Question

Dear Mitch,

The Fourth of July is coming up, and I still haven't come across the answer you promised to the final 4^{th} of July question which you gave with multiple choices. Did I just miss it? If you haven't answered it yet, can you soon?

Thanks,

Scott F

Pontiac, MI

Answer

Dear Scott,

You haven't missed it; I just hadn't posted it, so now here it is:

First, for those who missed the question, here it is once again:

At a 4^{th} of July barbecue, 1/4 of the guests ate only hamburgers, 2/3 of the guests ate only hot dogs, and no guest had both hot dogs and hamburgers. If the remaining five guests had absolutely nothing to eat because they were too busy making sure their children were eating and behaving, then how many guests were at the barbecue?

CHOICES:

A) 25

B) 30

C) 45

D) 50

E) 60

O.K. Now, I posted this particular question for a couple of reasons I did not reveal. First, it is a personalized version of a classic question that has been on standardized tests so many times that it is surprising to still see it appear on the biggest of those tests and* still* perplex so many intelligent students.

They key here, as you may have already guessed, is that it is in multiple choice format, which is something I rarely do on this website (but which I do enjoy tremendously because of the almost humorous 'game' aspect to them).

Watch:

The very first part of the question begins with the phrase "1/4 of the guests..."

Always remember that for multiple choice questions the answer choices do not appear AFTER you've done all your work; NO, they are there the whole time and should be considered part of the question, NOT SEPARATE, as they give you so many clues so much faster than your calculator will.

Back to that clause: 1/4 of the guests....

That's the beginning, as we said, but then at the very *end* they ask how many guests could have been there in total?

Choices:

A) 25

B) 30

C) 45

D) 50

E) 60

Try taking 1/4 of the first choice, 25.

Good?

NO.

Why?

"HELP!!!!! BLOODSHED!!!! STOPPPPPP PLEASE!!! CAN"T TAKE PAIN DON'T-CHOP ME UP!!!!!" is what you might hear, because 25 people cannot be divided into four even groups without some horrifying dismemberment. That is not the kind of party we want. So cross off choice A.

Next, take 1/4 of the 30 people attending in choice B.

Good?

NO!!

Why?

"HELP!!!!! BLOODSHED!!!! STOPPPPP PLEASE!!! CAN"T TAKE PAIN DON'T-CHOP ME UP!!!!!" is what you might hear.

(1/4 of 30 is 7 1/2.)

How can we tell so fast? 1/4 is one-half of one-half. (THINK: When the pizza man (or woman) puts your pie in the box and cuts it into four slices, usually he/she cuts it once down the center, making two slices, then gives the pie a half-turn and cuts across the two slices so each of the halves is cut in half, producing four slices.)

1/2 of 30 is 15, and then 1/2 of 15 is 7 1/2.

Not good.

Cross off choice B

Choice C, 45.

Well, since 4 goes into 44 evenly, 45 would lead to a "HELP!!!!! BLOODSHED!!!! STOPPP PLEASE!!! CAN"T TAKE PAIN DON'T-CHOP ME UP!!!!!" situation.

Cross off choice C.

Choice D, 50.

Good?

NO!

Why?

1/4 of 50 is 12 1/2 . How do we know? Simply divide 50 by 4.

You get 12.5.

"HELP!!!!! BLOODSHED!!!! STOPPP PLEASE!!!!..."

Which leaves us with one choice, choice D.

Let's see:

1/4 of 60 = 15. Yes, so far, so good.

Then you plug it into the rest of the story and se if it works.

1/4 at only hamburgers (15)

2/3 ate only hot dogs (2/3 of 60 = 40)

The remaining 5 guests had nothing...

(15 guests + 40 guests = 55 guests, which would indeed leave 5 guests remaining -- and the story works!)

I have a rule which I call "The people, livestock, and balloons rule". Whenever you have a question about things that cannot and should not be divided up into pieces, such as people, livestock (living animals) or inflated balloons at a children's party, check out the fractions to see if there's going to be screaming.

__NOTE__: If the question were not about people but were about pounds of coffee, for example, then sure you could take a third of a pound and not hear much screaming!

Happy Holiday!

And keep the math safe!

Hope this helps,

Mitch