*Zeyu Ma*

You are given the following information:

- 10 cows eat all the grass that starts on a 2-acre field, together with all the grass that grows on the field in a total of 5 days
- 11 cows eat all the grass that starts on a 2-acre field, together with all the grass that grows on the field in a total of 4 days

How many days will it take to eat all the grass if there are 15 cows on a 3-acre field?

**Scroll down for the solution!**

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**Method 1**

We assume that every cow eats the same quantity of grass every day and the grass has a constant growth rate, i.e. the quantity of grass that grows per acre per day is constant. At the start when the cows begin to eat, the quantity of grass per acre is the same.

There is a traditional way to solve this problem which is to list all the variables we don’t know and then try to formulate equations to solve them. Here, we don’t know how much a cow eats per day – let’s call this x (note we don’t care about the units here, maybe it’s x kg or x trucks – they are ultimately irrelevant). Also, let’s call the quantity of grass which grows per acre per day y, and the initial quantity of grass per acre z.

We begin by using the relations we know to list equations: initial grass + grass grown = total amount eaten. We know that 10 cows eat all the grass on a 2-acre field together with all the grass that grows on the field in 5 days. Therefore, we have:

2𝑧 + 5 · 2𝑦 = 10 · 5 𝑥

2z + 10y = 50x

Our second piece of information tells us that 11 cows eat all the grass on a 2-acre field together with all the grass that grows on the field in 4 days, which becomes:

2𝑧 + 4 · 2𝑦 = 11 · 4 𝑥

2z + 8y = 44x

We now express y and z in terms of x and then calculate the answer to the question. Substracting the equations gives:

2y = 6x, so 𝑦 = 3𝑥.

And now substituting into the second equation we have:

2z + 8(3x) = 44𝑥, or 2z = 20x which gives z = 10x.

We assume the 15 cows will eat all the grass on the 3-acre field in 𝑎 days. Formulated as an equation this gives:

3𝑧 + 𝑎 · 3𝑦 = 15 · 𝑎 · 𝑥

And now substituting for y and z in terms of x we find:

3(10x) + a(9x) = 15ax

30 + 9a = 15a, or 30 = 6a, which gives a = 5.

This method clearly works, however, what if we hate solving equations… is there an easier way to think about this question?

**Scroll down for method 2!**

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**Method 2**

First, imagine the scenario where there was no grass at the beginning, what would happen? If the cows eat slower than the grass grows, they will never consume all of the grass and if the cows eat faster than the grass, they will be hungry on the very first day. Therefore, there must be a specific number of cows per acre which eat the same quantity of grass per day as the grass grows per day.

Continuing with this idea, we can divide the cows into two groups: one group of cows that consume the grass as soon as the grass grows, and the other cows that consume the initial quantity of grass gradually. The number of cows per acre which eat the same quantity of grass per day as the grass grows is constant. Thus, the time when the cows eat all the grass depends on the number of cows in the second group. The more cows in this group, the faster the grass is eaten. Therefore, we can conclude that the time is inversely proportional to the number of cows in this group.

In this question, for example, the time is 5:4, so the ratio of the cows who eat the initial grass to those that eat the new grass is 4:5. Because the area of the field is the same, the numbers of cows who eat grass as it grows is also the same. The difference between 10 and 11 cows is caused by the difference in the number of cows who eat the initial grass. Thus, in the first case, there are 4 cows who eat the initial grass and 6 cows who eat grass as it grows. In the second case, there are 5 cows who eat the initial grass and 6 cows who eat the grass as it grows. From this we can know the rate at which grass grows in a 2-acre field equals the rate at which 6 cows eat, so the rate of growth per day per acre is the rate of eating per day per 3 cows.

Also, the quantity of grass on the 2-acre field at the start equals the quantity 5 cows eat in 4 days, or 4 cows eat in 5 days, so the initial quantity of grass is 10 cows per day per acre. In the case of the actual question we are asked to solve, there are 3 acres of field, so we need 9 cows eating as the grass grows and the initial grass can support 30 cow days, i.e. the other 6 cows will spend 5 days eating this grass. Thus, the answer is 5.

Now you may be wondering what happens if there are different areas of field in the two cases we are given information about? Well, we can convert them into the quantity corresponding to the same field. For example, the case where 20 cows eat all the grass in a 4-acre field together with all the grass that grows on the field in 5 days, is equivalent to the case where 10 cows eat all the grass on a 2-acre field together with all the grass that grows on the field in 5 days.