saintforlife wrote:
ABC
+BCB
CDD
In the addition shown above, A, B, C, and D represent the nonzero digits of three 3-digit numbers. What is the largest possible value of the product of A and B ?
A. 8
B. 10
C. 12
D. 14
E. 18
I'm happy to help with this.
First, look at the one's column. C + B produces a one's unit of D, and we don't know whether anything carries to the ten's place.
But now, look at the ten's place --- B + C again produces a digit of D --- that tells us definitively that nothing carried, and that B + C = D. B & C have a single digit number as a sum.
Now, C = A + B, because again, in the hundreds column, nothing carries here.
Notice that, since C = A + B and D = B + C, B must be smaller than both B and C. We want to make the product of A & B large, so we want to make those individual digits large. Well, if we make B large, then that makes C large, and then B + C would quickly become more than a one-digit sum, which is not allowed. Think about it this way. Let's just assume D = 9, the maximum value.
D = 9 = B + C = B + (A + B) = A + 2B
We want to pick A & B such that A + 2B = 9 and A*B is a maximum. It makes sense that B would be smaller.
Try A = 7, B = 1. Then A + 2B = 9 and A*B = 7
Try A = 5, B = 2. Then A + 2B = 9 and A*B = 10
Try A = 3, B = 3. Then A + 2B = 9 and A*B = 9
Try A = 1, B = 4. Then A + 2B = 9 and A*B = 4
Indeed, as B gets bigger, the product gets less. This seems to imply that the biggest possible product is 10. This corresponds to A = 5, B = 2, C = 7, and D = 9, and the original addition problem becomes
527
+272 799
Thus, the maximum product is 10, and answer =
(B).
Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)