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A group of 10 new mothers in a support group, each with a single baby, compare the lengths of their pregnancies. The difference between the longest and shortest pregnancy is 32 days, and the group's average pregnancy duration is 275 days. Additionally, all 10 of the babies were born in August of 2024. Given this information, what is the earliest possible date that one of the babies was conceived?

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  • $\begingroup$ Does this account for the two week false count? Human pregnancies are usually counted from last menstruation instead of conception. Of course, that two weeks isn't fixed, the mothers might not have been consistent in usage. Worse is the possibility of a frozen fertilized embryo, which could give use YEARS before the birth. $\endgroup$
    – David G.
    Commented Jul 10 at 1:45

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Assuming pregnancy lasts an integer number of days and the average of 275 days is exact:

An earlier date of conception means a longer pregnancy and an earlier birthday. The birthday should be as early as possible, so it is 2024/08/01. Since maximum length = minimum length + 32 days, we want to maximise the minimum length. If the minimum length is 272 or more days, the maximum length is at least 304 days. The average length is at least $\frac{272\times9+304}{10}>275$ days. This is not possible so the minimum length is at most 271 days. The maximum length is at most 303 days. For the length of pregnancies, one lasted 303 days, one lasted 279 days, all others are 271 days. So 303 days is indeed possible and the earliest date of conception is 2023/10/03.

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I'm going to assume that new mothers comparing their respective gestational lengths use the difference between two dates rather than whole-number or fractional day counts; and that the average they found was rounded. I think those are reasonable real-life assumptions.

Because we want to maximize one gestation while keeping the average fixed, we want to minimize the others. So, assuming that the shorter gestations have a length of $x$ days, we have $$\frac{9(x)+1(x+32)}{10}=275$$ to the nearest integer, so that $x=272$. Then the longest gestation was $304$ days, and the baby was conceived, at earliest, early in the day on October 2, 2023.

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