Have you ever stumbled upon a math problem that looks simple, but then it throws a curveball that challenges your problem-solving skills? Today, we’ll tackle one that does just that—a puzzle where numbers play hide and seek within the rules of arithmetic.
The Problem:
A + B = A * B = A / B
What are the values of A and B?
THE ANSWER:
The answer to our puzzle is: A equals one-half (1/2), and B equals negative one (-1).
EXPLANATION
Let’s unravel this numerical mystery step by step. The equations given are:
- ( A + B = A * B )
- ( A * B = A / B )
At a quick glance, these equations suggest that A and B are partners in an arithmetic dance, matching each step in addition to their moves in multiplication and division. The key to solving this puzzle lies in understanding the unique properties of numbers when it comes to equality in both addition and multiplication.
Firstly, let’s focus on the second equation, ( A * = A / B ). This equation is the chaperone that guides us to the realization that ( B ) must be a number that, when squared, gives us 1. This is because ( A * B ) is also represented as ( A / B ). In other words, ( B^2 = 1 ). This gives us two options for B: it can either be 1 or -1.
Now, if we take a moment to consider B as 1, we quickly run into a contradiction. Plugging 1 into the first equation, ( A + 1 = A * 1 ), would imply that ( A = 0 ). However, substituting back into our second equation, ( A * 1 = A / 1 ), would not hold true if A were 0.
This leads us to our only logical option: B must be -1. With B firmly set as -1, the first equation transforms into ( A – 1 = -A ). This equation is the ballroom where A is revealed to be one-half, gliding elegantly to balance the equation.
Now that we have our dancers identified, let’s see how they perform when put back on the dance floor of our original equations. Plugging A as 1/2 and B as -1 into the first equation gives us:
[ 1/2 – 1 = 1/2 * (-1) ]
[ -1/2 = -1/2 ]
It’s a perfect match! And for the second equation:
[ 1/2 * (-1) = 1/2 / (-1) ]
[ -1/2 = -1/2 ]
Again, it’s a flawless performance. The puzzle is complete, with A and B having danced their way to a solution that satisfies both equations.
In this mathematical dance, A and B have shown that even a seemingly straightforward puzzle can have depth and complexity. By exploring each step and considering the properties of numbers, we’ve uncovered the solution and, in the process, sharpened our problem-solving skills.