Subject: 2 mathematical problems

Dear Math Central,

I am a trainee teacher. I would appreciate it very much if you could help me solve the following two problems which I think are meant for students at middle grade.

1. If a hen and a half lays an egg and a half in a day and a half, how many hens are needed to lay a dozen eggs in one day?
2. There are fewer than 200 passengers on a train. If they get off in pairs, one passenger will be without a partner. If they get off in groups of 3 or 4, there will still be one passenger left by himself. However, if they get off in groups of 5, no one will be left by themselves. How many passengers are there on the train?

Yours sincerely,
Faye

For your first problem it is helpful to make a chart for hens(H), eggs(E) and days(D) as follows:

To make the units easier first try to eliminate some fractions.

H E D 1 1/2 1 1/2 1 1/2 3 3 1 1/2 12 12 1 1/2 . . . 18 12 1

<-- Twice the hens gives twice the eggs in the same time. <-- Likewise <-- 1 1/2 times as many hens needed to speed up the process.

There is a problem here with this solution as a valid row in the table is

H	E	D
1	1	1 1/2

and there is a legitimate argument to say that no number of hens can lay a dozen eggs is one day since it takes each hen a day and a half to lay an egg.

Let the number be n then we know n<200 and n is a multiple of 2 plus 1, a multiple of 3 plus 1, a multiple of 4 plus one and a multiple of 5. Thus

n is in	{1, 3, 5, 7, ... 199}
n is in	{1, 4, 7, 10, ... 199}
n is in	{1, 5, 9, 13, ... 197}
and
n is in	{5, 10, 15, 20, ... 195}

n is in	{5, 15, 25, 35, ... 195}
n is in	{10, 25, 40, 55, ... 190}
n is in	{5, 25, 45, 65, ... 185}
and
n is in	{5, 10, 15, 20, 25, ... 195}

It is now easy to see that one answer is 25. To find others add the lcm of {2, 3, 4, 5}, which is 60, to get 85 and 145 as the other possibilities less than 200.

Cheers, Penny