Three years ago, average age of 'Amiť, 'Bittu' and 'Chitu' is 27 years. Four years hence, ratio of Amit and Chitu's age is 7: 10. If Bittu is 6 years younger than Chitu, then find present age of 'Amit"?

Options

A. 24 years
B. 27 years
C. 30 years
D. 28 years

Answer

To find the present age of Amit, we can follow these steps:

### Step 1: Calculate the total age of Amit, Bittu, and Chintu three years ago.
Given that the average age of Amit, Bittu, and Chintu three years ago was 27 years, we can calculate their total age at that time.

\[ \text{Total age three years ago} = \text{Average age} \times \text{Number of people} = 27 \times 3 = 81 \text{ years} \]

### Step 2: Calculate their total present age.
Since this total was three years ago, we need to add 3 years for each person to find their total present age.

\[ \text{Total present age} = 81 + 3 \times 3 = 81 + 9 = 90 \text{ years} \]

### Step 3: Set up the equations based on the information provided.
Let Amit's age be \(X\), Bittu's age be \(Y\), and Chintu's age be \(Z\).

From the total present age, we have:
\[ X + Y + Z = 90 \quad \text{(1)} \]

### Step 4: Use the ratio of Amit and Chintu's age four years hence.
Four years from now, the ages will be:
- Amit: \(X + 4\)
- Chintu: \(Z + 4\)

The ratio of Amit's age to Chintu's age is given as 7:10:
\[ \frac{X + 4}{Z + 4} = \frac{7}{10} \]

Cross-multiplying gives:
\[ 10(X + 4) = 7(Z + 4) \]
\[ 10X + 40 = 7Z + 28 \]
Rearranging this, we get:
\[ 10X - 7Z = -12 \quad \text{(2)} \]

### Step 5: Use the information about Bittu's age.
We know that Bittu is 6 years younger than Chintu:
\[ Y = Z - 6 \quad \text{(3)} \]

### Step 6: Substitute equation (3) into equation (1).
Substituting \(Y\) from equation (3) into equation (1):
\[ X + (Z - 6) + Z = 90 \]
\[ X + 2Z - 6 = 90 \]
\[ X + 2Z = 96 \quad \text{(4)} \]

### Step 7: Solve the system of equations.
Now we have two equations (2) and (4):
1. \(10X - 7Z = -12\) (from step 4)
2. \(X + 2Z = 96\) (from step 6)

From equation (4), we can express \(X\) in terms of \(Z\):
\[ X = 96 - 2Z \quad \text{(5)} \]

Substituting equation (5) into equation (2):
\[ 10(96 - 2Z) - 7Z = -12 \]
\[ 960 - 20Z - 7Z = -12 \]
\[ 960 - 27Z = -12 \]
\[ -27Z = -12 - 960 \]
\[ -27Z = -972 \]
\[ Z = \frac{972}{27} = 36 \]

### Step 8: Find \(Y\) and \(X\).
Using \(Z = 36\) in equation (3):
\[ Y = Z - 6 = 36 - 6 = 30 \]

Now substitute \(Z\) back into equation (5) to find \(X\):
\[ X = 96 - 2(36) = 96 - 72 = 24 \]

### Conclusion:
The present age of Amit is \(X = 24\) years.