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I am nursing/pumping milk for my son and want to figure out how much longer I have to nurse/pump to be able to give him breast milk until he turns 1. I feel like this can be figured out via math but my brain isn't working well enough these days to do it. Here is the necessary information.

  • As of tomorrow (8/15) he will turn 1 in 97 days.

  • He nurses/bottle feeds 4 times each day

  • by the end of today I will have 142 bags of frozen milk for him (1 bag = 1 bottle)

  • each weekday I pump 2 times producing 3 bags (2 bags at 11:30 & 1 bag at 3:30)

  • I exclusively nurse on the weekends for now

I will need to gradually quit pumping/nursing. I will first drop the 3:30 pumping (and will bottle feed at 3:30 on weekends) followed by the 11:30 pumping (and will bottle feed on weekends). At that point I will be nursing morning and night 7 days a week and bottle feeding 7 days a week at 11:30 and 3:30. From here I will first drop the morning nursing and replace with a bottle and will finally drop the evening nursing and will replace with a bottle. At that point he will be strictly bottle fed from my frozen supply until he turns 1.

So the question is: when can I drop my 3:30 pumping, then my 11:30 pumping, then my morning nursing and finally my night nursing so that I have enough frozen milk to make it to my son's first birthday?

user1729
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Michelle
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  • I think that this question should be asked to a doctor/pediatrist. I do not know how much does it make sense to "estimate" the process using the given data, without considering other factors. – Avitus Aug 14 '13 at 17:39
  • From a mathematical point of view you will need to fulfil the equation that $(freeze/thaw; milk)=(fresh ;milk)$ and exactly this is from a biochemical point of view not true/equal. Hence no mathematical solution can be provided for the equation does not hold. – al-Hwarizmi Aug 14 '13 at 18:38
  • There are no other factors to consider...I simply need to do the math but don't want to have to do trial and error to get the right answer. I would currently need 388 bags of milk to get him through to a year if I only used frozen milk. I only have 142. So I need to figure out how much longer I nurse/pump 4 times per day, then 3 times per day then 2 times per day, then 1 per day and finally 0...all while he is still having milk 4 times per day. Obviously, right now, daily supply is still greater than demand as I freeze 3 new bags a day and he only goes through 2 bags per day. – Michelle Aug 14 '13 at 19:49

1 Answers1

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(I realise this is too late to help the OP, but more than one person expressed an interest in this question, so it may have value to others. My aim is that it should have an answer. Also, the details of the question are pretty complicated, so I'm simplifying a bit to give something slightly more general.)

First I want to phrase this as a general maths question.

You have an event of some kind occurring at a particular starting frequency $F_0$, and want to change the frequency in steps to an ending frequency $F_1$. I will assume these frequencies are an integer number of instances per unit of time, and the step size is $1$. Over the transition period, the desired total number of instances of the event is $T$. I will also assume that the step changes occur at regular intervals. Moreover, I will assume $F_0> F_1$ (otherwise we can swap them over).

Take $t_s$ to be the time interval between steps. Then the total number of times the event will occur is $$ t_s\times F_0+t_s\times (F_0 -1)+\cdots +t_s\times (F_1+1)=t_s\sum_{k=F_1+1}^{F_0}k=t_s \left(\sum_{k=0}^{F_0} k - \sum_{k=0}^{F_1} k\right)=t_s\left(\frac{F_0(F_0+1)}{2}-\frac{F_1(F_1+1)}{2}\right). $$ In this case, we want to be sure to reach $T$, so overshooting is preferable to undershooting. Therefore we want to find the minimal integer value of $t_s$ for which $$ T\leq t_s\left(\frac{F_0(F_0+1)}{2}-\frac{F_1(F_1+1)}{2}\right). $$ That is, we calculate $$ T\left(\frac{F_0(F_0+1)}{2}-\frac{F_1(F_1+1)}{2}\right)^{-1} $$ and then round up.

Now for the specific case in question here.

For $97$ days at $4$ feeds a day, a total of $388$ bags of milk are needed. Since $142$ have already been created, another $T=246$ are required over the steping-down process. The initial frequency is $F_0=5$ bags produced per day (I've simplified here by ignoring the difference at weekends, when in fact only $4$ bags worth of milk are produced). The final frequency is $F_1=0$ bags per day. Thus $$ \frac{F_0(F_0+1)}{2}-\frac{F_1(F_1+1)}{2}=\frac{5\times 6}{2}-\frac{0\times 1}{2}=15 $$ and so $$ t_s\geq 246/15=16.4. $$ Our answer then is, roughly, that reducing the milk produced by 1 bag every 17 days should work, with the first drop occurring in 17 days time. (I have simplified the question by treating the two bags of milk produced at 11:30 as independent.)

Jessica B
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