At a certain hospital, 89 children were born in the month of June. If more children were born on the fifteenth of June than on any other day in June, what is the least number of children that could have been born on the fifteenth of June?

The number of days is not given, is it possible to solve this problem without it?

There are 30 days in June, and 89 children born.

Suppose 60 of the 89 were spread over 30 days evenly, so every day, there are two newborns, leaving 28 to be distributed over 29 days.

If the remainder is spread out over the remaining days, so some have 3 and some have 2 newborns a day, then the least number of newborns on the 15th is 4 for this to happen (best case).

On the other hand, do not confuse with the problem of "what is the least number of children born on the 15th so that no matter what happens, there are more newborns on the 15th than any other day?"
In this case, the answer would be (89+1)/2=45 (worst case)

thanks! :)

You're welcome!

Yes, it is possible to solve this problem without knowing the total number of days in the month of June. Let's break it down.

To find the least number of children that could have been born on the fifteenth of June, we need to consider the minimum number of children born on all the other days in June.

Since the hospital had a total of 89 children born in June, we know that there are less than or equal to 89 children born on the fifteenth of June. To find the least number, we need to consider the maximum number of children born on the other days in June.

Since we don't have any specific information about the number of children born on the other days, we can assume that the number of children born on those days is evenly distributed. This means that we can divide the remaining children (89 minus the number born on the fifteenth) equally among the other days.

Let's say x represents the least number of children that could have been born on the fifteenth of June. Then, the number of children born on the other days would be (89 - x) / (Total number of days in June - 1).

We can assume the total number of days in June is 30, which is the maximum number of days in a month. So, the number of children born on the other days would be (89 - x) / 29.

Now, since we want to find the least number of children born on the fifteenth of June, we want to minimize the value of x. In this case, it would be achieved by maximizing the number of children born on the other days. We can assume the maximum number of children to be 89 - 1 = 88 (since we need at least one child born on the fifteenth). Therefore, we get:

(89 - x) / 29 <= 88
89 - x <= 88 * 29
89 - x <= 2552

To minimize x, we want to make the right side of the inequality as small as possible. Therefore:

x = 89 - 2552
x = -2463

Since we can't have a negative number of children, the least number of children that could have been born on the fifteenth of June is 0.

In conclusion, the least number of children that could have been born on the fifteenth of June, without any additional information, is 0.