Kim finds a 1-meter tree branch and marks it off in thirds and fifths.

hey karishma thanks for the reply
have u found the distinct sizes by finding out the difference between them?

Yes I have. Difference between 3/15 and 5/15 is 2/15.
Between 5/15 and 6/15 is 1/15.
Between 6/15 and 9/15 is 3/15.
and then they repeat...

Responding to a pm:



This is what you are doing: You are marking off at 1/3, 2/3 and 3/3
and at 1/5, 2/5, 3/5 etc

Or we can say that we are marking at 5/15, 10/15 and 15/15
and at 3/15, 6/15, 9/15 etc so that we can easily compare.


Now the first piece you get is 3/15 in length, next one is 2/15 and the next one is 1/15. Now think, will you get pieces of length 4/15, 5/15 etc? No because you mark off at every 3/15 (you mark at 3/15, 6/15, 9/15, 12/15 and 15/15). So it is not possible to get any piece which is greater than 3/15.

The length of distinct pieces combined = 1/15 + 2/15 + 3/15 = 6/15
The fraction of original branch that remains = 1 - 6/15 = 9/15 = 3/5

Hi Bunuel- I took the total length to be 10 meter (length can be anything as question only asks for fraction) so the markings are 1,3,5,6,9,10 the unique lengths are 3,2,1 so 10-6 = 4 remains so fraction remaining is 4/10 = 2/5. Could you please correct me? Also the question says 1 piece of every unique length so the piece with the length 2 also can be take right?

10m is not a good choice in this case, because 10 is not divisible by 3. So the cuts cannot be at 3, 6, and 9, but should be at 3and1/3, 6and2/3. Also, the margins are 0 and 10, 1 is not a margin.
The best to take in this case is a length of 15m, it is both divisible by 3 and 5.

Distinct lengths would be: 3, 5-3=2 and 10-9=1 meters long pieces, so she removes total of 3+2+1=6 meters, so 15-6=9 meters remains, which is 9/15=3/5 of the original branch,

Answer: E.

Similar question to practice:
on-the-number-line-above-the-segment-from-0-to-1-has-been-104204.html

Hope it helps.[/quote]

Hi Bunuel,

See above highlighted. Shouldn't we first consider 6-5= 1 as the distinct lenghth. Is there any specific reason you jumped to 10-9??

The point is that 1-meter long piece is possible, so it does not matter whether you get that from 6-5 or 10-9.

Image attached.

LCM of 3 and 5 is 15, so consider the rope as 15 units long.
Rope is divided into 3rd's and 5th's. (3rd's in blue and 5th in green)

The unique length pieces are marked in light green color in the image as 3 , 2 and 1
So, after 6units all the lengths are repeating.
So, part of the rope which will be left = (15-6)/15
= 9/15 = 3/5
So, answer will be E.

Hope it helps!

Kindly refer attachment:
Equalise the denominator:
One thirds (marked in black) & one fifths (marked in blue)
7 pieces created & differences calculated
Removed the "distinct" lenghts & added the rest
3/15 + 1/15 + 2/15 + 3/15
= 9/15
Answer = E = 3/5

Hi Bunuel

Please help me regarding this question

As per your explanation, the branch would be cut at 3, 5, 6, 9, 10, and 12 meters (in red are given fifths of the length and in black thirds of the length). So , distinct lengths should be: 3, 5-3 = 2 , 10-9 =1 AND 6 ??
So she removes total of 3+2+1+6 =12 meters, so 15-12=3 meters remains, which is 3/15=1/5 of the original branch ?

Thanks

Consider the diagram below:
---|--|-|---|-|--|---

Notice that we don't have a 6-meter long piece there.

Hope it's clear.

Hi Bunuel, please excuse my ignorance, but even after going through all the explanations, I cannot make out what does the question mean exactly by distinct lengths?

0---|--|-|---|-|--|---15

When you cut 15-meter piece at 3, 5, 6, 9, 10, and 12 meters you get 3 distinct lengths:

3 meters: from 0 to 3, from 6 to 9, and from 12 to 15;
2 meters: from 3 to 5 and from 10 to 12;
1 meter: from 5 to 6 and from 9 to 10.

Check similar questions to practice:
https://gmatclub.com/forum/on-the-number ... 04204.html
https://gmatclub.com/forum/a-straight-pi ... 45031.html
https://gmatclub.com/forum/a-100-meter-s ... 50349.html

So distinct length actually referring to distinct differences between lengths once cut? Like the post above me I saw distinct as "unique" the way it is with factoring and this really threw me and I'm still struggling to fully comprehend the meaning of distinct in relation to this problem.

Distinct does mean unique, the same way as it is used with factors.

12 = 2*2*3
How many distinct prime factors does 12 have? It has 2 distinct prime factors: 2 and 3
There are two 2s but since we need just the distinct prime factors, we ignore the other 2.

Similarly, how many distinct lengths are there after you cut the branch into pieces? You get 3 distinct lengths: 3, 2 and 1. There are multiple pieces of the same length but since we need pieces of distinct length only, we leave the other pieces.

Hi fskilnik,

I got yet another doubt for the day:)

In the above question ,my approach was

LCM OF 3 and 5 is 15, so assumed the line to be 15 m long
Then marked it 1/3rds and 1/5ths => markings i get are 3,5,6,9,10,12,15 . On breaking them at every markings, only the portion between 12 and 15 remains => (15-12)/15 = 3/15 =1/5 is what I got. Please help me as figure where Im going wrong.

Thankyou again

Hi, yashna36!

Ok!

Nice idea! Please note there are three different lengths "between markings" available: 3 (3-0, for instance) , 2 (5-3, for instance) and 1 (6-5, for instance).

According to the question stem, you must take out ONE of each, hence you take out 3+2+1 = 6.

The answer is what is left = 15-6 = 9 and going back to the original unit, we have 9/15 = 3/5 as the correct answer.

I hope you got it!

Regards and success in your studies!
Fabio.