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# If K is the sum of reciprocals of the consecutive integers

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If K is the sum of reciprocals of the consecutive integers [#permalink]  07 Jan 2013, 02:05
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If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

[Reveal] Spoiler:
How do we decide between 1/6 and 1/8
[Reveal] Spoiler: OA

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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  07 Jan 2013, 02:14
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fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

Given that $$K=\frac{1}{43}+\frac{1}{44}+\frac{1}{45}+\frac{1}{46}+\frac{1}{47}+\frac{1}{48}$$. Notice that 1/43 is the larges term and 1/48 is the smallest term.

If all 6 terms were equal to 1/43, then the sum would be 6/43=~1/7, but since actual sum is less than that, then we have that K<1/7.

If all 6 terms were equal to 1/48, then the sum would be 6/48=1/8, but since actual sum is more than that, then we have that K>1/8.

Therefore, 1/8<K<1/7. So, K must be closer to 1/8 than it is to 1/6.

Similar question to practice from OG: m-is-the-sum-of-the-reciprocals-of-the-consecutive-integers-143703.html

Hope it helps.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  07 Jan 2013, 02:15
2
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fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

I believe a good approximation would be to take the mean, reciprocal of that and multiply by 6 (No of numbers being added)

= $$\frac{6}{45.5}$$ which is closest to $$\frac{6}{48}$$ (\frac{1}{6} would be $$\frac{6}{36}$$ and $$\frac{1}{10}$$would be $$\frac{6}{60}$$) and hence $$\frac{1}{8}$$
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  07 Jan 2013, 02:17
Bunuel wrote:
fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

Given that $$K=\frac{1}{43}+\frac{1}{44}+\frac{1}{45}+\frac{1}{46}+\frac{1}{47}+\frac{1}{48}$$. Notice that 1/43 is the larges term and 1/48 is the smallest term.

If all 6 terms were equal to 1/43, then the sum would be 6/43=~1/7, but since actual sum is less than that, then we have that K<1/7.

If all 6 terms were equal to 1/48, then the sum would be 6/48=1/8, but since actual sum is more than that, then we have that K>1/8.

Therefore, 1/8<K<1/7. So, K must be closer to 1/8 than it is to 1/6.

Similar question to practice from OG: m-is-the-sum-of-the-reciprocals-of-the-consecutive-integers-143703.html

Hope it helps.

I solved till that part then couldn't decide can you explain that step in detail.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  07 Jan 2013, 02:18
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fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

The numbers are $$1/43 + 1/44+ 1/45 + 1/46 + 1/47 + 1/48$$.
The easiest method is to find smart numbers.
If you consider each of the numbers as $$1/42$$, then there sum will be $$6/42$$ or $$1/7$$. Remember that since we chose a higher number than those given, hence the actual sum will be smaller than $$1/7$$.
Now consider each of the numbers $$1/48$$. Then in such case, the sum will be $$6/48$$ or $$1/8$$. Remember that since we chose a smaller number than those given, hence the actual sum will be greater than $$1/8$$.
Therefore the sum lies between $$1/7$$ and $$1/8$$. Hence among teh answer choices, the sum is closest to $$1/8$$.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  08 Jan 2013, 01:29
1
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Expert's post
fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

Hi,

Well all other approaches are correct. Here is one more. A little less calculation intensive.

From 1/43 to 1/48, there are 6 #.

=> We can infer that sum of 6 # of 1/40s > Sum of (1/43+/1/44+......+1/48) > Sum of 6 # of 1/50s

=>So, 6/40 > Sum of (1/43+/1/44+......+1/48) > 6/50

=> 1/6.66 > Sum of (1/43+/1/44+......+1/48) > 1/8.33

=> 1/6.66 > 1/ (6.66< Denominator < 8.33) > 1/8.33

Only option available is C. Answer is 1/8.

-Shalabh Jain
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  08 Jan 2013, 04:08
egmat wrote:
fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

Hi,

Well all other approaches are correct. Here is one more. A little less calculation intensive.

From 1/43 to 1/48, there are 6 #.

=> We can infer that sum of 6 # of 1/40s > Sum of (1/43+/1/44+......+1/48) > Sum of 6 # of 1/50s

=>So, 6/40 > Sum of (1/43+/1/44+......+1/48) > 6/50

=> 1/6.66 > Sum of (1/43+/1/44+......+1/48) > 1/8.33

=> 1/6.66 > 1/ (6.66< Denominator < 8.33) > 1/8.33

Only option available is C. Answer is 1/8.

-Shalabh Jain

Thanks this made it clear I was confused between those 2 options.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  08 Jan 2013, 22:24
Expert's post
fozzzy wrote:
egmat wrote:
fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

Hi,

Well all other approaches are correct. Here is one more. A little less calculation intensive.

From 1/43 to 1/48, there are 6 #.

=> We can infer that sum of 6 # of 1/40s > Sum of (1/43+/1/44+......+1/48) > Sum of 6 # of 1/50s

=>So, 6/40 > Sum of (1/43+/1/44+......+1/48) > 6/50

=> 1/6.66 > Sum of (1/43+/1/44+......+1/48) > 1/8.33

=> 1/6.66 > 1/ (6.66< Denominator < 8.33) > 1/8.33

Only option available is C. Answer is 1/8.

-Shalabh Jain

Thanks this made it clear I was confused between those 2 options.

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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  26 Apr 2013, 13:30
2
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What is the sum of $$\frac{1}{43}+ ... +\frac{1}{48}$$?

$$\frac{1}{43}(1+\frac{43}{44}+\frac{43}{45}+\frac{43}{46}+\frac{43}{47}+\frac{43}{48})$$
we can rewrite as: $$\frac{1}{43}(1+1+1+1+1+1)=\frac{6}{43}$$

6/43 is something more than 7, so is colse to 8
$$\frac{6}{43}=(almost)\frac{1}{8}$$
C
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  26 Apr 2013, 13:44
Expert's post
danzig wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

a) $$\frac{1}{12}$$

b) $$\frac{1}{10}$$

c) $$\frac{1}{8}$$

d) $$\frac{1}{6}$$

e) $$\frac{1}{4}$$

I would look at it like this

K < 6/43 approx 1/7
K > 6/48 approx 1/8

So K is a value between 1/7 and 1/8 so C
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  26 Apr 2013, 14:06
Bunuel wrote:
fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

Given that $$K=\frac{1}{43}+\frac{1}{44}+\frac{1}{45}+\frac{1}{46}+\frac{1}{47}+\frac{1}{48}$$. Notice that 1/43 is the larges term and 1/48 is the smallest term.

If all 6 terms were equal to 1/43, then the sum would be 6/43=~1/7, but since actual sum is less than that, then we have that K<1/7.

If all 6 terms were equal to 1/48, then the sum would be 6/48=1/8, but since actual sum is more than that, then we have that K>1/8.

Therefore, 1/8<K<1/7. So, K must be closer to 1/8 than it is to 1/6.

Similar question to practice from OG: m-is-the-sum-of-the-reciprocals-of-the-consecutive-integers-143703.html

Hope it helps.

Bunuel, I understand your method. However, how can we know that the distance between k and 1/8 is shorter than the distance between k and 1/6. For example, if k were almost 1/7, we would have to calculate the distance between 1/8 and 1/7 and also the distance between 1/7 and 1/6.
I make this comment because the GMAT Prep explains that point, but it does that in a complex way.
Thanks!
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  26 Apr 2013, 14:56
My take

k = 1/43 +1/44 + 1/45 + 1/46 + 1/47 + 1/48 approx = 3/45 + 3/48 = 3/45 (1+45/48) = 1/5 (1 + 15/19) = 1/5 (1 + 15/20) = 1/5 (1+3/4) = 7/4*1/5 = 7/9 higher than 7/10 and lower than 8/10

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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  26 Apr 2013, 15:52
1/43 , 1/44 , 1/45 , 1/46 , 1/47 , 1/48

Hypothetically , assume all 6 Numbers to be 1/43
Hence sum of it would result in = 6/43 ( almost equals 1/7 , or slightly less than 1/7 - )

If all 6 nos were to be 1/48 ... Sum of which would result in 6/48 = 1/8 .

Hence answer should lie between 1/7 and 1/8

The only option that satisfies this criteria is option C i.e 1/8

Hope that helps
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  26 Apr 2013, 16:13
My method:

$$\frac{1}{43}$$ through $$\frac{1}{48}$$ are all very close to $$\frac{1}{50}$$ (we are dealing with very small fractions at this point, so the differences are nearly none)

So I added all six together as $$\frac{1}{50}$$ each, giving a total of $$\frac{6}{50}$$. This reduces to $$\frac{3}{25}$$, which is near $$\frac{3}{24}=\frac{1}{8}$$

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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  27 Apr 2013, 04:14
Expert's post
danzig wrote:
Bunuel wrote:
fozzzy wrote:
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

How do we decide between 1/6 and 1/8

Given that $$K=\frac{1}{43}+\frac{1}{44}+\frac{1}{45}+\frac{1}{46}+\frac{1}{47}+\frac{1}{48}$$. Notice that 1/43 is the larges term and 1/48 is the smallest term.

If all 6 terms were equal to 1/43, then the sum would be 6/43=~1/7, but since actual sum is less than that, then we have that K<1/7.

If all 6 terms were equal to 1/48, then the sum would be 6/48=1/8, but since actual sum is more than that, then we have that K>1/8.

Therefore, 1/8<K<1/7. So, K must be closer to 1/8 than it is to 1/6.

Similar question to practice from OG: m-is-the-sum-of-the-reciprocals-of-the-consecutive-integers-143703.html

Hope it helps.

Bunuel, I understand your method. However, how can we know that the distance between k and 1/8 is shorter than the distance between k and 1/6. For example, if k were almost 1/7, we would have to calculate the distance between 1/8 and 1/7 and also the distance between 1/7 and 1/6.
I make this comment because the GMAT Prep explains that point, but it does that in a complex way.
Thanks!

Even if K=1/7, still the distance between 1/8 and 1/7 is less than the distance between 1/7 and 1/6.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  29 Apr 2013, 00:55
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I did in this way
$$\frac{1}{43} + \frac{1}{44} + \frac{1}{45} + \frac{1}{46} + \frac{1}{47} + \frac{1}{48}$$

$$= (\frac{1}{43} + \frac{1}{48}) + (\frac{1}{44} + \frac{1}{47}) + (\frac{1}{45} + \frac{1}{46})$$ .... Grouping the denominator's whose addition is same (91)

$$= \frac{1}{24} + \frac{1}{24} + \frac{1}{24}$$ (Approx)

$$= \frac{3}{24}$$ (Approx)

$$= \frac{1}{8}$$
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Last edited by PareshGmat on 03 Aug 2014, 22:01, edited 1 time in total.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  13 Oct 2013, 00:51
K = 1/43 + 1/44 + 1/45 + 1/46 + 1/47 + 1/48

(48 + 43)/2 = 91/2 --> this is the denominator of my middle number

Therefore, K = 6 * 1/(91/2) = (6 * 2)/91 = ~2/15.x = ~1/8.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  02 Jan 2014, 07:06
If we see 1/43 to 1/48 each is greater than 2%. So sum will be slightly greater than 2*6= 12%
Now only option C is slightly more than 12%.
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  28 Jan 2015, 10:50
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Re: If K is the sum of reciprocals of the consecutive integers [#permalink]  28 Aug 2015, 00:27
One way would be to find the middle terms.
Since total terms is 5. Middle term will be 3rd term. i.e. 1/45. Which should be the approximate (but less) than original mean.
1/45 * 5 = 1/9. So you know that the sum will be very lose to 1/9 but just a little more. 1/8 is the closest and also the correct answer.
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Re: If K is the sum of reciprocals of the consecutive integers   [#permalink] 28 Aug 2015, 00:27

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