# School task for cooperative work

Alice can eat a cake in 10 minutes, a jar of jam in 13 minutes and drink a pan of milk in 14 minutes. Bob can do the same in 6, 6 and 7 minutes. What is the shortest time they can have breakfast with a jar of jam, a pan of milk and a cake?

I have two solutions with different answers:

1-st way.

Alice eats 1/10 of cake in a minute and Bob eats 1/6 of cake in a minute, so they eat 1/10+1/6=4/15 of cake in a minute. So they need 15/4 minutes to eat all the cake.

Alice eats 1/13 of jam in a minute and Bob eat 1/6 of jam in a minute, so they eat 1/13+1/6=19/78 of jam in a minute. So they need 78/19 minutes to eat all the jam.

Alice drinks 1/14 of milk in a minute and Bob drinks 1/7 of milk in a minute, so they drink 1/7+1/14=3/14 of milk in a minute. So they need 14/3 minutes to drink all the milk.

Common time is 15/4 + 78/19 + 14/3 = 2855/228 or 12 minutes and 119/228.

2-nd way.

Alice eats all breakfast for 10 + 13 + 14 = 37 minutes. She eats 1/37 of whole breakfast in a minute.

Bob eats all breakfast for 6 + 6 + 7 = 19 minutes. He eats 1/19 of whole breakfast in a minute.

Together they eat 1/37 + 1/19 = 56/703 of breakfast in a minute.

So they need 703/56 minutes for eating all breakfast or 12 minutes and 31/56.

• I take it they can help each other eat? (Eat each other's food?) Commented Nov 4, 2020 at 18:27
• The second way is definitely wrong. You're assuming they can eat each item at the same rate but that's not true. When you combine the total time it takes to eat things, you're getting an average rate for the whole meal, but as they're eating, certain things will be ate faster or slower than others so you can't use the 2nd approach. Commented Nov 4, 2020 at 18:59

As the question does not say that they both have to eat/drink each item, here is how we can optimize.

Bob finishes a jar of jam in $$6$$ minutes. Then starts drinking milk.

Alice starts with eating cake and finishes in $$10$$ minutes.

At the end of $$10$$ minutes, the jam is finished, the cake is finished and they are left with $$\frac{3}{7}$$ of the milk (As Bob finishes $$\frac{4}{7}$$ in $$4$$ minutes after finishing jam).

Together they can drink $$\frac{3}{14}$$ of milk in a minute so in $$2$$ minutes, they can finish rest of the milk ($$3/7$$).

So total of $$12$$ minutes.

• And I would add that the reason Bob starts with Jam instead of cake while Alice does jam is because of comparative advantage. Commented Nov 4, 2020 at 19:03
• @CottonHeadedNinnymuggins well said! In fact Bob starting with milk while Alice starts with cake and then finishing the jar of jam is very close $(1/19)$ mins more. Commented Nov 4, 2020 at 19:07