**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 father’s age would be 50 years.

Hence father would be \(\frac{50}{20}\) = 2.5 times of daughter’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 – 10)

Therefore, (x + 20) = 10 (x – 10)

Solving the equation, we get x + 20 = 10x – 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 Nisha’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 – 5).

Romi’s age before 5 years = (y – 5)

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

(y + 15 – 5) = 3 (y – 5)

(y + 10) = (3y – 15)

2y = 25 ⇒ 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

Honey’s age = 2x and Piyush’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, Honey’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 ⇒ 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, mother’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 son’s age is (x + 10)

After ten years: Mother’s age will be (3x + 10) +10 and son’s age will be (x + 10) + 10. Given that, mother’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: Saransh’s age = (50 – x) and Nazma’s age = (40 – 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 ⇒ 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 ⇒ 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 Sara’s age 4 years ago and Vaishali’s age after 4 years is 1: 1. Presently, the ratio of their ages is 5: 3. Find the ratio between Sara’s age 4 years hence and Vaishali’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.

Sara’s age 4 years ago = 5x – 4

Vaishali’s age after 4 years = 3x + 4

Ratio of Sara’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 Sara’s age 4 years hence and Vaishali’s age 4 years ago.

Sara’s age: (5x + 4)

Vaishali’s age: (3x – 4)

Putting the value of x, we get:

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

**Answer:** Option C

**Explanation:**

Let John’s present age be x

John’s age before 5 years = (x – 5)

John’s age after 10 years = (x + 10)

We are given that, John’s age after 10 years (x + 10) is 5 times his age 5 years back (x – 5)

Therefore,

(x + 10) = 5 (x – 5)

Solving the equation, we get

x + 10 = 5x – 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 Rahul’s age 5 years ago = (y + 15 – 5)

Rohan’s age before 5 years = (y – 5)

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

(y + 15 – 5) = 3 (y – 5)

(y + 10) = (3y – 15)

2y = 25

y = 12.5

Rohan’s age = 12.5 years

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

**3. One year ago, ratio of Harry and Peter age’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

Harry’s age = 5x and Peter’s age = 6x

One year ago, their age was 5x and 6x. Hence at present, Harry’s age = 5x +1 and Peter’s age = 6x +1

After 4 years,

Harry’s age = (5x +1) + 4 = (5x + 5)

Peter’s age = (6x +1) + 4 = (6x + 5)

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

\(\frac{Harry’s Age}{6}\) = \(\frac{Peter’s Age}{7}\)

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

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

X = 5

Peter’s present age = (6x + 1) = (6 x 5 + 1) = 31 years

Harry’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: Sharad’s age = (60 – x) and Santosh’s age = (80 – x)

Ratio of their ages x years ago was 4 : 6

\(\frac{(60 – x)}{(80 – x)}\) = \(\frac{4}{6}\)

6(60 – x) = 4(80 – x)

360 – 6x = 320 – 4x

x = 20

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

**5. 5 years ago, sister’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 sister’s age was 5 times the age of her brother.

Therefore,

(34 – x) – 5 = 5 (x – 5)

34 – x – 5 = 5x – 25

5x + x = 34 – 5 +25

6x = 54

x = 9

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

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