On Friday I posed this puzzle….

Yesterday I bought a strip of multi-coloured cloth.  One third and one quarter of it is black, and the other 8 metres are red.  How long is the strip?

If you have not tried to solve it, have a go now.  For everyone else, the answer is after the break.

If x is the length of the cloth, then

x/3 + x/4 + 8 = x

or

4x + 3x + 96= 12x

and thus x=19.2 m

Did you solve it?  Any other answers?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for theKindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. JCB dumpster says:

No 96 doesn’t equal 12x

1. Typo then 4x + 3x + 96=12x => x=96/5 => x=19.2

2. Alison says:

The statement “One third and one quarter of it is black” can be interpreted as only 1/3 of the cloth is black (because if 1/3 of it is black, 1/4 is also black).
That would make the cloth 12m long (x/3 +8 = 8 ->2x/3 =8 ->x=12)

1. Michael Sternberg says:

It could be interpreted that way, but only by ignoring vox populi. Ever bought, say, “a pound and a half” of some product?

3. The second line of the formulas is clearly wrong

4x + 3x = 96= 12x (which would give x = 8)

x/3 + x/4 + 8 = x yields

x – x/3 – x/4 = (12 – 4 – 3)x/12 = 5x/12 = 8

therefore x = 96/5 = 19.2

1. Michael Sternberg says:

I’d call it an excusable typo. “= 96” should be “+ 96”.

On a classroom blackboard, that’d be easily (and possibly quietly 😉 fixed with a smudge of a finger.

4. -M- says:

Got it, did it this way:

1/4 + 1/3 = 3/12 + 4/12 = 7/12 (seven twelfths of the cloth is black, five twelfths of the cloth is red)

5/12 = 8/length

length = 8/5*12 = 19.2

1. Rudi says:

Me too

5. simon says:

Obviously only a typo cos should be
4x + 3x + 96=12x
Calm down dear!

6. Rusty says:

5/12 of x = 8
Therefore x = 8/5 x 12 =19.2

1. Rusty says:

Sorry:

5/12 of x = 8
Therefore x = 8/5 * 12 =19.2

7. Jenifer says:

Yep, that’s what I got. 😀

8. Michael Sternberg says:

Yep; no surprises.

I went the mental route of (1/3 + 1/4) = 7/12, hence x × 5/12 = 8m.

Let me take the opportunity to rail a bit against the loosey-goosey handling of units in Richard’s equations above. “8” should be written as “8 m”. Otherwise it’s easy to confuse dimensionless numbers (“3” and “4”) with physical quantities, canceling and losing or mangling units in the process.

It is good practice to handle givens in the calculation of a problem as symbolic parameters all the way through, and substitute in the numeric values at the end. This makes for more reproducible steps and more robust handling of units, metric or otherwise.

Calling the cloth remainder length r, we can write up the problem as:
(1/3 + 1/4) x + r = x,
r = (1 – 7/12) x, hence
x = 12/5 r.

And since today’s value of r = 8 m, we get the stated solution as (12/5 × 8) m, naturally in the same unit.

9. Hugh Janus says:

Sloppy editing RIchard, and not for the first time.

10. Stephen says:

I got the answer above, but in doing so, I also got the answer that the strip is 8m long. The colours could be changing along the length, rather than along the width, or it could be a checkered pattern with very large squares!

11. drsuzanne says:

Gosh. I thought the strip of cloth was 8m long with longitudinal coloured stripes. Wouldnt that work too?

1. Michael Sternberg says:

No, no, no. You see, not only would that make not much of a puzzle but we are clearly always encouraged to think laterally here.

2. Anonymous says:

Oh, I though I was the one thinking laterally 😉

From the wording of the question either of us could be right, or even Stephen above who pictured a chequered pattern.

I think the real solution is that we can’t deduce anything about the length of the piece of fabric because we don’t have enough information.

3. drsuzanne says:

Actually Anonymous (above) was me! Sorry.
Suzanne

P.S. Typo “though” should read “thought”

1/4 = 3/12
1/3 = 4/12
=> 1/4 + 1/3 = 7/12
=> 5/12 of the length = 8m
=> (12 x 8) /5 = 19.2

13. Charles Sullivan says:

Not being a mathmathician, I realized that 5/12 = 8m, so I just tried to figure out what 1m was, by dividing 8 by 5, then I added it all up.

14. pdurrant says:

But that’s not a puzzle! That’s just an arithmetic problem.

It’s only a puzzle if the answer is 12m.

15. Lazy T says:

what no twist?

1. Sometimes the twist is that there is no twist.

2. SimonP says:

But that means there is a twist, oh no. My brain explodes with the paradoxical nature of it all.

16. sjames1958 says:

The problem I had was parsing this “One third and one quarter of it is black” Is that 1/3 + 1/3*1/4?
You could say that x/3 + 8 = x is legitmate. If 1/3 of it is black, then already 1/4 of it is black as well. But getting x/3 + x/4 + 8 = x from that phrase of English?

17. Francis says:

And what if the stripes were in the other direction? The cloth would be 8 meter long! That’s another valid solution.

18. rimbaud3000 says:

Got it.

19. Nope. It says “…and the OTHER 8 metres are red,” clearly excluding the black portion. If he’d just said, “…and 8 metres are red,” you’d have a point.

1. drsuzanne says:

Good point, I didn’t spot “the OTHER 8 metres”. I think that makes my otherwise elegant 8m solution less likely to be right.

2. Stephen says:

Not necessarily. It could be interpreted as meaning the “remaining” 8 metres, or the “rest of” the 8 metres. Part of the difficulty is that the “one third”, and “one quarter” parts don’t specifically refer to lengths, widths, areas, or anything else.

3. -M- says:

It does refer to length, because saying “the other 8 meters are red” refers to that the first black part must also be considered length.

20. Julia says:

I didn’t have any problems with the wording of this one. But then I’m not a mathematician.

21. Mervulon says:

The mathematical solution as stated here is the obvious answer, but I prefer the 8 metr solution, wherein the multi-colors are stripes running the length of the fabric, (as is indeed more likely, since it constitutes more of a ‘pattern’ one would see in material). In this
solution, one could cut away the 1/3 and 1/4 black widths of the cloth, leaving the remainig 5/12 red width portion at the original 8 meters. This is a legitimate solution: 8 meters are black, another 8 meters are black, ” and the other 8 metres are red.”

22. So, Richard, is your new puzzler source an Algebra 1 textbook? My daughter is studying 6th grade math and solved this one in her head at dinner.

1. rmbajpai says:

That shows most of the grown ups are not smarter than a fifth grader.

23. Anonymous says:

Richard, would you like me to audit your Q&A’s because, unless you are making deliberate mistakes to get debate going, you have got rather sloppy of late.

24. Anonymous says:

There’s only one way to find out – FIGHT!!!!

25. AL says:

Be patient. This man has now occupied the throne of global puzzle psychology king and should be allowed to eat drink and be merry to celebrate along with a little inattention to detail, which might be viewed as part of the puzzle.

1. Non Anonymous says:

“This man has now occupied the throne of global puzzle psychology king” – he can’t even get the answers right.
In the great man’s words – DREAM ON

26. it could be a mobius strip, in which case it would be 9.6m long…

1. Yeah but you wouldn’t have different colours on each ‘side’…

27. Richard,

could we please have a Friday puzzle that isn’t math-related?

Thanks

28. Eddie Says:

April 18, 2012 at 2:53 pm
Yeah but you wouldn’t have different colours on each ‘side’…