1054 is divided into three parts such that 2 times the first part , 3 times the second part and 5 times the third part are equal .Find each of the three parts .

Options

A. `450,360,244`
B. `350,456,248`
C. `510,340,204`
D. `325,470,259`

Answer

To solve the problem of dividing 1054 into three parts such that 2 times the first part, 3 times the second part, and 5 times the third part are equal, we can follow these steps:

### Step 1: Define the parts
Let:
- First part = \( x \)
- Second part = \( y \)
- Third part = \( z \)

### Step 2: Set up the equations
According to the problem, we have the following relationships:
1. \( 2x = 3y \)
2. \( 2x = 5z \)

We also know that the sum of the three parts is equal to 1054:
\[ x + y + z = 1054 \]

### Step 3: Express \( y \) and \( z \) in terms of \( x \)
From the first equation \( 2x = 3y \), we can express \( y \) as:
\[ y = \frac{2x}{3} \]

From the second equation \( 2x = 5z \), we can express \( z \) as:
\[ z = \frac{2x}{5} \]

### Step 4: Substitute \( y \) and \( z \) into the sum equation
Substituting \( y \) and \( z \) into the equation \( x + y + z = 1054 \):
\[ x + \frac{2x}{3} + \frac{2x}{5} = 1054 \]

### Step 5: Find a common denominator
The common denominator for 3 and 5 is 15. We can rewrite the equation:
\[ x + \frac{10x}{15} + \frac{6x}{15} = 1054 \]

### Step 6: Combine the terms
Combining the terms gives:
\[ x + \frac{16x}{15} = 1054 \]
This can be rewritten as:
\[ \frac{15x}{15} + \frac{16x}{15} = 1054 \]
\[ \frac{31x}{15} = 1054 \]

### Step 7: Solve for \( x \)
To find \( x \), multiply both sides by 15:
\[ 31x = 1054 \times 15 \]
\[ 31x = 15810 \]
Now divide by 31:
\[ x = \frac{15810}{31} \]
\[ x = 510 \]

### Step 8: Find \( y \) and \( z \)
Now that we have \( x \), we can find \( y \) and \( z \):
1. For \( y \):
\[ y = \frac{2x}{3} = \frac{2 \times 510}{3} = \frac{1020}{3} = 340 \]

2. For \( z \):
\[ z = \frac{2x}{5} = \frac{2 \times 510}{5} = \frac{1020}{5} = 204 \]

### Final Answer
The three parts are:
- First part \( x = 510 \)
- Second part \( y = 340 \)
- Third part \( z = 204 \)