Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem13

11&12 Video Solutions 2010 problem13

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Question number 13. Certain whole numbers x and y satisfy the equation 2x is equal to 5y. One of the following numbers is equal to the sum x plus y. Which is it?  So we use the information we have in the problem to simplify our relationships here.  If we write 2x is equal to 5y we can divide both sides by 5 and we obtain y is equal to 2 fifths x. So then x plus y is equal to x plus 2 fifths x which is equal to 7 fifths x.  And x we can solve for now in terms of the sum x plus y is 5 sevenths x plus y.  Now we know that since x and y are both whole numbers it must follow that x plus y is divisible by 7.  Because our equation here clearly tells us that the sum here x plus y after multiplying by 5 and dividing by 7 gives us x.  And x is a whole number so x plus y must be divisible by 7. And so then we should check which of the candidate choices for the sum x plus y are divisible by 7.  And the test for divisibility by 7 of which there are several tests goes like this. You take your number and you group the digits in blocks of 3. So here 008 is just 8.  Here 007 is the number 7. So we group the first three digits from the right and just read that off as a number.  Then we group the next three digits from right to left and read that off as a number. So we have nothing nothing or 002 that's a 2. Here we have the same situation a 2.  And then we subtract but in the reverse order. So 7 minus 2 would be 5 and 8 minus 2 would be 6.  And then we check which number is divisible by 7. After doing this operation subtracting alternating blocks of 3 like that from right to left.  If the end result is divisible by 7 then the original number is divisible by 7. So here D is not divisible by 7. E is not divisible by 7. So let's move on to C.  We have 009 so that's 9. 9 minus 2 is 7 here and 7 is divisible by 7. So this is our candidate solution here. Now let's just verify that we have in fact the right solution.  So we check our work quickly. We have that X is equal to 5 sevenths times 2009. And after dividing by 7 and multiplying by 9 we have that X is equal to 1435.  Which tells us then that Y after subtracting from 2009, 1435 is equal to 574. And so then we can also see that solving the equation initially in the other order  we can have X is equal to 5 halves Y. So X plus Y is equal to 5 halves Y plus Y or 7 halves Y.  So Y is equal to 2 sevenths times X plus Y. And indeed 574 is equal to 2 sevenths times 2009. So everything checks out.  We could have just left the answer C after checking that 2009 is divisible by 7 but this is a nice check over here that we are indeed correct. So the answer is C, 2009.

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