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Jo adds up all the positive integers from 1 to 100. Kate does a similar thing with the first 100 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 100 values. What is the positive difference between Jo's sum and Kate's sum?

 Jul 28, 2019
 #1
avatar+104937 
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Jo's sum is easy   = (100)(101)/2  =  50 * 101   =  5050

 

Kate  would first start with 5  [ since 1 - 4 ]  are rounded to 0

Starting with 5 thru 14  she will have ten 10s

15 - 24   she will have  ten 20s

25 - 34 she will have  ten 30s

.

.

.

85 - 94 she will have ten 90s

95 - 100   she will have six 100s

 

So.....her sum is

 

10  [10  + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90]  + 6 [100] =

 

10  [ (10) (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) ] + 6[100]  =

 

10 (10)  [  sum of the first nine positive integers ] + 6 [ 100]

 

100  [  (9)(10) / 2 ]  + 6[100]  =

 

100 [ 45]  + 6[100] =

 

100 [ 45 + 6 ]  =

 

100 [ 51 ] =

 

5100

 

So....the positive difference is    5100 - 5050  =  50

 

 

cool cool cool

 Jul 28, 2019

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