What is the minimum time needed for Shubrata and Amlan to eat 252 jackfruits?

Question

Once Shubrata and Amlan ate 50 jackfruits in total for 3 hours and 5 hours respectively. Another time they ate 200 jackfruits for 26 hrs and 10 hrs respectively. Now what is the minimum time needed for them to eat 252 jackfruits?

Source: Bangladesh Math Olympiad 2016 Junior Catagory

I can not understand the ratio of the time in this question. The second statement makes me more confused.

Answer 1

Let $x,y$ denote the number of jackfruits Shubrata and Amlan have eaten per hour. Then $3x+5y=50$ We also know that $26x+10y=200 \implies 13x+5y=100$.

Hence, $100-13x = 50-3x \implies 10x = 50 \implies x=5$ and $5y=50-15=35 \implies y=7$

Now, we're looking for the minimum amount of time it takes these two people to eat $252$ jackfruits. We're looking for time $t$ such that $5t+7t=252$. Hence, $12t=252$ and $t=21$ hours.

Answer 2

Say that Shubrata can eat $x$ jackfruits per hour and Amlan can eat $y$ jackfruits per hour. So you have:

$$3x+5y=50$$

$$26x+10y=200$$

Can you proceed from here? The answer is 21 hours.

For those curious, here is how jackfruit looks like :)

Answer 3

I would interpret the wording of such a question this way:

On the first occasion, Shubrata spent $3$ hours eating jackfruits and Amlan spent $5$ hours eating jackfruits. Together they ate $50$ jackfruits on this occasion, that is, if Shubrata ate $S$ jackfruits and Amlan ate $A$ then $S + A = 50.$

On the second occasion, Shubrata spent $26$ hours eating jackfruits and Amlan spent $10$ hours eating jackfruits. Together they ate $200$ jackfruits on that occasion.

I think we're meant to assume that each person has the ability to eat some number of jackfruits per hour, and the same person can go on hour after hour eating the same number of jackfruits, although a different person might be able to eat a different number of jackfruits each hour. Then from the two given occasions we can figure out how many jackfruits Shubrata eats each hour and how many Amran eats each hour.

Finally, I suppose that "minimum time" means that at some instant we supply the $252$ jackfruits, so that Shubrata and Amlan can eat them, after $T$ hours we observe that all the jackfruits have been eaten, and we want this to be done in such a way that $T$ is as small as possible. This is done when both Shubrata and Amlan spend the entire $T$ hours eating jackfruit so that no time is wasted.

Mind you, jackfruit is tasty, but I do not think anyone can eat so many jackfruit as either of these two, hour after hour, and not burst.