4 goats or 6 sheeps can graze a field in 50 days . 2 goats and 9 sheeps can graze the field in s

Options

A. 100 days
B. 75 days
C. 50 days
D. 25 days

Answer

To solve the problem step by step, we need to use the relationship between the number of goats and sheep and the time taken to graze the field.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We know that 4 goats or 6 sheep can graze a field in 50 days. This means that the work done by 4 goats in 50 days is equal to the work done by 6 sheep in the same time.

2. **Finding the Work Rate:**
- Let's denote the work done by 1 goat in one day as \( G \) and the work done by 1 sheep in one day as \( S \).
- The total work done by 4 goats in 50 days is:
\[ \text{Work by 4 goats} = 4G \times 50 = 200G \]
- The total work done by 6 sheep in 50 days is:
\[ \text{Work by 6 sheep} = 6S \times 50 = 300S \]
- Since both expressions represent the same amount of work, we can equate them:
\[ 200G = 300S \]
- Simplifying this gives:
\[ \frac{G}{S} = \frac{300}{200} = \frac{3}{2} \]
- This means that 1 goat does the work of 1.5 sheep.

3. **Setting Up the Equation for 2 Goats and 9 Sheep:**
- Now, we need to find out how many days \( s \) it takes for 2 goats and 9 sheep to graze the field.
- The work done by 2 goats in one day is:
\[ 2G \]
- The work done by 9 sheep in one day is:
\[ 9S \]
- The total work done by 2 goats and 9 sheep in one day is:
\[ 2G + 9S \]

4. **Substituting \( G \) in Terms of \( S \):**
- From the ratio \( G = \frac{3}{2}S \), we can substitute \( G \) in the equation:
\[ 2G + 9S = 2\left(\frac{3}{2}S\right) + 9S = 3S + 9S = 12S \]

5. **Finding the Total Work:**
- The total work required to graze the field is \( 300S \) (as calculated from 6 sheep grazing for 50 days).
- If 2 goats and 9 sheep do \( 12S \) work in one day, the number of days \( s \) required to complete the work is:
\[ s = \frac{\text{Total Work}}{\text{Work done in one day}} = \frac{300S}{12S} = 25 \text{ days} \]

### Final Answer:
Thus, 2 goats and 9 sheep can graze the field in **25 days**.

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