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OG 13 - PS 158

This topic has 2 expert replies and 5 member replies
gdshamain Senior | Next Rank: 100 Posts Default Avatar
Joined
09 Jan 2014
Posted:
41 messages

OG 13 - PS 158

Post Sun May 04, 2014 10:45 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    158. In the first week of the year,Nancy saved $1.In each
    of the next 51 weeks,she saved $1 more than she
    had saved in the previous week. What was the total
    amount that Nancy saved during the 52 weeks?
    (A) $1,326
    (B) $1,352
    (C) $1,378
    (D) $2,652
    (E) $2,756

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    Post Sun May 04, 2014 11:24 am
    Hi gdshamain,

    This question describes a sequence of numbers: 1, 2, 3,…..52 and asks you to add them all up.

    Adding the numbers up in order would take way too much time to be practical. There are actually a couple of ways to quickly add up these terms; here's one way called "bunching":

    When adding up a group of numbers, the order of the numbers does not matter. I can group the numbers into consistent sub-groups:

    1+52 = 53
    2+51 = 53
    3+50 = 53
    4+49 = 53
    etc.

    So every group of 2 terms sums to 53. There are 52 total terms, so that means that there are 26 sets of 2 terms.

    26(53) = 1378

    Final Answer: C

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

    _________________
    Contact Rich at Rich.C@empowergmat.com

    Post Sun May 04, 2014 11:30 am
    Quote:
    In the first week of the year, Nancy saved $1. In each of the next 51 weeks, she saved $1 more than she had saved in the previous week. What was the total amount that Nancy saved during the 52 weeks?

    A. $1,326
    B. $1,352
    C. $1,378
    D. $2,652
    E. $2,756
    Here's a similar approach with a slight TWIST at the end.

    We want to add 1+2+3+4+...+51+52
    So, let's add them in pairs, starting from the outside and working in.
    1+2+3+4+...+51+52 = (1+52) + (2+51) + (3+50) + . . .
    = 53 + 53 + 53 + ....

    How many 53's are there in our new sum?
    Well, there are 52 numbers in the sum 1+2+3+..+52, so there must be 26 pairs, which means there are 26 values in our new sum of 53 + 53 + 53 + ....

    So, what does (26)(53) equal?
    Fortunately, if we examine the answer choices, we see that we don't need to calculate (26)(53)

    Why not?
    Notice that when we multiply (26)(53), the units digit in the product will be 8 (since 6 times 3 equals 18).

    Since only 1 answer choice (C) ends in 8, the correct answer must be C

    Cheers,
    Brent

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    Post Sun May 04, 2014 12:09 pm
    Hi gdshamain,

    There's also a more-formal arithmetic approach for sequences that are made up of evenly-spaced terms:

    1) Take the average of the smallest and largest terms: (1+52)/2 = 26.5
    2) Count up the total numbers of terms: 52
    3) Multiply the two values: (26.5)(52)

    You'll end up with the exact same answer: C

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

    _________________
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    confused13 Senior | Next Rank: 100 Posts Default Avatar
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    17 Dec 2013
    Posted:
    97 messages
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    2 members
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    Post Sun May 04, 2014 12:27 pm



    Last edited by confused13 on Sun May 11, 2014 11:17 pm; edited 1 time in total

    gdshamain Senior | Next Rank: 100 Posts Default Avatar
    Joined
    09 Jan 2014
    Posted:
    41 messages
    Post Mon May 05, 2014 10:05 pm
    Thank you for the response.

    Post Tue Jul 07, 2015 3:37 am
    gdshamain wrote:
    158. In the first week of the year,Nancy saved $1.In each
    of the next 51 weeks,she saved $1 more than she
    had saved in the previous week. What was the total
    amount that Nancy saved during the 52 weeks?
    (A) $1,326
    (B) $1,352
    (C) $1,378
    (D) $2,652
    (E) $2,756
    Solution:

    Let's first set up the pattern of Nancy's savings. The first week she saved $1, the second week she saved $2, the third week she saved $3, and so forth. Therefore, the total amount of money she will have saved at the end of 52 weeks will be: $1 + $2 + $3 + $4 + … + $52. The pattern is obvious, but the arithmetic looks daunting because we need to add 52 consecutive integers. To shorten this task, we can use the formula: sum = average x quantity.

    We know that Nancy saved money over the course of 52 weeks, so our quantity is 52.

    To determine the average, we add together the first amount saved and the last amount saved and then divide by 2. Remember, this technique only works when we have an evenly spaced set.

    The first quantity is $1 and the last is $52. Thus, we know:

    average = (1 + 52)/2 = 53/2

    Now we can determine the sum.

    sum = average x quantity

    sum = (53/2) x 52

    sum = 53 x 26 = 1,378

    Answer:C

    Note: If we did not want to actually multiply out 26 x 53, we could have focused on units digits in the answer choices. We know that 26 x 53 will produce a units digit of 8 (because 6 x 3 = 18), and the only answer choice that has a units digit of 8 is answer choice C.

    _________________
    Jeffrey Miller Head of GMAT Instruction

    seandcarey Newbie | Next Rank: 10 Posts Default Avatar
    Joined
    07 Jul 2015
    Posted:
    6 messages
    Post Tue Jul 07, 2015 5:57 pm
    Another way to do it is

    N*(N+1)/2

    In this case N is equal to 52.

    (52*53)/2 = 1378

    If it were 100
    100 * 101 / 2 = 5050

    It is very useful for expected value questions also. For example, what is the EV of the roll of a single die.

    EV = Sum of outcomes / # of outcomes
    6*7 = 42/2 = 21. 21/6 = 3.5

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