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avatar+107 

After two numbers are removed from the list

 

 

$$9,~13,~15,~17,~19,~23,~31,~49,$$

 

 

the average and the median each increase by 2 . What is the product of the two numbers that were removed?

 Apr 20, 2019
 #1
avatar+42 
+1

We know that the median right now is 18 because (17+19)/2=18. So the median has to be 18+2 or 20. That means one of the numbers is either 17 or 19 because if you don't remove one of those numbers the median won't change. Now we take a look at are other clue. The average is increased by two. The average without removing the 2 numbers is (9+13+15+17+19+23+31+49)/8=22. That means the average after you take out the 2 numbers is 22+2=24. And that means the sum of all the numbers after you get rid of the 2 is 24*6=144 because if the average is 24, you multiply the average by the number of numbers in the set to find the total sum. Before, you get rid of the two numbers the sum is 9+13+15+17+19+23+31+49=176. 176-144=32 which means the two numbers you get rid of add up to 32. If you get rid of 17, you have to get rid of 15 because 32=17+15. But the median is then (19+23)/2=21 so you choose 19. If you choose 19 you need to also choose 13 because 32=19+13. and the median is (17+23)/2=20 so 19 and 13 are correct. The product is 247.

 Apr 20, 2019
 #2
avatar+101838 
+1

9, 13, 15, 17, 19, 23, 31, 49

The sum of these numbers  = 176

And we have 8 numbers.....so    .....the original average  =  176 / 8  = 22

And the original median = (17 + 19) / 2  =  36/2 = 18

So....the new median must be 20

 

So....after removing two numbers....the average is 24.....this means that the sum of the new numbers must be  24 (6)  = 144

 

So...the sum of two numbers removed must be   176 - 144  =   32

These pairs sum to 32

(15,17)....if we take these out...the new median is (19 + 23) /2  =  21

(13,19).....if we take these out.....the new median is (17 + 23) / 2  = 20

(9, 23)

 

So...the numbers must have been  13  and 19.....and their product  = 247

 

 

 

cool cool cool

 Apr 20, 2019

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