1162 is divided into three parts such that 4 times the first part, 5 times the second part and 7 times the third part are equal. Find the parts.

Options

A. 490, 392, 280
B. 492, 392, 278
C. 493, 329, 340
D. 393, 290, 360

Answer

To solve the problem of dividing 1162 into three parts such that 4 times the first part, 5 times the second part, and 7 times the third part are equal, we can follow these steps:

### Step 1: Define the Parts
Let the three parts be:
- First part = \( a \)
- Second part = \( b \)
- Third part = \( c \)

According to the problem, we have:
\[ 4a = 5b = 7c \]

### Step 2: Express Each Part in Terms of a Common Variable
Let \( k \) be the common value such that:
- \( 4a = k \) → \( a = \frac{k}{4} \)
- \( 5b = k \) → \( b = \frac{k}{5} \)
- \( 7c = k \) → \( c = \frac{k}{7} \)

### Step 3: Sum of the Parts
The total sum of the parts is given as:
\[ a + b + c = 1162 \]
Substituting the expressions for \( a \), \( b \), and \( c \):
\[ \frac{k}{4} + \frac{k}{5} + \frac{k}{7} = 1162 \]

### Step 4: Find a Common Denominator
The least common multiple (LCM) of 4, 5, and 7 is 140. We can express each fraction with a denominator of 140:
\[ \frac{35k}{140} + \frac{28k}{140} + \frac{20k}{140} = 1162 \]
Combining the fractions:
\[ \frac{35k + 28k + 20k}{140} = 1162 \]
This simplifies to:
\[ \frac{83k}{140} = 1162 \]

### Step 5: Solve for \( k \)
To find \( k \), we can cross-multiply:
\[ 83k = 1162 \times 140 \]
Calculating the right-hand side:
\[ 1162 \times 140 = 162680 \]
Now, divide both sides by 83:
\[ k = \frac{162680}{83} = 1960 \]

### Step 6: Find Each Part
Now that we have \( k \), we can find each part:
- First part \( a \):
\[ a = \frac{k}{4} = \frac{1960}{4} = 490 \]
- Second part \( b \):
\[ b = \frac{k}{5} = \frac{1960}{5} = 392 \]
- Third part \( c \):
\[ c = \frac{k}{7} = \frac{1960}{7} = 280 \]

### Final Answer
The three parts are:
- First part = 490
- Second part = 392
- Third part = 280