HOT Competition 31 Aug/8AM: Let’s assume that 328 children are born in

Let’s assume that 328 children are born in 1 year. Among these children, if October has the most number of births only, what is the least possible value of the number of births in October?

Distributing evenly we get 27 month and 4 extra. but since we need least number we add 2 more than avg and distribute the other two amon remaining.
so 29 is least highest others will 27(9)+28(2)

Let’s assume that 328 children are born in 1 year. Among these children, if October has the most number of births only, what is the least possible value of the number of births in October?

We have to first evenly divide 328 into 12 months.
=328/12
=27 births per month
Plus 4 balance births need to be distributed among the months
As October has the only maximum value among the group and we have to minimize the maximum value
So we can distribute 2 births separately into two months and 2 births to October

It means October has 29 births
2 months has 28 births
9 months has 27 births

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So, first of all let's understand the question. It's asking about a "CAN BE TRUE" case, and not a "MUST BE TRUE" case.
What is the minimum number of births we CAN have in October, such that it STILL CAN BE the month with the greatest birth. In other words, even if there was 1 birth less in October, some other month would NECESSARILY HAVE TO become equal to or greater than the number of births in Oct, to maintain a final sum of 328. (Note the important difference here. It's not a MUST BE TRUE. This distinction will become clearer when we get the answer)

So, 328 births in 12 months ==> 27.33 births per month (since births can't be decimal, let's say 27 births in 11 of the months and the rest in October).
Note that the condition for minimizing the number in October is same as the condition for maximizing it in the other months (which still keeping them under Oct). Now if we are maximizing all others, they should all have the same number of births.

This brings us to:
11x27 + Oct = 328
=> Oct = 31

Note that, Oct is still 4 more than any other month. We can afford to give some birth away to other months, while still maintaining Oct on top.
So,
11x27 + 31 = 328
10x27 + 28 + 30 = 328 (1 birth moved from Oct to 1 other month)
9 x 27 + 2 x 28 + 29 = 328 (2 births moved from Oct to 2 other months)

Note that we can't move any more births away from Oct (else it will become 28, and won't be the ONLY month with maximum births).
So, our answer is 29 (Answer C)

Note: If Oct has 29, it's NOT always necessary that it MUST have maximum number of births. There may be a case where some other month has same or more births. But, that case ISN'T NECESSARILY TRUE. On the other hand, if Oct had 28 or lesser, IT WILL NECESSARILY NOT HAVE maximum number of births.

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It is given that "only October can have most number of births". Now let us try out with the least of the options in an incremental manner (since the question wants us to find the least number, we can stop looking further once a number fits in).

A. 27 - INCORRECT : Out of 328 children, 27 are already born in October and October has most number of births. All the other months must have at least 1 birth less for this option to be correct. Remaining births are 328-27 = 301. If 26 (just one less than 27) children are born in 10 of the remaining months, there would be total 260 births of the remaining 301, leaving 41 for the 11th month. 41>27 - Doesn't satisfy the condition.

B. 28 - INCORRECT : Out of 328 children, 28 are already born in October and October has most number of births. All the other months must have at least 1 birth less for this option to be correct. Remaining births are 328-28 = 300. If 27 (just one less than 28) children are born in 10 of the remaining months, there would be total 270 births of the remaining 300, leaving 30 for the 11th month. 30>28 - Doesn't satisfy the condition.

C. 29 - CORRECT : Out of 328 children, 29 are already born in October and October has most number of births. All the other months must have at least 1 birth less for this option to be correct. Remaining births are 328-29 = 299. If 28 (just one less than 29) children are born in 10 of the remaining months, there would be total 280 births of the remaining 299, leaving 19 for the 11th month. 19<29 - Satisfies the condition.

Since we have found the value, we need not look further.

This is the approach of elimination that I could think of. If some one could think of an approach to directly reach to the answer, I would really appreciate it if you could share that.

Test choice;

A. If the least possible value of the number of births in October is 27, the maximum value of the number of births in other months can be 26.
--> 26*11=286 children are born in 11 months, except October.
--> The number of children are born in October is 328-286=42.
--> Then, we know that the maximum value of the number of births in 11 months can be more than 26, and the least possible value of the number of births in October is not 27.
A. OUT.

B. If the least possible value of the number of births in October is 28, the maximum value of the number of births in other months can be 27.
--> 27*11=297 children are born in 11 months, except October.
--> The number of children are born in October is 328-297=31.
--> Then, we know that the maximum value of the number of births in some months can be more than 27 to make the least possible value of the number of births in October lower than 31, so the least possible value of the number of births in October is not 28.
B. OUT.

C. If the least possible value of the number of births in October is 29, the maximum value of the number of births in other months can be 28.
--> 28*11=308 children are born in 11 months, except October.
--> The number of children are born in October is 328-308=20. Something's wrong here because the no. in October need to be the highest!!
--> Then, we have to decrease the number in other months, and the solution below is work.
Jan - Sep ; 27 babies/month ; sum=27*9=243 babies
Nov - Dec ; 28 babies/month ; sum=28*2=56 babies
Oct ; 29 babies/month ; sum=29 babies
Total 243+56+29=328
ANS C.

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Answer is C 29
Let us solve through option
A.27
Oct =27*1=27+26*11= 313
B.28
Oct= 28*1=28+27*11=325
C.29
Oct= 29*1=29+28*11=337
So we can manage this option and we need the least so don't look further

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The answer is (C).


We want to know *least* possible value of the number of birth in October, and October has the most number of birth only.
This is similar with one of day 28th’s math question about the distribution of candies.
We can distribute babies equally 12 months, and with remained babies, we can find least number of babies.

-> \(\frac{328}{12}=27\frac{4}{12}\).
-> When you distribute 4 babies to 1per each 4 different months, October is not a month with the most number of babies.
-> When you distribute 4 babies to 2 for October, and 1 per each 2 different months, October has 29 number of babies, other two months have 28 each, and rest of months have 27 babies each. Therefore, October is the month with the most number of babies, having at least 29 babies.


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Let’s assume that 328 children are born in 1 year. Among these children, if October has the most number of births only, what is the least possible value of the number of births in October?

Given:

1) 328 children are born in 1 year
2) Most number of children are born in October.

In this question, we have to calculate the least value of the number of births in October, therefore we have to maximise values of all other months except Oct.

Divide 328 by 12, Quotient= 27 and remainder =4

Now divide 27 to each month,
January=27
Feb =27
March=27
April=27
May =27
June =27
July=27
Aug=27
Sep=27
Oct=27
Nov=27
Dec=27

We are left with Remainder = 4, now if we distribute remaining four equally to sept, oct , nov and dec
Sep=28
Oct=28
Nov=28
Dec=28

Then Oct is not maximum

Now, to maximum Oct transfer one birth from any month to Oct

Oct =29 (Answer)

You get 9 months with 27 births, 2 months with 28 births, and 29 births in October as shown above.

The answer is C.

A. 27
B. 28
C. 29
D. 30
E. 31

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