9x - 10) = 4(10x - 10)$
$45x - 50 = 40x - 40$
Rearranging the terms to solve for $x$:
$45x - 40x = 50 - 40$
$5x = 10$
$x = 2$
Rakesh's present age is $10x = 10 \times 2 = 20 \text{ years}$.