^{[This is a harder problem ... ]}

‘Please help me with my homework, Dad! You’re a farmer – you should be able to do this one!’

Farmer Brown read the problem: The grass in the meadow grows thick and fast. If 70 cows eat it all up in 24 days, and 30 cows eat it all up in 60 days, how many cows would crop the grass in 96 days?

‘Well,’ said Farmer Brown learnedly, ‘96 = 4 x 24, so 1/4 of 70 is 17 1/2 cows ... . No, no! 96 / 60 is ... . If 70 cows take 24 days, then 30 cows should take 56 days. But the problem says 60 ... . Ask your mother, boy!’

With some wisdom, Mrs Brown observed that perhaps the fact that the grass was growing was of some importance.

Without this assumption the problem does not make sense, and is insoluble. And with this assumption ...?!

Hint 1

This looks like a problem on simple proportion, but clearly it is not (as Farmer Brown found out!)

Hint 2

Set the problem up algebraically, noting that there will be an initial amount of grass in the meadow, and some grown grass each day.

Solution

If this were a simple proportion problem, we could derive the answer from just the information on the 70 cows. This is what would happen if the grass was assumed to be not growing. Clearly this is a more complicated problem.

Suppose that A denotes the amount of grass in the meadow initially. In fact, it will be easier to assume that A = 1. (There is no problem in doing this. We just choose a (probably weird!) unit of volume which makes this so. Suppose too that the meadow produces volume g of grass each day.

Now let us build in the given information. In 24 days, the amount of grass produced will be 1 + 24g. This is eaten by 70 cows in exactly 24 days. Hence the amount eaten by each cow in one day is 1 + 24g divided by (70 x 24). Similarly, since 30 cows eat up all the grass in 60 days, the amount eaten by each cow is also given by 1 + 60g divided by (30 x 60). If we equate these two expressions, we should be able to find a value for g. Thus:

   1 + 24g    =    1 + 60g .    (*)
    70 x 24        30 x 60

Cross multiplying, and collecting together the terms in g, and then simplifying gives g = 1/480. Substituting this value into either side of equation (*), we find that the amount each cow consumes per day is 1/1600.

Finally, suppose that N cows eat up all the grass in 96 days. Arguing as before, we get

   1 + 24g   =   1 .
    96 x N       1600

Hence 96N = 1600(1 + 24/480), giving N = 20 as the required number of cows.

Although the arithmetic is a bit messy here, this problem shows that we need to be flexible in our thinking; there may be factors involved which we do not see at first.

Hint 1

Hint 2

Solution