Grades 5-6 Video Solutions 2015-Level 5&6 Video Solutions 2015 problem20

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Question number 20. In this sum, the same letters represent the same digits, and different  letters represent different digits. What digit does the letter x represent? Here in our sum,  we see that x represents a one-digit number, and then we have a two-digit number with all digits the same, and those are represented by y. The sum of these is a three-digit number  with all digits the same, represented by the letter z. So we can have rough estimates on the size of these numbers. We know that x over here, being a single-digit number, is  less than or equal to 9, and y here, being a two-digit number, yy being a two-digit number,  is at most 99. So, in that case, we can add them up and say something about the three-digit number zzz. x plus x plus the double-digit number yy, that's 9 plus 9 plus 99, or 117.  And since that is less than 222, we conclude that z has to be equal to the number 1. So, zzz is in  fact 111. So we have that information. Let's mark it here in red. z is the digit 1. Now, what about  y? Is y really the digit 9? So that is our question here. Is y equal to 9? Well, what if it wasn't?  What if it was 8? If y is equal to 8, then we have yy plus x plus x, that could be equal to 88,  plus 9, plus 9, which comes out to 106. And 106 is less than 111, which is our answer. We are sure  of that. So y is not equal to 8, and it is true that y is in fact equal to 9. So we have the next  piece of information, y is equal to 9. And with that, we can solve for x. So let me do that over  here. Solve for x. We have twice x plus 99, which is yy, and that's equal to 111. And that gives us 2x  is equal to 12, or x is equal to 6. So we have discovered that the letter x represents the digit 6, and the answer here to our problem is therefore e6.

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