How to account for the time required to take a nutrition suggestion?

Question

The problem is as follows:

At a certain clinic doctor's suggest Kelly to take a diet consisting of $50\,g$ of mueslix each five hours and $40\,g$ of berries each four hours. If she began consuming both kinds of the food suggested and in total she has to take a total of $930 \,g$. How long measured in hours would she need to complete the recommended diet?

The alternatives in the book are as follows:

$\begin{array}{ll} 1.&\textrm{45 hours}\\ 2.&\textrm{42 hours}\\ 3.&\textrm{44 hours}\\ 4.&\textrm{43 hours}\\ 5.&\textrm{40 hours}\\ \end{array}$

What I've attempted to do to solve this problem was to account for the whole treatment as follows:

She began taking the mueslix and the berries at first so she began taking $40+50=90$ grams and it is mentioned in the problem that the dose of mueslix is $50$ grams each five hours and the dose of berries is $40$ grams each four hours, thus $t$ would account for the time in hours.

$40+50+\left(\frac{50}{5}\right)t+\left(\frac{40}{4}\right)t=930$

Solving this $t=42\,h$

However my book indicates that the answer is $44$ hours. What could I be doing wrong?. Can someone help me?.

Answer 1

Interesting word problem, at first I thought this will easy to do with diophantine equation. So I began solving that Linear diophantine equation by two variables and I got two practical combinations of (mueslix,berries) - $(9,12)$ and $(13,7)$ (Not writing the process because that's irrelevant here)

Since, mueslix is taken every 5 hour, so total time would be $9\times5=45$ and for berries $12\times4=48$ hours. However I did a mistake here, I didn't count the initial intake, i.e. I took the amount 0 at first, so subtracting 1 from the ordered pair will do. And hence the correct hours will be $40$ for mueslix and $44$ for berries. Which is the answer.

Your solution is better. But what you couldn't catch is that, the intake is done only in whole amounts, or in other words, 50 g of mueslix adda up only after 5 hours, same for berries.

The function that you made is a linear continuous function, a straight line. It must be discrete. And to obtain that change $\frac{t}{T} \rightarrow \left[\frac{t}{T}\right]$, $T$ denotes the time interval.

The equation becomes, $$40+50 +40\left[\frac{t}{4}\right]+50\left[\frac{t}{5}\right]=930$$ $$4\left[\frac{t}{4}\right]+5\left[\frac{t}{5}\right]=84$$

Rearrange the equation as$$5 \left[\frac{t}{5}\right]=4\left(21-\left[\frac{t}{4}\right] \right)$$

Both sides need to integers, and $\left[\frac{t}{4}\right]<21$ and $t\ge5$ and $\left(21-\left[\frac{t}{4}\right] \right)$ must be a factor of 5.

It can be $20, 15,10$ and $5$. If is $\left[\frac{t}{4}\right]=1$ then $4 \le t <7$ and for this range of $t$, $\left[\frac{t}{5}\right]$ can be just 1.

Can you check all possible cases now?

Answer 2

Although both foods are started at the same time, the 5-hour schedule for one and the 4-hour schedule for the other will get them unsynchronized. Fortunately, after 20 hours, they're back in sync --- A moment before the 20-hour mark, Kelly will have had 4 doses of mueslix (totaling 200 gm) and 5 doses of berries (also totaling 200 gm) and will be about to take a synchronized pair of foods, just as at the beginning. The next 20 hours go the same way, so just before the 40-hour mark, Kelly will have eaten a total of 800 gm and again be about to eat a sychronized pair. At the 40-hour mark, Kelly eats one dose of each food, meaning 90 gm, so the total is now up to 890 gm. To reach the desired total of 930, Kelly needs another 40 gm. So the goal will be achieved 4 hours later by eating 40 gm of berries. So the total time is 44 hours.