Question

Dear Mitch,

It is almost Thanksgiving, so would you please post the answers to the recent group of Thanksgiving problems you posted?

I liked them a lot and my students appreciate your sense of humor.

Have a good holiday.

Mr. Engene

Answer

__ANSWERS____ to the NOVEMBER 16 ^{th} Thanksgiving questions (With each question presented again above its answer__):

1. For Thanksgiving dessert, a bakery made special pies. They come in two sizes: small and large. Both sizes are circular in shape, and both sizes are the same thickness, density, and fluffiness.

The small pie has a diameter of 10", and the large has a diameter of 20".

If the 10" pie is priced at $3.00, how much should the 20" pie sell for? (Assume the bakery is managed by someone who uses a logical system for pricing the items in his shop).

Here are your choices:

(A) $6.00

(B) $8.00

(B) $10.00

(D) $12.00

(E) $15.00

First, I'll tell you the most popular answer. Then I will explain how to arrive at the correct one.

Most popular: Choice A, $6.00. Why? I can only assume it is because 20 inches is twice the 'size' as 10 inches, and two times three dollars is six dollars.

But Circles don't really work like that.

First, you should always pause for a moment to ask yourself this: What is the question really ABOUT? Here, the question is about the amount of food you are getting in one pie compared to another. Since the problem states that the pies are the same thickness, that factor becomes what math people call a 'constant', meaning it doesn't change, and so your thinking should be simplified from the math of 'volume' to the math of 'area'. (Area is always one step easier than volume because you only have to consider two dimensions, not three.)

So now you go to the formula for the area of a circle. That is the famous

π r².

R stands for radius, which is the measurement from the circle's center to the edge, or half the diameter. So the area of the 10 inch diameter pie is π5² = 25π.

Here there is the natural temptation to start converting 25π into a 'number' by inserting 3.14... for the other constant here, pi, but DON'T. HOLD OFF and let's see if that's necessary.

Now, the larger one has a diameter of twenty inches, so its radius is 10 inches. That pie has an area that measures π10² = 100π

100 π is FOUR TIMES THE SIZE of 25π

Therefore, if the small pie sells for $3.00, the large should sell for four times that amount, or $12.00 (choice D).

I hope this explanation helps.

2. For Thanksgiving, a neighborhood grocery store in Clayton, Missouri was giving customers one free pound of stuffing with every 5 pounds the customer buys. Mrs. McFergass Donahaus Rebitz McIntire III entered the store. She was well-known in the area for one thing : she enjoyed her stuffing. Oh, she'd invite you to dinner if you were in need of a place to go but don't touch her stuffing – at least not if you'd like to ever get another invitation...

So, that day when Mrs. McFergass Donahaus Rebitz McIntire III entered the store she bought only

stuffing and left the store with 30 pounds of the stuff (stuffing). If she spent $75.00, how many dollars does each pound of stuffing sell for?

(A) 2

(B) 3

(C) 6

(D) 9

(E) 20

(HINT: Think of this as a ratio question....)

Okay, I don't know if you ever read it, but a while back, a* long *while back, I presented ratios and how I think one can think of them so that they become immediately clear. Once you figure out how to find things in our archive of past Q&A's, it's fairly easy to find such archived pieces, but here is a quick version of my concept:

Think of ratio questions as questions about shrink-wrapped packages. If you are not sure what the term 'shrink-wrapped' means, it refers to those packages in stores of more than one item that are not sold separately and cannot be separated because they have been sealed together using a plastic that has been heated closed around them. (The packaging could also contain cardboard, wire, etc. but the idea is the same: if you want one of the items you have to buy all of them packaged together whether you need them or not.) For example, batteries often come in shrink-wrapped packages, so that it is easy to purchase a package of two or four or eight, but not always easy to find one battery by itself (unless it is a rather large battery, particularly one for an automobile). Or you may wish to purchase one tennis ball but find that they only come in packs of three.

Okay, ratios are like that: groups of items that always come in a certain kind of a group. In this problem, the groups (or shrink-wrapped packages") are formed like this: (5 pounds of stuffing + one pound of FREE stuffing)

So you cannot separate that group to get the one free pound by itself or rearrange it with other groups to make it half regular pounds and half free pounds. They are grouped together in six pound 'packs'.

The problem states that the woman left the store with thirty pounds. That means that she must have taken 5 of these six-pound 'packs'. Stacked, they would look (sort of) like this:

(5 pounds + 1 Free) = 6 pounds

(5 pounds + 1 Free) = 6 pounds

(5 pounds + 1 Free) = 6 pounds

(5 pounds + 1 Free) = 6 pounds

(5 pounds + 1 Free) = 6 pounds

And since 1 pound in each pack was free, altogether she received 5 pounds free and paid for 25 of the thirty pounds.The problem states that the total amount she spent was $75.00.

To get the price per pound you divide the total cost by the number of pounds she actually paid for.So you have $75.00 divided by 25 pounds.=

75/25 = 3

She paid $3.00 for each pound she bought. Answer choice B.

Hope this explanation helps.

3. In the 8^{th} grade social studies community service class, the students were required to either cook a turkey or a make a soup; the food when then be distributed to local families who otherwise would have very little for the holiday. If a student wished to, he or she could make *both* a turkey *and* a soup, but if a student didn't make either then he/she would not be released from the darkened classroom, alone, until the holiday weekend was over.

23 students each made a soup, 27 students each made a turkey. If there were 41 students in all, how many students made a soup but not a turkey?

This problem is a very popular type of question that seems tricky at first but with a little practice becomes very straightforward.

The first thing you probably notice is that when you add up the number of students who are in the turkey-making group with the number of students who are in the soup-making group you end up with a number that is larger than the total number of students altogether. That seems to confuse some people. All it means is that some of the students are being counted twice because they are in both groups (recall, the problem states that a student may be in both groups, but cannot be in neither).

The easiest way to do such a question is to draw a Venn Diagram. Remember those? Now... as I mentioned in another recent post, we are still in the midst of ironing out a minor technical difficulty that recently arose in our site's software, and, as a result, the Venn diagrams that are to accompany this question's answer will not appear here, so to see the original diagrams that accompanied this question's answer when it originally appeared on this website when we were still in our infancy in 2007, here's what you do:

Go to our Q&A archive, which you can easily access from our homepage, etc., and scroll down to the bottom of any page to see the horizontal list of other pages of past Q&A's, then click on #23, then scroll down to the second Q&A from the bottom, which is the one entitled "Thanksgiving Math for Middle School: Answer #1.' (November 06, 2007). Then scroll down to the second part of the answer, which you shouldn't miss, due to the breathtaking Venn diagrams adorning the screen.

NOTE: If you did not make it to those Venn diagrams, then simply use your imagination, pretend you are seeing them (or draw them yourself), and use the remaining instructions/explanation contained below (though devoid of the illustrations to which they refer).

. . . label one of the circles with a T (for turkey-makers).

Label the other circle with an S (for soup-makers).

Next, you figure out how many students are being double-counted. That you get by adding up the two groups and seeing how many more 'counts' you have than there would be if each students was only counted once (the difference).

So, 23 students are in the soup-maker group, and 27 are in the turkey-maker group.

23 + 27 = 50.

The problem states that there were 41 students in all.

Therefore, since 50 – 41 = 9, it means that nine students were being counted twice.

In a Venn diagram, the number of students who are in both categories gets written in the intersecting portion.

So you have 9 in the middle.

Then, you subtract that number from the total number of turkey-makers, and you get 27- 9 = 18.

You write that number in the part of the circle labeled T that is not intersecting, so students are not double-counted if they are not in both groups.

You do the same thing with the number of students in the soup-group.

23 - 9 = 14.

And you write 14 in the circle labeled with an S, but not the intersecting part of the circle.

At this point it is easy to check if you have arrived at all the correct numbers, but I'll leave that you to you to figure out! (For now).

Next you reread the question to find out exactly what is being asked of you. Here they are asking how many students are making soup but not a turkey.

SO: you look at the part of your Venn diagram that shows how many are in the 'S' circle but NOT the intersecting portion. And that part says: 14.

So the answer is 14 students are making soup but not turkey.

With or without the diagrams, I hope this explanation helps.

4. If 10 pounds of stuffing cost d dollars, how many pounds of stuffing can be purchased for 3 dollars?

(A) 30d

(B) (3d)/10

(C) 30/d

(D) d/30

(E) (10d)/3

ANSWER: C, 30/d

How?

Like this:

Think of the slang: Instead of asking how much something *costs*, people often ask, "How much ** is** that?"

And, in math, "is" is represented by an equal sign, as in 3 + 4 *is* 7, or 3 + 4 = 7.

So, we have:

10 = d dollars

But, more helpful than knowing how much a person can get for* d *dollars, we'd like to know how much one gets for* one* dollar.

So, since the way to turn anything into a one is by dividing it by itself, we divide each side by d to get "something = one dollar".

Like this:

10/d = 1 dollar

And to find out how many can be bought for 3 dollars, as the questions asks, we multiply that one dollar by 3, and do the same to the other side, as math requires.

And we get 30/d = 3 dollars.

I hope this explanation helps.

5. If p pounds of sweet potato pie costs s cents, 10 pounds of that pie will cost

(A) (ps)/10 cents

(B) 10ps cents

(C) (10s)/p cents

(D) (10p)/s cents

(E) (s + p +10) cents

ANSWER: C, (10s)/p cents

Similar to above,

P = s cents

p/p = s/p cents

1 = s/p cents

So,

10 = (10s)/p cents

I hope this explanation helps.

6. One week before Thanksgiving last year a gourmet food store reduced the price of their "family-sized" turkey dinner by half the regular price, and then, 3 days later, when the chef became concerned that he had prepared more than they were likely to sell, had the owner reduce the sale price by 10%. The final price is what percent of the original price?

(A) 5%

(B) 10%

(C) 25%

(D) 40%

(E) 45%

ANSWER: E, 45%.

When dealing with percentage questions where they do not give you a number, choose 100. First reduction: 100 becomes 50.

Second reduction: 10% of 50 = 5, so 50 goes to 45.

45 out of the original 100 is 45%.

I hope this explanation helps.

7. On the night before Thanksgiving, Stanley S., who, although 22-years-old, still lives with his parents, sneaked into the kitchen and ate one-fourth of the pumpkin pie his mother had spent all day preparing. The following morning, he woke up early (having cleverly set his alarm clock), tip-toed downstairs, slipped back into the kitchen, and helped himself to one-half of what was left of the pie. What fraction of the entire pie did Stanley eat before the holiday dinner?

(A) 1/2

(B) 7/8

(C) 3/8

(D) 5/8

(E) 3/4

ANSWER: Choice D, 5/8.

Why? Draw a circle and divide it up into 8 equal slices. Then take away one-fourth and you are left with 6 slices. Then take away half of the remaining 6 and you get 3 slices. 3 of the original 8 *remaining*. He** ate **5/8ths of the pie.

And, finally, I hope this explanation helps!

Happy holiday!,

Mitch