Question

A and B rent a pasture for 10 months by contributing equal rent; A puts in 80 cows for 7 months and leaves. How many cows can B put in for the remaining 3 months, if he pays half as much rent again as A?

• A. \(250\)
• B. \(280\)
• C. \(275\)
• D. \(200\)
• E. \(120\)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is a problem based on partnership and joint utility.
The total rent paid by each person is directly proportional to the number of animals they graze and the amount of time the animals graze.
The phrase "contributing equal rent" describes their initial generic plan to rent the pasture, but the actual finalized payment ratio is specified by the condition that B "pays half as much rent again as A".
Step 2: Key Formula or Approach:
Rent of A : Rent of B = (Number of cows of A \( \times \) Months of A) : (Number of cows of B \( \times \) Months of B).
"Half as much again as A" means \( 1 + \frac{1}{2} = \frac{3}{2} \) times A's rent.
Step 3: Detailed Explanation:

Let the rent paid by person A be denoted by \( R_A \) and the rent paid by person B be \( R_B \).

The problem states that B pays "half as much rent again as A".

This idiom means B pays A's rent plus an additional half of A's rent. Thus, \( R_B = R_A + 0.5 R_A = 1.5 R_A \).

This gives us the ratio of their rents: \( \frac{R_A}{R_B} = \frac{1}{1.5} = \frac{2}{3} \).

We know that the share of rent depends on the "cow-months" used by each individual.

Person A puts in 80 cows for a period of 7 months.

A's usage in cow-months = \( 80 \text{ cows} \times 7 \text{ months} = 560 \text{ cow-months} \).

Let the number of cows put in by person B for the remaining 3 months be \( x \).

B's usage in cow-months = \( x \text{ cows} \times 3 \text{ months} = 3x \text{ cow-months} \).

According to the partnership principle, the ratio of their usage must equal the ratio of their rent payments.

Setting up the proportion: \( \frac{560}{3x} = \frac{2}{3} \).

To solve for \( x \), we can cross-multiply: \( 560 \times 3 = 3x \times 2 \).

This simplifies to \( 1680 = 6x \).

Dividing both sides by 6 gives \( x = \frac{1680}{6} \).

Calculating the division yields \( x = 280 \).

Therefore, B must put in 280 cows for the remaining 3 months to justify the rent ratio.

Step 4: Final Answer:
B puts in 280 cows for the remaining 3 months.