A certain ball team has an equal number of right and left

[griffin.811]

Hi,
My question pertains to the CAT exam question below:

A certain ball team has an equal number of right- and left-handed players. On a certain day, two-thirds of the players were absent from practice. Of the players at practice that day, one-third were left handed. What is the ratio of the number of right-handed players who were not at practice that day to the number of left-handed players who were not at practice?

1/3
2/3
5/7
7/5
3/2

Correct answer: (B) 5/7

Q: I deally I would have selected 18, and this wuestion would have been a breeze, however, I selected 12. My thought process was I want a number divisible by 3 and by 2 because we also have equal teams.

Using 12, I get a ratio of about 3.34:4.67. How can I get from this point to the correct answer? Also how does everyone else know to pick 18 (I feel like I'm the only one who selected something different)

If it's helpful, I used a double set matrix for this with the headings "right, left" and "present, absent".
Thanks

[tim]

Picking smart numbers is a great idea here, but the smartest number to pick would be one that is a multiple of 9, because you are dealing with two denominators of 3 multiplied together. This is why 12 caused trouble but 18 would be fine. Pay careful attention to denominators when you are picking smart numbers, as this will often give you the best results. When in doubt, pick numbers with a lot of factors, such as 60 or even 360.

[griffin.811]

Thanks Tim, that makes sense.

[jlucero]

Glad it helped.

[jadorexox]

Picking smart numbers is a great idea here, but the smartest number to pick would be one that is a multiple of 9, because you are dealing with two denominators of 3 multiplied together. This is why 12 caused trouble but 18 would be fine. Pay careful attention to denominators when you are picking smart numbers, as this will often give you the best results. When in doubt, pick numbers with a lot of factors, such as 60 or even 360.

I picked 60 for this problem but ended up with a not so pretty number

20 * (1/3)

Could you provide more insight for how to chose numbers?

[Sage Pearce-Higgins]

Sorry for the delay in replying to your question. The idea of taking a multiple of nine for a smart number comes from the fact that 2/3 of the players were absent (so 1/3 were present), and 1/3 of those present were left-handed. From this we can see that 1/3 of 1/3 is going to give us 1/9, so if the number of players is a multiple of 9, then we're going to be dealing with nice integers.

However, if you didn't see this straight away, don't despair! With practice we can get pretty quick at using smart numbers so that we have time to adjust if we need to. If I'd picked 30 at the beginning, then once I got to difficult fractions I'd start again with another smart number. Second time round things would be quicker. What would probably help here is a tree diagram dividing the players into present / not present, then branching into left / right-handed. Taking time to organize the information before starting my calculations would help me see the best smart numbers.

Finally, to really crack this problem, I'd recommend that you try, just for practice, doing it without smart numbers. Take the total number of players to be "x" and see where that takes you. Sure, it will be tougher and longer, but it might give you some insight into how the problem works.