Math CentralQuandaries & Queries


Question from Kenneth:


I want to determine which quantities are directly and inversely proportional in order to determine the answer for the following:.

If 4 cooks can bake 8 pies in 6 hours, 2 cooks can bake how many pies in 4 hours?

Answer: 2 2/3 pies

Can someone fully explain what I need to know in order to determine what is directly and inversely proportional in the example above?

I thank you for your reply.


The person who wrote this problem is assuming (absurdly for small numbers such as these) that the number of pies baked is directly proportional to the number of cooks and directly proportional to the time.

I have to ask - has the person who made this question up ever baked a pie? For a kitchen-scale baking project like this, even a second cook will make almost no difference to the time. (And baking 2/3 of a pie? What is that even supposed to mean?)

Assume we're baking from scratch. Most of the time involved in baking a pie is oven time and resting the crust - a million cooks will not speed this up, and having only one cook will not slow it. Some other parts of the task are not on the "critical path" - they can be done during necessary delays. The only scenario I can think of in which 8 pies require 6 hours involves a small oven that can hold 2 pies, each baking taking around an hour - that time is implausibly long for three bakings. If eight pies are made in five, six, or seven bakings, the oven must be able to hold two pies; so four bakings would do. Eight single-pie bakings would take too long, at least in a conventional oven. So, four bakings of two pies each.

We need around one hour to rest the dough. You just need to know this. Or look it up.

Fillings can be prepared while the dough is resting, later crusts rolled while earlier pies are baking. These are not on the critical path and do not affect the time.

Let's assume making and rolling the dough take between five minutes (very short) and half an hour (very long) each. Then a baking takes at least an hour, but the other steps leave less than three hours; so there can be at most two bakings.

So four hours allows two bakings, or four pies.

With pre made frozen pie shells and canned filling, the problem's different; setup time is almost zero. So eight pies in six hours means slower baking. Possible baking schedules that need 6 hours for 8 pies are 2 pies at a time, for a rather long 90 minutes, or a 1-pie convection oven that does it in 45 minutes. The first of these allows 4 pies to be baked in 3 hours, the second 5 in 3 hours 45 minutes. In either case, the slack time cannot be used.

Good Hunting!

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