math problem

[xyyz]

Here's a problem I found on abcnews.com along with the solution.

I follow the solution until one part, where I get lost.

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A Tricky Problem


A mathematics question is not inappropriate in a discussion of the SAT so let me include here a tricky one. In the correctly solved additions below, each of the five letters represents a different digit, EA being a two-digit number. What is the value of B + D if

A C
+B +D
-- --
C EA

Solution: Combining the two additions yields A+B+C+D=C+EA. If we cancel the C's from both sides of this equation, we obtain A+B+D=EA, and thus B+D=EA-A. The two-digit number EA equals 10xE+A, and so EA-A equals (10xE+A)-A, or simply 10xE. Since the digit E must be 1, (B+D)=10x1 or just plain 10. There are other approaches as well.

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I don't understand how the following conclusion is reached "Since the digit E must be 1..." E can also be 2-9. It seems like a huge jump to me.

I emailed the author to have him explain by saying "how ya figure?"

Hopefully, he'll respond soon... cause I've been mulling this damn thing for 3 hours... it's prevented my from sleeping and the damn sun's up now.

 

[d0ofy]

E has to equal 1 because C and D are single digit, and the sum of any two single digits is <20. Therefore, C+D=1x.

Edit:
Actually, C+D < 19, since the highest values would be 9+9=18.
Hope this helps you get some sleep.

 

[MereMortal]

C+D<18, since C and D represent different digits.

 

[xyyz]

it doesn't say that b or d are single digit numbers... why must this be assumed?

 

[MrCodeDude]

C+D<18, since C and D represent different digits.

No, it's C+D<18. Because why can't C = D? Are you saying that two variables can not be equal?
-- mrcodedude

 

[MrCodeDude]

it doesn't say that b or d are single digit numbers... why must this be assumed?

Because it implies that EA is a two digit number. Therefore, each variable represents a digit. So if you have ABC, it'd be a three digit number.
So the range for EA is:
10 < EA < 99

 

[RossGr]

<< each of the five letters represents a different digit, EA being a two-digit number >>



But it DOES say that each letter repersents a digit. The decimal digits are defined to be the set {0,1,2,3,4,5,6,7,8,9} so there are no assumption to be make it is cearly stated in the problem that not only are the letters digits they are different.

 

[RossGr]

Opps sorry about the double post!