Imagine This

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Imagine you are given a question that describes the following situation:

There is an isosceles triangle inscribed in a circle with radius = 5. The center of the circle is at (0,0). The triangle is symmetrical about the y-axis.

Do you have a picture in your head about what that would look like? If the answer is no, it could be because you’re unfamiliar with some of the terms that were described. Perhaps you’ve forgotten what an isosceles triangle is (a triangle with two sides of equal length) or perhaps you don’t quite remember which one is the y-axis (it’s the vertical one). If that’s the case, it’s time to go back and review some of the core mathematical terms that you’ll need to know in order to be successful on your test. The Barron’s video course¬†for your test would be a great place to start.

However, even if you got that far, there’s potentially another problem. My guess is that most of you have imagined something like this:

But did you also consider that this green triangle is a possibility as well?

The common advice we give when figures in the coordinate plane is that you should draw a picture. This is sound insofar as your brain can much more easily interpret the graphical information than a set a words given to describe that information. However, draw a picture can get you into trouble when you need to draw the picture or even the pictures.

It’s a word of caution that I hope you’ll remember. When you’re given some graphical information in word form it’s great to translate that into a picture. However, make sure you really take the time to dissect what all of the information means and could mean, so that you aren’t overlooking a part of the solution.

Good luck and happy studying!

 The 12 Days of Christmas

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It’s that time of year when you’re hearing all kinds of holiday music everywhere you go. Maybe the music reminds you of old memories with family. Maybe the music stirs generosity in your heart. Or maybe the music causes you to contemplate mathematical problems.

Okay, so unless you’re me, you problem don’t fall into that last category. But, now that you’re reading I started wondering how many total gifts are given in the 12 Days of Christmas song? If you’re not familiar with it, here’s a link to the lyrics.

The answer to this question isn’t so important to me as the process. How would you go about figuring out a question like that? Would you go gift by gift and count on your fingers and toes? Would you go day by day and sum the 12 totals? Would you be able to divine some other solution?

Since this really isn’t a test-type question that you’re likely to face, this is more an exercise in mathematical thinking. How would you go about it?

Once you’re done figuring that out, take a look at this excellent solution to the problem I proposed.

Isn’t math great?

 Thanksgiving Leftovers Part 2

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Auntie Donna makes a special drink for the Thanksgiving family gathering. All the kids love it. Little do they know that it’s just watered down grocery store punch. She has purchased 9 liters of punch from the store. If she wants to make punch that is 0.6 times as strong as the original, how much drink can she make?

A) 12 liters

B) 15 liters

C) 18 liters

D) 27 liters

E) 54 liters

In order to make punch that is 0.6 times as strong as the original that means that for every 1 liter of mixed drink there are 0.6 liters of punch. Consequently, that means there are 0.4 liters of water in every liter of mixed drink. That’s a ratio of 3 parts punch to 2 parts water. Since Donna purchased 9 liters of punch, she needs to add 6 liters of water for a total of 15 liters of mixed drink.

 Thanksgiving Leftovers Part 1

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Auntie Gwen makes Thanksgiving dinner for her extremely large family every year. A few days after Thanksgiving she makes a very special casserole with the leftovers. The recipe is very specific. It calls for an exact ratio of 5 cups of mashed potatoes to 4 cups of stuffing to 2 cups of gravy to 1 cup of cranberry sauce. This is a very beloved family recipe so Gwen always tries to make as much as she possibly can.

This year there are 26 cups of leftover mashed potatoes, 14 cups of leftover stuffing, 8 cups of leftover gravy and 5 cups of leftover cranberry sauce. If she makes as much casserole as she can with those leftovers, how many cups of casserole can she make?

A) 36

B) 42

C) 48

D) 60

E) 66

This is a classic ratios problem. The first thing we need to do is find what I believe is called the limiting reagent in chemistry. Basically, if we use the ingredients in the given proportion which one will be run out of first? To do that we take the original ratio 5:4:2:1 and see how many times the base recipe we can make. For instance, since we have 26 cups of mashed potatoes and the recipe calls for 5 cups, we can make 5.2 times the original recipe. Testing the other recipes we can find that we can make 3.5 recipe’s worth of stuffing, 4 recipe’s worth of gravy and 5 recipe’s worth of cranberry sauce. That tells us that stuffing is the first ingredient that will run out.

Now we know that we can make 3.5 times the original recipe. The original recipe makes 12 cups of casserole, so 3.5 * 12 = 42 cups of casserole.

(Note: my guess is that recipe makes a terrible-tasting recipe so I don’t suggest you try to make it)