In the realm of mathematical puzzles, there are few as intriguing as problems involving age. The age-old conundrum of a father being twice as old as his daughter and their ages being intertwined through time has captured the curiosity of many. In this article, we will delve into this classic age problem and use algebra as our trusty tool to unlock the mystery. By the end, you’ll confidently know how to determine the present age of the father.
Setting the Stage
Let’s start by stating the problem: “A father is twice as old as his daughter. If 20 years ago, the age of the father was 10 times the age of the daughter, what is the present age of the father?”
To approach this problem systematically, we’ll introduce variables to represent the unknown ages. Let ‘D’ be the present age of the daughter and ‘F’ be the present age of the father. With these variables in place, we can translate the given information into mathematical expressions.
Translating Clues into Equations
- “A father is twice as old as his daughter.”
This statement gives us our first equation:
F = 2D
Now, let’s consider the second statement:
- “20 years ago, the age of the father was 10 times the age of the daughter.”
This statement leads to our second equation:
(F – 20) = 10(D – 20)
With these two equations in hand, we have a system of equations that we can solve to find the values of ‘F’ and ‘D.’
Solving the Equation System
To solve this system, we’ll employ the method of substitution. First, we’ll substitute the value of ‘F’ from equation 1 into equation 2:
(2D – 20) = 10(D – 20)
Next, we’ll distribute the 10 on the right side to eliminate the parentheses:
2D – 20 = 10D – 200
Now, to isolate the ‘2D’ term on the left side, we’ll subtract ‘2D’ from both sides:
-20 = 8D – 200
To get ‘8D’ by itself, we’ll add 200 to both sides:
180 = 8D
Now that we’ve found ‘D,’ which is the daughter’s age, we can proceed to determine the father’s age using equation 1:
F = 2D F = 2 * 22.5 F = 45
The Revelation: Present Age of the Father
There you have it! After methodically solving the age puzzle, we’ve determined that the present age of the father is 45 years.
In conclusion, age problems like these may initially seem perplexing, but with the power of algebra, we can dissect the clues, create equations, and unveil the answers. So, the next time you encounter a similar age-related riddle, you’ll be equipped with the tools to decipher it confidently.