**Algebra** is a very powerful branch of Mathematics which can be used to solve the Problems on Ages. Algebra helps in **transforming word problems into mathematical expressions** in the form of equations using variables to denote unknown quantities or parameters and thus, providing numerous techniques to solve these mathematical equations and hence, determining the answer to the problem. **Identifying key information, organizing information, using mathematical expressions** to assume unknown values and thus solving mathematical expressions for the unknown values will help us identify solutions.

**Answer:** Option C

**Explanation:**

Let the daughter’s age be x and father’s age be 4x.

So as per question, 4x + 5 = 3(x + 5). So x = 10.

Hence present age of daughter is 10 years and present age of father is 40 years.

So after 5 + 5 = 10 years, daughter age would be 20 years and fatherâ€™s age would be 50 years.

Hence father would be \(\frac{50}{20}\) = 2.5 times of daughterâ€™s age.

**2. What is Aman’s present age, if after 20 years his age will be 10 times his age 10 years back?**

**Answer:** Option C

**Explanation:**

Let Aman’s present age be x

Aman’s age before 10 years = x – 10)

Aman’s age after 20 years = (x + 20)

We are given that, Aman’s age after 20 years (x + 20) is 10 times his age 10 years back (x â€“ 10)

Therefore, (x + 20) = 10 (x â€“ 10)

Solving the equation, we get x + 20 = 10x â€“ 100

9x = 120, x = 13.3 years

**3. Nisha is 15 years elder to Romi. If 5 years ago, Nisha was 3 times as old as Romi, then find Nishaâ€™s present age.**

**Answer:** Option B

**Explanation:**

Let age of Romi be y

Nisha is 15 years elder than Romi = (y + 15). So Nisha’s age 5 years ago = (y + 15 â€“ 5).

Romi’s age before 5 years = (y â€“ 5)

5 years ago, Nisha is 3 times as old as Romi

(y + 15 â€“ 5) = 3 (y â€“ 5)

(y + 10) = (3y â€“ 15)

2y = 25 â‡’ y = 12.5

Romi’s age = 12.5 years

Nisha’s age = (y + 15) = (12.5 + 15) = 27.5 years.

**4. One year ago, the ratio of Honey and Piyush ages was 2: 3 respectively. After five years from now, this ratio becomes 4: 5. How old is Piyush now?**

**Answer:** Option C

**Explanation:**

We are given that age ratio of Honey: Piyush = 2: 3

Honeyâ€™s age = 2x and Piyushâ€™s age = 3x

One year ago, their age was 2x and 3x.

Hence at present, Honey’s age = 2x +1 and Piyush’s age = 3x +1

After 5 years, Honeyâ€™s age = (2x +1) + 5 = (2x + 6)

Piyush’s age = (3x +1) + 5 = (3x + 6)

After 5 years, this ratio becomes 4: 5. Therefore,

\(\frac{(2x+6)}{(3x+6)}\) = \(\frac{4}{5}\)

10x + 30 = 12x + 24 â‡’ x = 3

Piyush’s present age = (3x + 1) = (3x 3 + 1) = 10 years

Honey’s present age = (2x + 1) = (2x 3 + 1) = 7 years

**5. Ten years ago, the age of mother was three times the age of her son. After ten years, motherâ€™s age will be twice that of his son. Find the ratio of their present ages.**

**Answer:** Option D

**Explanation:**

10 years ago, age of mother was three times the age of her son. Say, the age of son was x and mother’s age was 3x.

At present: Mother’s age is (3x + 10) and sonâ€™s age is (x + 10)

After ten years: Mother’s age will be (3x + 10) +10 and sonâ€™s age will be (x + 10) + 10. Given that, motherâ€™s age is twice that of son after ten years.

(3x + 10) +10 = 2 [(x + 10) + 10]

(3x + 20) = 2[x + 20]

Solving the equation, we get x = 20

(3x + 10): (x + 10) = 70: 30 = 7: 3.

**Answer:** Option A

**Explanation:**

Here, we have to calculate: How many years ago the ratio of their ages was 3:2. Let us assume x years ago

At present: Saransh is 50 years and Nazma is 40 years

x years ago: Saranshâ€™s age = (50 â€“ x) and Nazma’s age = (40 â€“ x)

Given, the ratio of their ages was 3:2

\(\frac{(50-x)}{(40-x)}\) = \(\frac{3}{2}\)

Solving, we get: x = 20

Therefore, the answer is 20 years.

**2. The ratio of the present ages of Pranav and Qureshi is 4:5. Five years ago, the ratio of their ages was 7:9. Find their present ages? (In years)**

**Answer:** Option A

**Explanation:**

Their present ages be 4X and 5X.

5 years ago, the ratio of their ages was 7:9, then (4X – 5) : ( 5X – 5) = 7:9

X = 45 – 35 â‡’ X = 10.

Their present ages are: 40, 50.

**3. A man said to his son, “I was one-third of your present age when you were born”. If the present age of the man is 48 years, find the present age of the son.**

**Answer:** Option D

**Explanation:**

Present age of the son be P, he was born P years ago.

The age of the man was: (48 – P).

His age when the son was born should be equal to \(\frac{1}{3}\) of P.

(48 – P) =\(\frac{1}{3}\) P â‡’ P = 36

**4. Dinesh is younger to Roshan by 9 years. If their ages are in the respective ratio of 4:5, how old is Dinesh?**

**Answer:** Option A

**Explanation:**

Let Roshan’s age be x years.

Then, Dinesh ‘s age = (x – 9) years.

\(\frac{(x – 9)}{x}\) = \(\frac{4}{5}\)

x = 45 Hence, Dinesh’s age = (x – 9) = 36 years.

**5. The ratio of Saraâ€™s age 4 years ago and Vaishaliâ€™s age after 4 years is 1: 1. Presently, the ratio of their ages is 5: 3. Find the ratio between Saraâ€™s age 4 years hence and Vaishaliâ€™s age 4 years ago.**

**Answer:** Option B

**Explanation:**

Currently, the ratio of their ages is 5: 3. Suppose, their ages are: 5x and 3x.

Saraâ€™s age 4 years ago = 5x â€“ 4

Vaishaliâ€™s age after 4 years = 3x + 4

Ratio of Saraâ€™s age 4 years ago and Vaishali’s age after 4 years is 1 : 1

Therefore,

\(\frac{(5x – 4)}{(3x + 4)}\) = \(\frac{1}{1}\)

Solving, we get x = 4

We are required to find the ratio between Saraâ€™s age 4 years hence and Vaishaliâ€™s age 4 years ago.

Sara’s age: (5x + 4)

Vaishali’s age: (3x â€“ 4)

Putting the value of x, we get:

\(\frac{(5x + 4)}{(3x â€“ 4)}\) = \(\frac{3}{1}\) = 3:1

**Answer:** Option C

**Explanation:**

Let Johnâ€™s present age be x

Johnâ€™s age before 5 years = (x – 5)

Johnâ€™s age after 10 years = (x + 10)

We are given that, Johnâ€™s age after 10 years (x + 10) is 5 times his age 5 years back (x â€“ 5)

Therefore,

(x + 10) = 5 (x â€“ 5)

Solving the equation, we get

x + 10 = 5x â€“ 25

4x = 35

x = 8.75 years

**2. Rahul is 15 years elder than Rohan. If 5 years ago, Rahul was 3 times as old as Rohan, then find Rahul’s present age.**

**Answer:** Option B

**Explanation:**

Let age of Rohan be y

Rahul is 15 years elder than Rohan = (y + 15). So Rahulâ€™s age 5 years ago = (y + 15 â€“ 5)

Rohanâ€™s age before 5 years = (y â€“ 5)

5 years ago, Rahul is 3 times as old as Rohan

(y + 15 â€“ 5) = 3 (y â€“ 5)

(y + 10) = (3y â€“ 15)

2y = 25

y = 12.5

Rohanâ€™s age = 12.5 years

Rahulâ€™s age = (y + 15) = (12.5 + 15) = 27.5 years

**3. One year ago, ratio of Harry and Peter ageâ€™s was 5 : 6 respectively. After 4 years, this ratio becomes 6 : 7. How old is Peter? **

**Answer:** Option C

**Explanation:**

We are given that age ratio of Harry : Pitter = 5 : 6

Harryâ€™s age = 5x and Peterâ€™s age = 6x

One year ago, their age was 5x and 6x. Hence at present, Harryâ€™s age = 5x +1 and Peterâ€™s age = 6x +1

After 4 years,

Harryâ€™s age = (5x +1) + 4 = (5x + 5)

Peterâ€™s age = (6x +1) + 4 = (6x + 5)

After 4 years, this ratio becomes 6 : 7. Therefore,

\(\frac{Harryâ€™s Age}{6}\) = \(\frac{Peterâ€™s Age}{7}\)

\(\frac{(5x + 5)}{(6x + 5)}\) = \(\frac{6}{7}\)

7 (5x + 5) = 6 (6x + 5)

X = 5

Peterâ€™s present age = (6x + 1) = (6 x 5 + 1) = 31 years

Harryâ€™s present age = (5x + 1) = (5 x 5 + 1) = 26 years

**4. Sharad is 60 years old and Santosh is 80 years old. How many years ago was the ratio of their ages 4 : 6? **

**Answer:** Option C

**Explanation:**

Here, we have to calculate: How many years ago the ratio of their ages was 4 : 6

Let us assume x years ago

At present: Sharad is 60 years and Santosh is 80 years

x years ago: Sharadâ€™s age = (60 â€“ x) and Santoshâ€™s age = (80 â€“ x)

Ratio of their ages x years ago was 4 : 6

\(\frac{(60 â€“ x)}{(80 â€“ x)}\) = \(\frac{4}{6}\)

6(60 â€“ x) = 4(80 â€“ x)

360 â€“ 6x = 320 â€“ 4x

x = 20

Therefore, 20 years ago, the ratio of their ages was 4 : 6

**5. 5 years ago, sisterâ€™s age was 5 times the age of her brother and the sum of present ages of sister and brother is 34 years. What will be the age of her brother after 6 years?**

**Answer:** Option C

**Explanation:**

We are given, 5 years ago sisterâ€™s age was 5 times the age of her brother.

Therefore,

(34 â€“ x) â€“ 5 = 5 (x â€“ 5)

34 â€“ x â€“ 5 = 5x â€“ 25

5x + x = 34 â€“ 5 +25

6x = 54

x = 9

Future age (after 6 yrs) = (x + 6) = (9 + 6) = 15 years

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