Ratio of present age of Veer , Sameer, Divyaraj , Ayush and Sumit is`14:15:13:12:16` and sum ofage of Veer, Divyaraj & Sumit four years hence will be 44 years more than sum of present age of Sameer & Ayush . Find the ratio of age of Veer, Sameer , Divyaraj , Ayush and Sumit after 10 years ?

Options

A. `19:20:18:16:21`
B. `19:20:16:17:21`
C. `19:20:22:17:21`
D. `19:20:18:17:21`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and derive the required ages and their ratio after 10 years.

### Step 1: Define the present ages based on the given ratio
The present ages of Veer, Sameer, Divyaraj, Ayush, and Sumit are in the ratio 14:15:13:12:16. We can express their ages in terms of a variable \( x \):
- Veer's age = \( 14x \)
- Sameer's age = \( 15x \)
- Divyaraj's age = \( 13x \)
- Ayush's age = \( 12x \)
- Sumit's age = \( 16x \)

### Step 2: Set up the equation based on the given condition
According to the problem, the sum of the ages of Veer, Divyaraj, and Sumit four years hence will be 44 years more than the sum of the present ages of Sameer and Ayush.

The ages four years hence will be:
- Veer's age in 4 years = \( 14x + 4 \)
- Divyaraj's age in 4 years = \( 13x + 4 \)
- Sumit's age in 4 years = \( 16x + 4 \)

The sum of their ages in 4 years will be:
\[ (14x + 4) + (13x + 4) + (16x + 4) = 43x + 12 \]

The sum of the present ages of Sameer and Ayush is:
\[ 15x + 12x = 27x \]

According to the problem:
\[ 43x + 12 = 27x + 44 \]

### Step 3: Solve the equation
Now, we will solve for \( x \):
\[ 43x + 12 = 27x + 44 \]
Subtract \( 27x \) from both sides:
\[ 16x + 12 = 44 \]
Subtract 12 from both sides:
\[ 16x = 32 \]
Divide both sides by 16:
\[ x = 2 \]

### Step 4: Calculate the present ages
Now that we have \( x \), we can calculate the present ages:
- Veer's age = \( 14x = 14 \times 2 = 28 \)
- Sameer's age = \( 15x = 15 \times 2 = 30 \)
- Divyaraj's age = \( 13x = 13 \times 2 = 26 \)
- Ayush's age = \( 12x = 12 \times 2 = 24 \)
- Sumit's age = \( 16x = 16 \times 2 = 32 \)

### Step 5: Calculate the ages after 10 years
Now we will calculate their ages after 10 years:
- Veer's age after 10 years = \( 28 + 10 = 38 \)
- Sameer's age after 10 years = \( 30 + 10 = 40 \)
- Divyaraj's age after 10 years = \( 26 + 10 = 36 \)
- Ayush's age after 10 years = \( 24 + 10 = 34 \)
- Sumit's age after 10 years = \( 32 + 10 = 42 \)

### Step 6: Write the ratio of their ages after 10 years
The ages after 10 years are:
- Veer: 38
- Sameer: 40
- Divyaraj: 36
- Ayush: 34
- Sumit: 42

Now, we can express the ratio:
\[ \text{Ratio} = 38 : 40 : 36 : 34 : 42 \]

### Step 7: Simplify the ratio
To simplify the ratio, we can divide each term by 2:
\[ \text{Ratio} = 19 : 20 : 18 : 17 : 21 \]

Thus, the final ratio of the ages of Veer, Sameer, Divyaraj, Ayush, and Sumit after 10 years is:
\[ \text{Final Ratio} = 19 : 20 : 18 : 17 : 21 \]