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Re: What is the sum of odd integers from 35 to 85, inclusive?
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03 Jul 2016, 12:08

35

27

Bunuel wrote:

What is the sum of odd integers from 35 to 85, inclusive?

A) 1,560 B) 1,500 C) 1,240 D) 1,120 E) 1,100

Number of odd integers = (85-35)/2 + 1 = 50/2 + 1 = 26 Sum of odd integers = (35+85)/2 * 26 = 60 * 26 = 1560 Answer A
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1. We're dealing with consecutive integers so I can find the average by: \(\cfrac { First\quad Number+Last\quad Number }{ 2 }\) 2. To find the range in a set of numbers for which both end points are inclusive, we: \(\left( Hightest\quad Number-Lowest\quad Number \right) +1\)

I don't understand why you can simply divide the range by 2 in \(\cfrac { \left( Hightest\quad Number-Lowest\quad Number \right) }{ 2 } +1\) to find the number of odd numbers. Would it be the same if I wanted to find the number of even numbers?

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Re: What is the sum of odd integers from 35 to 85, inclusive?
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01 Jul 2017, 12:18

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Notice that by adding pairs of numbers we get a sum of 120. We need to find the number of integers between 35 and 85 inclusive. The number of integers is \(\frac{(85-35)}{2}+1\) = 26

So we add 120 , 26 times to get 120*26, dividing this by 2 , we get 1560.
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What is the sum of odd integers from 35 to 85, inclusive?

Quote:

Number of odd integers = (85-35)/2 + 1 = 50/2 + 1 = 26

Can you explain why did you divide by 2? If I want to count no. of terms from 1 to 10, I would perform (Last term - First term) + 1. But is not above approach only valid for consecutive terms and sequence starting with 1?

Quote:

Sum of odd integers = (35+85)/2 * 26 = 60 * 26 = 1560

I guess you used that for consecutive numbers, mean = median = (First term + last term) / 2. Let me know if we are on same page
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let me try to provide reply to your queries, Your query: Can you explain why did you divide by 2? Actually we divide by 2 because the difference between the consecutive odd numbers = 2.

In fact we have to count the multiple of any number say 7, multiple of 7 between 7 and 98 both inclusive. it will be (98-7)/7 +1 Always divide by common difference ( this is as per Arithmetic progression , AP) As per you "If I want to count no. of terms from 1 to 10, I would perform (Last term - First term) + 1." you are right here, the common difference = 1. so dividing by 1 is not necessary. Your Query :I guess you used that for consecutive numbers, mean = median = (First term + last term) / 2. yes you are right, in fact, it is true for any set of equally spaced numbers (AP)

What is the sum of odd integers from 35 to 85, inclusive?

Quote:

Number of odd integers = (85-35)/2 + 1 = 50/2 + 1 = 26

Can you explain why did you divide by 2? If I want to count no. of terms from 1 to 10, I would perform (Last term - First term) + 1. But is not above approach only valid for consecutive terms and sequence starting with 1?

Quote:

Sum of odd integers = (35+85)/2 * 26 = 60 * 26 = 1560

I guess you used that for consecutive numbers, mean = median = (First term + last term) / 2. Let me know if we are on same page

Add the numbers in pairs, starting from the outside and working towards the middle, we get: 35 + 37 + 39 + 41 + . . . . 79 + 81 + 83 + 85 = (35 + 85) + (37 + 83) + (39 + 81) + . . . = (120) + (120) + (120) + (120) + ..... = 120 x some integer

So, the desired sum must be DIVISIBLE by 120 Only one answer choice is divisible by 120

Answer: A

This observation is interesting but I would take a separate approach after the first two steps:-

Add the numbers in pairs, starting from the outside and working towards the middle, we get: 35 + 37 + 39 + 41 + . . . . 79 + 81 + 83 + 85 = (35 + 85) + (37 + 83) + (39 + 81) + . . . = (120) + (120) + (120) + (120) + ..... = 120 x some integer

Some integer is 13 because the average of extreme values is 60. So for every odd number from 35 to 60 there will be a pair for it from 60 to 85. Its easy to count that the number of pairs will be 13. therefore the answer is 120*13. [Only reason I mentioned this approach was because sometimes dividing 5 numbers could be time consuming. Counting the number of pairs to 13 was simple and straight forward)

Edit: Formating

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let me try to provide reply to your queries, Your query: Can you explain why did you divide by 2? Actually we divide by 2 because the difference between the consecutive odd numbers = 2.

In fact we have to count the multiple of any number say 7, multiple of 7 between 7 and 98 both inclusive. it will be (98-7)/7 +1 Always divide by common difference ( this is as per Arithmetic progression , AP) As per you "If I want to count no. of terms from 1 to 10, I would perform (Last term - First term) + 1." you are right here, the common difference = 1. so dividing by 1 is not necessary. Your Query :I guess you used that for consecutive numbers, mean = median = (First term + last term) / 2. yes you are right, in fact, it is true for any set of equally spaced numbers (AP)

What is the sum of odd integers from 35 to 85, inclusive?

Quote:

Number of odd integers = (85-35)/2 + 1 = 50/2 + 1 = 26

Can you explain why did you divide by 2? If I want to count no. of terms from 1 to 10, I would perform (Last term - First term) + 1. But is not above approach only valid for consecutive terms and sequence starting with 1?

Quote:

Sum of odd integers = (35+85)/2 * 26 = 60 * 26 = 1560

I guess you used that for consecutive numbers, mean = median = (First term + last term) / 2. Let me know if we are on same page

I think you are missing one point here Always divide by common difference ( this is as per Arithmetic progression , AP) this is not true we have to divide by common difference of first and last multiple of the integer whose multiples are being found out in this case 7 and 98 are by chance multiples of 7 lets say multiples b/w 7 and 99 even then we do 98-7
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Re: What is the sum of odd integers from 35 to 85, inclusive?
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25 Jul 2019, 22:31

ironsheep wrote:

Why do you find the number of odd integers? Why then do you add one to it??

Hey ironsheep,

The question has been solved using the formula so that you can avoid the long calculations of adding each of the digits. The reason 1 is added is that in the set 35 & 85 both are included, hence if we don't add 1 while calculating the number of terms we will be missing 1 term

Re: What is the sum of odd integers from 35 to 85, inclusive?
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26 Jul 2019, 06:41

1

Bunuel wrote:

What is the sum of odd integers from 35 to 85, inclusive?

A) 1,560 B) 1,500 C) 1,240 D) 1,120 E) 1,100

a slightly different way sum of n consective odd integers ( n+1)^2/4 so sum of odd integers from 35 to 85 sum of integers 1 to 85 - sum of integers ( 1 to 33) (86*86)/4 - ( 34*34)/4 = 1849-289 ; 1560 IMO A
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Re: What is the sum of odd integers from 35 to 85, inclusive?
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26 Jul 2019, 16:33

The numbers are 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85. There are 26 of these numbers that are odd. 85 minus 35 equals 50 and that 50 equals both odd and even integers. 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85. 26th place is the median of those 50 numbers, but I am still confused on why it is 26 and not 25?? 1/2 of 50 is 25

gmatclubot

Re: What is the sum of odd integers from 35 to 85, inclusive?
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26 Jul 2019, 16:33