Math Problems, Proportions Problems

Problem of the Day: 12/25/12

Merry Christmas!

Mrs. Claus is baking post-Christmas cookies to treat all the elves for their hard work. She needs 3/4 cup sugar for every 20 cookies she makes, and she has 400,000 elves working in the North Pole (they take shifts). If every elf should get a cookie, how much sugar will she need?

Solution to yesterday’s problem:
A good way to solve our elvish problem from yesterday is to use dimensional analysis. If we express each piece of information as a ratio, then we can multiply these ratios to get the value we want, in this case how many days it would take for the elves to make 3 billion gifts.

First, we know that each elf can make one gift in one minute. Let’s write this as a fraction.
\frac{1 \textup{gift}}{1 \textup{minute}}
Next, we know that there are 60 minutes in an hour, and 24 hours in a day.
\frac{60 \textup{minutes}}{1 \textup{hour}}, \frac{24 \textup{hours}}{1 \textup{day}}
We also have 200,000 elves, so we can add that as a constant on the left.
200,000\times\frac{1 \textup{gift}}{1 \textup{minute}}\times\frac{60 \textup{minutes}}{1 \textup{hour}}\times\frac{24 \textup{hours}}{1 \textup{day}}
The units (minutes, hours) cancel out, leaving the following:
200,000 \textup{gifts}\times60\times\frac{24}{1 \textup{day}}
Multiplying, we get \frac{288 000 000 \textup{gifts}}{\textup{day}}

Now we want to figure out how many days it would take the elves to make 3 billion gifts. To do this, simply divide 3 000 000 000 by 288 000 000, which gives us 10.416666…

Of course, we can’t leave the answer like that. The question is asking on what day in December will the elves have to start making gifts. They’ll have to work all of the 24th, 23rd, 22nd, 21st, 20th, 19th, 18th, 17th, 16th, and 15th, and they’ll also have to work part of the 14th. So the elves will have to start on December 14th.

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