kitchen table math, the sequel: math writing, redux - what does this mean?

## Tuesday, August 31, 2010

### math writing, redux - what does this mean?

6. A gumball machine contains gumballs of 8 different colors, which are dispersed in a regularly repeating cycle. The fourth gumball in the cycle is red and the sixth gumball in the cycle is yellow. If 100 gumballs are dispensed from the machine, how many are not either red or yellow?

Acing the New SAT I Math
p. 236
After spending half the afternoon contemplating the position of subordinate clauses in sentences, I'm wondering whether I can read math at all.

What does this question mean?

If you were actually sitting in a room with this gumball machine, what would be coming out of it and in what sequence? When would you see the first red gumball, and when would you see the second? "When" meaning: after how many other gumballs have appeared?

I find the language in this book particularly challenging, by the way. I've never quite recovered from the watch 'gaining' 3 minutes per hour.

Obi-Wandreas, The Funky Viking said...

The gumballs come out in a regular pattern of 8 different colors. 100/8 = 12 1/2. This means that over the course of 100 gumballs the pattern repeats itself 12 times completely, and then gets halfway done (i.e. the first 4 in the pattern) before hitting 100.

In each complete repetition of the cycle, 1 is red, and 1 is yellow; therefore 6 are neither. 6*12 = 72.

In the half-completed cycle, one is red, and the yellow is not reached. 3 are therefore neither red nor yellow.

72+3 = 75.

Catherine Johnson said...

Well, that's sure the way I read it.

Allison said...

catherine,

What are you literally doing when you read a problem like this?

do you have pencil and paper? Are you writing down math equations? Drawing a picture? Rewriting the same sentences as the question has?

Would you have drawn 8 blanks and written R on the fourth and Y on the 6th?

Catherine Johnson said...

In this case, I wrote 8 blanks, putting an 'r' for red in the fourth and a 'y' for yellow in the 6th.

Then I divided 100 by 8, multiplied 6 by 12, and added 3 for the 3 non-read & non-yellow gumballs in the 4 remaining gumballs.

But that's not the book's answer, and it's not the book's solution.

Catherine Johnson said...

Which is why I ask what this problem means to you all.

Catherine Johnson said...

the 3 non-red

Michael Weiss said...

The problem is really badly worded. It says "The gumballs come out in a regularly repeating cycle" but it does not specify that the length of the cycle is 8 -- you are, I guess, supposed to infer that from the fact that there are eight colors, but it really ought to be specified explicitly. The problem could be fixed by inserting the single word "all", as in "all of which are dispensed..."

With that minor edit, I would visualize the problem as:

Gumballs are coming out. I see x, x, x, RED, x, YELLOW, x, x, where each x represents a gumball that's not yellow or red. Then it starts repeating.

Allison said...

Okay, I'm with you. I computed the same answer. So what does the book say the solution is?

Anonymous said...

My previous comment does not seem to have appeared, but was similar in content to Obi-Wandreas's and your interpretation.

Looking at it more closely, the question is a bit vague, as it does not say where in the cycle the process started, so there could be 1 more or 2 fewer---thus the right answer is 73 to 76.

Possibilities; the book did it wrong, you copied the problem wrong, or we're all asleep.

Catherine Johnson said...

Possibilities; the book did it wrong, you copied the problem wrong, or we're all asleep.

lollll---

Now I'm going to proofread the post again.

(If you don't mind, I won't post the solution quite yet - I'm curious what other people say - )

Catherine Johnson said...

I asked myself the same question about where in the cycle the sequence started.

Allison said...

Whatever the issue, it's senseless that the issue is where the repeating sequence starts. The gumball machine produces a repeating sequence; we're told the 4th is red. To now suggest that the gumball machine "started" with the "3rd" one would mean that the 2nd one is red--the only reasonable interpretation is that the 4th one is the 4th one from wherever it starts.

Catherine Johnson said...

OK, here's a problem:

The book's answer to the gumball problem is not 75.

It's 59.

How did the author(s) of the book arrive at 59?

Bonus point: How should the question have been worded to justify an answer of 59?

Catherine Johnson said...

To now suggest that the gumball machine "started" with the "3rd" one would mean that the 2nd one is red--the only reasonable interpretation is that the 4th one is the 4th one from wherever it starts.

That's what I came up with.

If the red one is the 4th, then the sequence of 100 has to start so that the first red gumball is number 4 in the sequence.

Obi-Wandreas, The Funky Viking said...

The book may have made the same stupid mistake that I made before my wife caught me.

100/4 = 25 instances of red

100/6 = 16 4/6 -> 16 instances of yellow.

25+16=41. 100 - 41=59.

Obi-Wandreas, The Funky Viking said...

So, the question would have to say that every 4th gumball is red, and every 6th is yellow. The question actually says that every 4th in the cycle, not every 4th at all, etc.

But Obi-Wandreas, how can gumball #12 be both red and yellow?

I came up with the following, for the bonus points to reword the question for an answer of 59. :-)

A gumball machine contains gumballs of 8 different colors, which are dispersed in a regularly repeating cycle. The fourth gumball in the cycle is red and the sixth gumball in the cycle is yellow. If 79 gumballs are dispensed from the machine, how many are not either red or yellow? (Pattern is xxxRxYxx repeated.)

A gumball machine contains gumballs of 7 different colors, which are dispersed in a regularly repeating cycle. The fourth gumball in the cycle is red and the sixth gumball in the cycle is yellow. If 82 (or 83) gumballs are dispensed from the machine, how many are not either red or yellow? (Pattern is xxxRxYx repeated.)

A gumball machine contains gumballs of 6 different colors, which are dispersed in a regularly repeating cycle. The fourth gumball in the cycle is red and the sixth gumball in the cycle is yellow. If 87 (or 88) gumballs are dispensed from the machine, how many are not either red or yellow? (Pattern is xxxRxY repeated.)

A gumball machine contains gumballs of 8 different colors, which are dispersed in a regularly repeating cycle. Every fourth gumball is red, and every sixth gumball that is not red is yellow. If 92 gumballs are dispensed from the machine, how many are not either red or yellow? (Pattern is xxxRxxYR repeated.)

Allison said...

I'd bet the problem was wrong Obi's way: it originally said "every 4th is red, every 6th is yellow", and "solved" the problem that way, but some editor caught that that didn't make sense, so they fixed it. Except they didn't fix the solution to match.

Catherine Johnson said...

The book may have made the same stupid mistake that I made before my wife caught me.

ding! ding! ding!

Yup --- that's the book's solution.

I had a heck of a time with it because by that point I'd had my two-unknowns collapse, and I had NO confidence I could read any math problem correctly EVER.

I went round and round, trying to figure out what I was missing...

I was hap-hap-happy when I saw you guys coming up with 75, too.

Catherine Johnson said...