Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem8

Video Transcription

Question number 8. In the addition on the right, different letters represent different digits. What is the value of C minus D plus E? In the addition here we have two numbers  that sum to 2010 and five digits here are used to obstruct these numbers. We know that they are five different digits, A, B, C, D, and E belong to the set containing 0, 1,  2, all the way up to 9. And we can look at the last column here and conclude that D plus B, either the sum is 0 or the sum is 10. So that's the only two cases where the 0 would  appear in the ones column. Now, in the first instance, the sum of 0 is obtained if D is equal to B and both of them are 0. So they're not different from one another and that is  simply not permitted in this problem so we cross that out. Now, the only remaining choice possibility is that D plus B is equal to 10 and in that case, the 1 is carried over  to the next column and we notice that after addition, that 1 travels down here and remains. So the only way that can happen is if C plus C is equal to 10 and the 0 here remains in  the second column together with the 1, we have a 1 and that tells us immediately that C is equal to 5. Now, that causes another 1 to be carried over to the third column and  then that 1 does not appear. So in blue here, we conclude that E plus B is equal to 9 so that with the extra 1 from carrying over, we have a 10, a 1 is carried over to the last  column and then here we can conclude that A is equal to 1. But we don't need A in our problem, we need a relationship between D and E since we already know that C is equal  to 5. So let's find the relationship between D and B, we have it right here, we can subtract these, the bottom equation from the top and after subtracting, we see that D minus E is  equal to 1. So that allows us to work on our problem, we have C minus D plus E, I will factor out a minus here and write this as C minus quantity, positive D, negative E and  then we can substitute. C was equal to 5 and D minus E, that expression is equal to 1 so we obtain the final result here which is 4. So the value of that expression C minus D  plus E is 4 and that's answer B.

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