Unclear Math Concept

Question

Hi Experts,

I stumbled in a question that I do not understand what it tests or what it means.

Which of the following fractions, when expanded, will lead to a decimal value that reoccurs most late?

A) 1/130

B) 37/140

C) 53/280

D) 23/70

E) 3/50

What 'the thing' will occur most late?

Thanks in advance

IanStewart · Answer

The phrase "reoccurs most late" isn't in English. I have no idea what the question means, because when you produce a repeating decimal, there's more than one thing that "reoccurs". For example, if you convert the fraction 557/999 to decimal, you get:

557/999 = 0.557557557557....

557/999 = 0.557557557557....

If I ask when this "reoccurs", I could be asking when we first see a digit repeated - then the answer would be two places after the decimal, because we have a second '5' there. But I could also be asking when the entire looping pattern of digits begins to repeat. Here the "557" is the repeating pattern of digits, and it only starts to "reoccur" four digits after the decimal.

So the question is so badly worded there's no way to tell what it's even asking. It's also testing math you'd never need on the GMAT (if it's asking when digits start reappearing, you need long division, which you never need on the GMAT, and if it's asking about how long the repeating sequence of digits is, the theory behind that is far beyond what the GMAT expects). So you should just ignore this question, and find more realistic questions for practice. What is the source?

Mo2men · Answer

Hi  IanStewart

Thanks for taking the time to reply. The source is Expert global.

I must mention that in its solution, they mentioned that we must look to the greatest prime factor in the denominator. I do not know if this means something to you.

IanStewart · Answer

If they're talking about divisors of the denominator, then the question must be talking about when in the decimal expansion the entire repeating set of digits appears for the second time, and not about when any individual digit repeats. It's definitely not a GMAT question, and there isn't any reason to look at it, but under that interpretation, the right answer will be 53/280. I don't know how anyone could answer this quickly without knowing that in the decimal expansions of both 1/7 and 1/13, you will find a repeating pattern that is 6 digits long -- and you certainly do not need to know that for the GMAT. You'll find a repeating pattern of six digits (eventually) in the decimal equivalent of fraction in the list besides 3/50, since each has a denominator you can obtain from 7 or 13 by dividing by 2s and 5s. If you know that the repeating pattern is six digits long, then it's just a matter of working out how many of the early digits in the decimal expansions of each fraction will not belong to that repeating pattern. You can do that in a few ways - it's possible long division is fastest if you know when you want to stop doing it for each decimal, but I did things like this for each fraction:

53/280 = 28/280 + 21/280 + 4/280 = 1/10 + 3/40 + 1/70 = 0.1 + 0.075 + (0.1)(1/7)

from which you can see that the first three digits of the decimal won't be part of the repeating pattern (because we're modifying them by adding 0.175) but then the repeating pattern will begin in a cycle of six digits, because that's the length of the repeating pattern in the decimal expansion of 1/7.

But it's honestly a completely pointless question to study if you're practicing for the GMAT, which is why I'm only glossing over a solution above.

Unclear Math Concept

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