A farmer divides his herd of n cows among his four sons, so that the first son gets one-half the herd, the secondson gets one fourth, the third son gets `1/5` and the fourth son gets 7 cows. Then the value of n is

Options

A. 240
B. 100
C. 180
D. 140

Answer

To solve the problem of how many cows the farmer has in total, we can break it down step by step.

### Step 1: Define the variables
Let \( n \) be the total number of cows the farmer has.

### Step 2: Determine the distribution of cows among the sons
- The first son receives \( \frac{1}{2}n \) cows.
- The second son receives \( \frac{1}{4}n \) cows.
- The third son receives \( \frac{1}{5}n \) cows.
- The fourth son receives 7 cows.

### Step 3: Set up the equation
The total number of cows given to all four sons must equal the total number of cows \( n \). Therefore, we can write the equation:
\[ \frac{1}{2}n + \frac{1}{4}n + \frac{1}{5}n + 7 = n \]

### Step 4: Find a common denominator
To combine the fractions, we need to find a common denominator. The least common multiple (LCM) of 2, 4, and 5 is 20. We can rewrite the fractions:
- \( \frac{1}{2}n = \frac{10}{20}n \)
- \( \frac{1}{4}n = \frac{5}{20}n \)
- \( \frac{1}{5}n = \frac{4}{20}n \)

### Step 5: Substitute back into the equation
Substituting these values into the equation gives us:
\[ \frac{10}{20}n + \frac{5}{20}n + \frac{4}{20}n + 7 = n \]

### Step 6: Combine the fractions
Combining the fractions on the left side:
\[ \frac{10n + 5n + 4n}{20} + 7 = n \]
This simplifies to:
\[ \frac{19n}{20} + 7 = n \]

### Step 7: Isolate \( n \)
To isolate \( n \), we can subtract \( \frac{19n}{20} \) from both sides:
\[ 7 = n - \frac{19n}{20} \]
This simplifies to:
\[ 7 = \frac{1n}{20} \]

### Step 8: Solve for \( n \)
Now, multiply both sides by 20 to solve for \( n \):
\[ n = 7 \times 20 = 140 \]

### Conclusion
The total number of cows \( n \) is 140.