[SOLVED]Sequences and series

Dhamnekar Winod

Active member
Hi,

A person has 40 litres of milk. As soon as he sells half a litre, he mixes the remainder with half a litre of water. How often can he repeat the process, before the amount of milk in the mixture is 50% of the whole?
Detailed explanation is appreciated.
Solution:

I am working on this problem. Meanwhile if any member of math help boards knows the correct answer, may reply to this question with correct answer.

MarkFL

Staff member
I would begin by letting $$M_n$$ be the amount of milk in the mixture (in L) after the $$n$$th step/transaction. So, we have:

$$\displaystyle M_0=40$$

$$\displaystyle M_1=39.5$$

Now, during the second transaction, we don't have 0.5 L of milk leaving, we have:

$$\displaystyle \frac{M_1}{M_0}\cdot\frac{1}{2}$$ liters leaving. This suggests to me that we may state:

$$\displaystyle M_n=M_{n-1}-\frac{1}{2}\cdot\frac{M_{n-1}}{M_{0}}=M_{n-1}\left(\frac{2M_0-1}{2M_0}\right)$$

What is the root of the characteristic equation?

Dhamnekar Winod

Active member
I would begin by letting $$M_n$$ be the amount of milk in the mixture (in L) after the $$n$$th step/transaction. So, we have:

$$\displaystyle M_0=40$$

$$\displaystyle M_1=39.5$$

Now, during the second transaction, we don't have 0.5 L of milk leaving, we have:

$$\displaystyle \frac{M_1}{M_0}\cdot\frac{1}{2}$$ liters leaving. This suggests to me that we may state:

$$\displaystyle M_n=M_{n-1}-\frac{1}{2}\cdot\frac{M_{n-1}}{M_{0}}=M_{n-1}\left(\frac{2M_0-1}{2M_0}\right)$$

What is the root of the characteristic equation?
So the answer to this question is $\frac{\ln{(0.5)}}{\ln{(0.9875)}}=55.1044742773$

MarkFL

Staff member
The characteristic root is:

$$\displaystyle r=\frac{2M_0-1}{2M_0}$$

And so the closed form is:

$$\displaystyle M_n=c_1\left(\frac{2M_0-1}{2M_0}\right)^n$$

Now, we know:

$$\displaystyle M_0=c_1$$

Hence:

$$\displaystyle M_n=M_0\left(\frac{2M_0-1}{2M_0}\right)^n$$

To answer the question, we now want to solve:

$$\displaystyle M_n=\frac{1}{2}M_0$$

$$\displaystyle M_0\left(\frac{2M_0-1}{2M_0}\right)^n=\frac{1}{2}M_0$$

$$\displaystyle \left(\frac{2M_0-1}{2M_0}\right)^n=\frac{1}{2}$$

$$\displaystyle n=\frac{\ln(2)}{\ln\left(\dfrac{2M_0}{2M_0-1}\right)}$$

Using $$M_0=40$$, we have:

$$\displaystyle n=\frac{\ln(2)}{\ln\left(\dfrac{80}{79}\right)}\approx55.10447\quad\checkmark$$

So, we find that on the 56th repetition of the process, the mixture will be less than 50% milk.