4 goats or 6 sheep can graze a field in 50 days. 2 goats and 3 sheep will graze it in
50 days
Solving Goats and Sheep Grazing Time Problems
This question is a classic example of a work-rate problem, where different entities (in this case, goats and sheep) contribute to completing a task (grazing a field) over a certain period. The key is to understand the relationship between the work rates of the different entities.
Understanding the Relationship Between Goats and Sheep Work Rates
We are given that:
- 4 goats can graze the field in 50 days.
- 6 sheep can graze the same field in 50 days.
Since both 4 goats and 6 sheep take the same amount of time (50 days) to complete the same task (graze the field), their total work capacity over that period must be equal. This means the total work done by 4 goats is equivalent to the total work done by 6 sheep.
We can express this equivalence in terms of their work rates:
\(\text{Work Rate of } 4 \text{ goats} = \text{Work Rate of } 6 \text{ sheep}\)
Finding the Equivalent Work Rate for a Smaller Group
To find the equivalent work rate for fewer animals, we can simplify this relationship. Dividing both sides of the equivalence by 2, we get:
\(\frac{\text{Work Rate of } 4 \text{ goats}}{2} = \frac{\text{Work Rate of } 6 \text{ sheep}}{2}\)
This simplifies to:
\(\text{Work Rate of } 2 \text{ goats} = \text{Work Rate of } 3 \text{ sheep}\)
This is a crucial relationship: 2 goats have the same work rate as 3 sheep.
Calculating Time for the Combined Group
We need to find the time it takes for 2 goats and 3 sheep to graze the field. The combined group consists of "2 goats and 3 sheep".
Using the equivalence we just found, we can substitute the work rate of 3 sheep with the work rate of 2 goats. So, the combined group's work rate is equivalent to:
\(\text{Work Rate}(2 \text{ goats} + 3 \text{ sheep}) = \text{Work Rate}(2 \text{ goats} + \text{equivalent of } 3 \text{ sheep})\)
\(\text{Work Rate}(2 \text{ goats} + 3 \text{ sheep}) = \text{Work Rate}(2 \text{ goats} + 2 \text{ goats})\)
\(\text{Work Rate}(2 \text{ goats} + 3 \text{ sheep}) = \text{Work Rate}(4 \text{ goats})\)
Alternatively, we could substitute the work rate of 2 goats with the work rate of 3 sheep:
\(\text{Work Rate}(2 \text{ goats} + 3 \text{ sheep}) = \text{Work Rate}(\text{equivalent of } 2 \text{ goats} + 3 \text{ sheep})\)
\(\text{Work Rate}(2 \text{ goats} + 3 \text{ sheep}) = \text{Work Rate}(3 \text{ sheep} + 3 \text{ sheep})\)
\(\text{Work Rate}(2 \text{ goats} + 3 \text{ sheep}) = \text{Work Rate}(6 \text{ sheep})\)
Both ways show that the combined group of 2 goats and 3 sheep has the same total work rate as 4 goats or 6 sheep.
Since the combined group has the same work rate as 4 goats, and we know that 4 goats take 50 days to graze the field, the combined group of 2 goats and 3 sheep will also take the same amount of time.
Conclusion on Grazing Time
Therefore, 2 goats and 3 sheep will graze the field in 50 days.
| Group | Time to Graze Field |
|---|---|
| 4 Goats | 50 days |
| 6 Sheep | 50 days |
| 2 Goats & 3 Sheep | ? |
From our analysis, the work rate of 2 goats & 3 sheep is equivalent to the work rate of 4 goats (or 6 sheep). Since they have the same work rate, they will take the same time.
Revision Table: Work-Rate Concepts
| Concept | Description | Relation to Problem |
|---|---|---|
| Work Rate | The amount of work done per unit of time. | Used to compare the efficiency of goats and sheep. |
| Total Work | The complete task to be done (grazing the field). | The total work is the same for all groups mentioned. |
| Equivalence | When different entities have the same work rate or complete the same work in the same time. | 4 goats are equivalent to 6 sheep in terms of total work/rate for this field. |
| Combined Work Rate | The sum of the work rates of individuals working together. | The work rate of (2 goats + 3 sheep) is the combined rate. |
Additional Information: Work-Rate Problems Tips
- Always try to find a relationship or equivalence between the different types of workers or entities involved.
- If \(M_1\) entities can do a work in \(D_1\) days, and \(M_2\) entities can do the same work in \(D_2\) days, then \(M_1 \times D_1 = M_2 \times D_2\), assuming the work rate per entity is constant within each type. This can help find equivalences or solve for unknown times/entities.
- In this problem, \(4 \text{ goats} \times 50 \text{ days} = 6 \text{ sheep} \times 50 \text{ days}\). This implies \(4 \text{ goats} = 6 \text{ sheep}\) in terms of their total work capacity over the same time, leading to the work rate equivalence \(2 \text{ goats} = 3 \text{ sheep}\).
- When entities work together, their work rates are usually added. For example, if a goat does \(w_g\) work per day and a sheep does \(w_s\) work per day, the combined work rate of 2 goats and 3 sheep is \(2w_g + 3w_s\). Using our equivalence \(2w_g = 3w_s\), the combined rate is \(2w_g + 2w_g = 4w_g\) or \(3w_s + 3w_s = 6w_s\).
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