Solving the Age Conundrum: How Old is A Now?
Have you ever encountered a puzzle that left you scratching your head? Let's dive into an intriguing problem involving two individuals, A and B. The challenge is to determine the current age of A based on the given conditions. This article will break down the problem step-by-step, utilizing algebraic equations to find the solution. By the end, you'll not only have the answer but also a clearer understanding of problem-solving techniques in algebra.
I. The Problem Statement
Given that A is twice as old as B, and five years ago, A was three times as old as B, how old is A now? This requires a bit of logical reasoning and algebraic manipulation to solve.
1. Let the Present Age of B be x
To start, we define the current age of B as x. Accordingly, the current age of A can be represented as x 16. Five years ago, A's age was (x 16) - 5 x 11, and B's age was (x) - 5 x-5. We know that A was three times as old as B at that time, leading to the equation:
(x 11 3(x-5))
2. Expanding and Simplifying the Equation
By expanding the right side of the equation, we get:
(x 11 3x-15)
Rearranging the terms to isolate x, we obtain:
(11 15 3x-x
26 2x
x 13)
This implies that B is currently 13 years old, and consequently, A is x 16 13 16 29 years old.
II. Detailed Problem Solving
A. Algebraic Representation
To solve this problem systematically, let's define:
(A text{the current age of A})
(B text{the current age of B})
From the problem statement, we have:
(A 2B)
Five years ago, their ages were:
(A-5 3(B-5))
B. Substitution and Simplification
Substitute (A 2B) into the second equation:
((2B)-5 3(B-5))
Expanding and simplifying:
(2B-5 3B-15)
Rearranging the terms:
(2B-3B -15 5)
(-B -10)
(B 10)
With B's age known, we can find A's current age:
(A 2B 2(10) 20)
Hence, A is currently 20 years old.
III. Conclusion
Through a series of logical steps and algebraic manipulation, we've determined the current age of A to be 20 years. This problem not only tests one's understanding of algebraic equations but also reinforces the importance of careful reasoning and systematic problem-solving in mathematics.