Adding Fractions

Question

Guys,

How would you approach to solve the following problem. I encountered this problem solving question in one of the tests - do not remember which one. I thought for about 1.5 mins about how to approach the problem but gave up after 2 mins guessed and moved on. However I would like to find out if there is a shortcut to approach such a problem. The problem looks innocuous but without knowing how to approach it, it will end up taking a lot of your time.

It's amazing that sometimes seemingly simple questions based on easy concepts stump you?

What is the sum of 1/11+1/12+1/13+1/14+.....+1/19 ?

hobbit · Answer

one common approach to such questions in gmat is to estimate the result... for this we must have the given 5 possible answers.

subhen · Answer

Thanks Hobbit, I tried to estimate as well but could not figure out which of the 5 choices was the closest. Unfortunately I don't remember the 5 choices. Thanks for the hint though.

hobbit · Answer

just to give an example of estimation with fractions...

1/11 through 1/19 are all greater than 1/20 hence their sum is greater than 9/20 = 0.45
also, each is less than 1/10 hence the result is less than 9/10=0.9

so the sum is between 0.45 and 0.9 (this "first order" estimation should rule out 1-3 answers)

a more precise estimation would be:
1/11 through 1/15 are greater (or equal) to 1/15, hence the sum of the 5 is greater than 5/15=0.333
1/16 through 1/19 are greater than 1/20 hence their sum is greater than 4/20 = 0.2

so the sum of all nine should be greater than 0.53

similarly, using 1/15 as an upper bound for 1/15 through 1/19, you can get an upper bound for the sum of all 9 fractions is less than 0.73

this is relatively good approximation which should leave you with the right answer and possibly one more choice.

subhen · Answer

Hobbit,

Thanks a lot for explaining how to approach such problems. I am convinced that this is the right approach to solve such problems. With this I am confident I could have eliminated 3-4 answer choices. 

Basically to answer such questions we need to find the upper bound and lower bound of adding such fractions and choose an answer which is in between the two.

subhen · Answer

Hobbit,

Using your hint, I will approach the problem as follows.

Divide the fractions into two:
1. 1/11+1/12+1/3+1/14+1/15 Let the sum be X
2. 1/16+1/17+1/18+1/19 Let the sum be Y

Each of the fractions in 1 is <1>= 1/15
Therefore, 5/10=1/2 > X > 5/15=1/3 i.e. 1/2 > X > 1/3 or 0.5 > X > 0.33

Each of the fractions in 2 is <1>1/20
Therefore, 4/16=1/8 > Y > 4/20=1/5 i.e. 1/8 > Y > 1/5 or 0.125 > Y > 0.2

Therefore, 0.53 < X+y < 0.625 or 8/15 < X+Y < 5/8

SimaQ · Answer

I would solve this problem using the avergae formula:

Thus, average=sum of terms/number of terms

We need a sum of terms.... so we have to know: number of terms*average....

The numbers given are consecutive so:

Number of terms = 19-11+1=9
Average=(19+11)/2=15 or in this case 1/15

Sum=9*(1/15)=9/15=3/5....

hobbit · Answer

nice, but wrong...

the average of 1/11 and 1/19 is not 1/15
(1/11+1/19)/2 = 15/209 = 0.07177
1/15 = 0.06666

the method you suggests work well if it is easy to calculate the true average of the terms.... but with fractions it is usually difficult.
in fact - knwoing that the method you suggest is relatively common, i'd put 3/5 as a deceptive wrong answer in the answer choices...

subhen · Answer

I agree with Hobbit. The method of finding the average and multiplying with the no. of terms might not work with fractions and definitely would not have worked with the question I got since none of the answers were close to 135/209 which is what you will get as the sum using this method.

Hence you need to use the approach suggested by Hobbit.

devilmirror · Answer

This is a hard question. I'd have to skip this question too. However, when looking at the question carefully, I found a way to get the answer quite fast. First dividing the number into 3 group Group 1: (1/11 + 1/12 + 1/13) Notice that 11*13 = (12-1)(12+1) = 144 - 1 = 143. I will use this knowledge to sum the first 3 numbers 1/11 + 1/13 = 1/(12-1) + 1/(12+1) = (11+13) / 143 = 24/143 This term is very close to 24/144 = 2/12 Thus Approx Group 1 = 2/12 + 1/12 = 3/12 Group 2: (1/14 + 1/15 + 1/16) Same approach: Approx Group 2 = 3/15 Group 3: (1/17 + 1/18 + 1/19) Same approach: Approx Group 3 = 3/18 We get G1 + G2 + G3 = 3/12 + 3/15 + 3/18 = 3(15 + 12 + 10)/180 = 37/60 Approx 0.61... Choose the answer choice that closes to the figure above. :twisted:

Devil must create this question to be tested on GMAT. Seriously...

SimaQ · Answer

Anyway, if you are asked for approximate answer my method works fine.... the answer is approximately 0,6...

Adding Fractions

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