Question

Day 17.
Okay, I'll be honest, I completely forgot that I hadn't posted the question for today. I am working on a few projects that made me forget it, sorry.
Also, thank you @ripinpeace for reminding me that it was fay 16 yesterday, forgot that too. XD

Today's question is this: If A, B, C and D are real numbers in the equation $\mathsf{ABC + BCB = CDD}$, what are the greatest possible values for them, such that each letter represents a digit?
(Difficulty: Hard)

I've been giving easy questions for the past few days so I thought of giving this one. DO NOT POST COPIED, SPAM OR IRRELEVANT ANSWERS, I shall not tolerate them.

Answer 1

$\huge\mathbb\fcolorbox{purple}{lavenderblush}{✰Solution}$

In the addition shown above, A, B,

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).

Hope my solution helps you..