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M is the sum of the reciprocals of the consecutive integers

This topic has 6 expert replies and 0 member replies
barksdale Junior | Next Rank: 30 Posts Default Avatar
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M is the sum of the reciprocals of the consecutive integers

Post Thu Jun 01, 2017 9:21 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Totally confused about the process of getting to a solution in the 2 minute average time. Guess on this question?

    M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

    (A) 1/3 < M < 1/2
    (B) 1/5 < M < 1/3
    (C) 1/7 < M < 1/5
    (D) 1/9 < M < 1/7
    (E) 1/12 < M < 1/9

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    Post Thu Jun 01, 2017 9:54 pm
    Hi barksdale,

    To start, your goal should NOT be to answer every Quant question in 2 minutes or less; your goal should be to be 'efficient' with however you choose to approach a prompt. Certain Quant questions will require upwards of 3 minutes of work, so 'limiting' yourself to a 2-minute timeframe might actually keep you from picking up some relatively easy points in the Quant section (which would limit your Quant Scaled Score and your Overall Score). That having been said, in this question you can take advantage of how the Answers are designed and avoid a calculation-heavy approach. Here's how:

    In this prompt, the answer choices are "ranges" that do NOT 'overlap' with one another. If you can define the upper and/or lower boundaries of the range, then you'll have the answer.

    We're asked to find the range that the sum of the reciprocals of the 100 integers from 201 - 300 (inclusive) would fall into. The smallest term in that sequence would be 1/300, so the sum would have to be GREATER than (100)(1/300) = 1/3. At this point, you should notice that there's only one answer with 1/3 as the LOWER END of the range - and there's no other option that's larger, so the only answer that's possible is...

    Final Answer: A

    As you continue to study, be mindful of how the answer choices are written - they can sometimes provide a huge hint into the fastest way to answer the question.

    GMAT assassins aren't born, they're made,
    Rich

    _________________
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    Thanked by: amontobin, barksdale
    Post Thu Jun 01, 2017 10:32 pm
    barksdale wrote:
    Totally confused about the process of getting to a solution in the 2 minute average time. Guess on this question?

    M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

    (A) 1/3 < M < 1/2
    (B) 1/5 < M < 1/3
    (C) 1/7 < M < 1/5
    (D) 1/9 < M < 1/7
    (E) 1/12 < M < 1/9
    Most GMAT questions involve logical thinking and may not be calculation-intensive. The key is to find the optimum approach to solve the question. This one does not deserve more than 1 minute if you hit the right approach.

    You are asked to calculate the value of 1/201 + 1/202 + 1/203 + 1/204 + ... + 1/300 = M; there are a total of 100 terms. The GMAT does expect you to compute the correct value. Moreover, the options are in the range. This gives a hint that we need a ballpark figure.

    Let's look at the first option: 1/3 < M < 1/2 . As per the option, M would be greater than 1/3 but less than 1/2. This gives us a clue to think of what could be the minimum and the maximum value of M.

    Let's analyze. Looking at M = 1/201 + 1/202 + 1/203 + 1/204 + ... + 1/300, we find that each successive term in the series is less than the previous term. 1/201 > 1/202; 1/202 > 1/203; 1/203 > 1/204, etc.

    We see that among 1/201, 1/202, 1/203, 1/204, ... 1/300, the term 1/201 is the maximum, while 1/300 is the minimum.

    Since there are 100 terms, the maximum value of M would be less than 1/201 + 1/201 + 1/201 + 1/201 + ... + 1/201 = 100/201 = ~100/200 = ~1/2.

    And the minimum value of M would be more than 1/300 + 1/300 + 1/300 + ... + 1/300 = 100/300 = 1/3.

    Thus, 1/3 < M < 1/2.

    The correct answer: A

    Hope this helps!

    Relevant book: Manhattan Review GMAT Number Properties Guide

    -Jay
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    Thanked by: barksdale
    Post Fri Jun 02, 2017 4:11 am
    Quote:
    M is the sum of the reciprocals of the consecutive integers from 201 to 300 inclusive. Which of the following is true?
    A) 1/3 <M 1/2
    B)1/5<M<1/3
    C)1/7 <M< 1/5
    D) 1/9 < M < 1/7
    E) 1/12 <M< 1/9

    We want to find 1/201 + 1/202 + 1/203 + . . . + 1/299 + 1/300

    NOTE: there are 100 fractions in this sum.

    Let's examine the extreme values (1/201 and 1/300)

    First consider a case where all of the values are equal to the smallest fraction (1/300)
    We get: 1/300 + 1/300 + 1/300 + ... + 1/300 = 100/300 = 1/3
    So, the original sum must be greater than 1/3

    Now consider a case where all of the values are equal to the biggest fraction (1/201)
    In fact, let's go a little bigger and use 1/200
    We get: 1/200 + 1/200 + 1/200 + ... + 1/200 = 100/200 = 1/2
    So, the original sum must be less than 1/2

    Combine both cases to get 1/3 < M < 1/2 = A

    Cheers,
    Brent

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    Thanked by: barksdale, gmatdestroyer13
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    Post Mon Jun 05, 2017 7:20 pm
    barksdale wrote:
    Totally confused about the process of getting to a solution in the 2 minute average time. Guess on this question?

    M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

    (A) 1/3 < M < 1/2
    (B) 1/5 < M < 1/3
    (C) 1/7 < M < 1/5
    (D) 1/9 < M < 1/7
    (E) 1/12 < M < 1/9
    I think the biggest issue that students have with questions like this is that they grab their pen and start trying to solve before figuring out what is really going on in the question.

    As the others have pointed out, this one isn't about actually adding reciprocals; it's about benchmarking and using common sense. But the only way to figure that out is to look at the answer choices and ask yourself: "why are they all ranges, and not an exact sum?" Clearly there is an exact value answer to this question, but that's not what the GMAT is looking for. If you grab your pen and start adding things before even looking at the answer choices, you would miss that point entirely!

    Here's some good general advice for all PS: put the pen down and read the whole problem + answer choices before you start solving!

    More on how and why to do that here: https://www.manhattanprep.com/gmat/blog/2016/03/10/want-to-do-better-on-gmat-quant-put-your-pen-down/

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    Post Mon Jun 05, 2017 11:37 pm
    Also, when in doubt, try one (or more) of the following:

    1) Try a smaller version of the same pattern to get a feel for what's going on;

    2) Approximate;

    3) Backsolve from the answers

    It's amazing how many GMAT problems respond to one of those three approaches.

    Thanked by: barksdale, gmatdestroyer13
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    Post Tue Jun 13, 2017 7:53 am
    barksdale wrote:
    M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

    (A) 1/3 < M < 1/2
    (B) 1/5 < M < 1/3
    (C) 1/7 < M < 1/5
    (D) 1/9 < M < 1/7
    (E) 1/12 < M < 1/9
    Let's first analyze the question. We are trying to find a potential range for M in which M is the sum of the reciprocals from 201 to 300 inclusive. Thus, M is:

    1/201 + 1/202 + 1/203 + … + 1/300

    Since we probably would not be expected to do such time-consuming arithmetic (i.e., to add 100 fractions, each with a different denominator), that is exactly why the answer choices are in the form of an inequality. Thus, we do not need to know the EXACT value of M. The easiest way to determine the RANGE of values for M is to use easy numbers that can be quickly manipulated.

    Notice that 1/200 is greater than each of the addends and that 1/300 is less than or equal to each of the addends. Therefore, instead of trying to add together 1/201 + 1/202 + 1/203 + … + 1/300, we are going to add 1/200 one hundred times and 1/300 one hundred times. These two sums will give us a high estimate of M and a low estimate of M. Again, we are adding 1/200 one hundred times and 1/300 one hundred times because there are 100 numbers from 1/201 to 1/300.
    Instead of actually adding each one of these values one hundred times, we will simply multiply each value by 100:

    1/300 x 100 = 1/3. This value is the low estimate of M.

    1/200 x 100 = 1/2. This value is the high estimate of M.

    We see that M is between 1/3 and 1/2.

    Answer: A

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