A farmer divides his herd of k cows among his four sons so that the first son gets one-half of the herd, the second son gets one-fourth, the third son gets one-fifth and the fourth son gets 9 cows. What is the value of k, the total number of cows?

Introduction / Context:
This question is based on fractional division of a whole quantity among several people. The farmer divides his herd in fractional shares plus a fixed number of cows. The key idea is that the sum of all those portions must equal the entire herd. This is a classic algebra problem involving fractions and a single unknown total.

Given Data / Assumptions:

• Total number of cows is k.
• First son gets one-half of the herd, that is (1/2)k cows.
• Second son gets one-fourth of the herd, that is (1/4)k cows.
• Third son gets one-fifth of the herd, that is (1/5)k cows.
• Fourth son gets 9 cows.
• The entire herd is distributed among the four sons.

Concept / Approach:
We interpret the division of cows as an equation where the sum of all the portions equals k. The first three portions are written as fractions of k, and the last portion is a fixed number. Summing the fractions gives a combined fractional multiple of k, and adding 9 cows yields k. Solving this equation reveals the total number of cows. Manipulating fractional coefficients is the central skill here.

Step-by-Step Solution:
Step 1: Express the distribution as an equation: (1/2)k + (1/4)k + (1/5)k + 9 = k.Step 2: Find a common denominator for the fractions 1/2, 1/4 and 1/5. The least common multiple of 2, 4 and 5 is 20.Step 3: Rewrite the fractions with denominator 20. Then 1/2 = 10/20, 1/4 = 5/20, 1/5 = 4/20.Step 4: Add these fractions: (10/20 + 5/20 + 4/20)k = (19/20)k.Step 5: The equation becomes (19/20)k + 9 = k.Step 6: Subtract (19/20)k from both sides to get 9 = k - (19/20)k = (1/20)k.Step 7: Solve for k: (1/20)k = 9 implies k = 9 * 20 = 180.

Verification / Alternative check:
Check the distribution for k = 180. First son gets 1/2 of 180 = 90 cows. Second son gets 1/4 of 180 = 45 cows. Third son gets 1/5 of 180 = 36 cows. Fourth son gets 9 cows. Sum them: 90 + 45 + 36 + 9 = 180 cows. This matches the total herd and confirms that k = 180 satisfies the conditions.

Why Other Options Are Wrong:
If k were 160, then 1/2, 1/4 and 1/5 portions plus 9 would not sum to 160. For k = 120 or 140 or 200, similar calculations show that the combined fractional shares plus 9 do not equal the total. Only k = 180 makes the equation (1/2 + 1/4 + 1/5)k + 9 = k true, because the combined fractional part is exactly 19/20 of k, leaving 1/20k equal to 9.

Common Pitfalls:
A common error is to add fractions incorrectly or to misunderstand that the fractions are of the same total k. Some students may treat the 9 cows as part of a fraction or forget to include it in the equation. Miscalculating the common denominator or adding 10/20, 5/20 and 4/20 incorrectly can also lead to wrong answers. Careful fraction arithmetic and checking with substitution can prevent these mistakes.

Final Answer:
The total number of cows in the herd is 180, so the correct option is 180.