Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem3

Video Transcription

Question number three, the sums of the numbers in both rows of the table here shown on the right are equal. What is the value of the asterisks here?  So what we can do is we can practice our formula for adding n consecutive digits in order to sum quickly these values.  So if we add 1 plus 2 plus 3 plus all the way up to some number n consecutively, the answer is n times n plus 1 divided by 2.  So for example, 1 plus 2 plus 3 all the way up to 10 would be 10 times 11 divided by 2 which is 5 times 11 and which is 55. So the top row is summing up to 55 plus 2010.  So that gives us 2065. Now the bottom row, what we have is the numbers 11 plus 12 plus all the way up to 20 which  we can write as the sum 1 plus 2 plus 3 all the way up to 20 minus 1 plus 2 plus 3 all the way up to 10.  And so here on the first terms we can use our formula that would be 20 times 21 divided by 2 and then we subtract 55 and that gives us 10 times 21 minus 55 which is equal to  210 minus 55 and that is equal to 155. So the bottom row is equal to 155 plus the asterisk and that has to be equal to the sum of the numbers in the top row which came out to 2065.  So now we subtract and the asterisk here, the missing value is equal to 2065 minus 155 and that gives us exactly 1910. So we have our answer. The answer is C, 1910.

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