A farmer divides his herd of n cows among his four sons so that the first son gets one-half the herd, the second son gets one-fourth, the third son gets one-fifth and the fourth son gets 7 cows. The value of n is

Options

A. 80
B. 100
C. 140
D. 180

Answer

To solve the problem of how many cows the farmer originally had, we can break it down step by step:

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

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

### Step 3: Set up the equation
The total number of cows distributed among the four sons should 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
The least common multiple of the denominators (2, 4, and 5) is 20. We can rewrite each term with a denominator of 20:
- \( \frac{1}{2}n = \frac{10}{20}n \)
- \( \frac{1}{4}n = \frac{5}{20}n \)
- \( \frac{1}{5}n = \frac{4}{20}n \)

Now, substituting these back into the equation gives:
\[ \frac{10}{20}n + \frac{5}{20}n + \frac{4}{20}n + 7 = n \]

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

### Step 6: Isolate \( n \)
To isolate \( n \), subtract \( \frac{19}{20}n \) from both sides:
\[ 7 = n - \frac{19}{20}n \]
This simplifies to:
\[ 7 = \frac{1}{20}n \]

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

### Conclusion
The total number of cows the farmer originally had is \( n = 140 \).

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