Hey guys! Ever find yourself scratching your head over those age-related math problems? Well, today we're diving deep into one that involves ratios and figuring out present ages. We'll break it down step-by-step, so you can confidently tackle similar questions in the future. Let's get started!
Decoding the Age Ratio Problem
In this section, we will thoroughly break down the age ratio problem, and here's the problem we're tackling:
The ratio of the ages of A and B, four years ago, was 4:5. Eight years from now, the ratio of the ages of A and B will be 11:13. What is the sum of their present ages? (a) 80 years (b) 96 years (c) 72 years (d) 76 years
This looks like a tricky one, right? But don't worry, we'll dissect it piece by piece. The key here is to translate the word problem into mathematical equations. We need to represent the unknown ages with variables and then use the given ratios to form equations. By carefully setting up the equations based on the information provided, we can use algebraic techniques to solve for the unknowns, ultimately leading us to the solution. This involves understanding how ratios change over time and how to represent past and future ages in terms of present ages. So, let's roll up our sleeves and get to work!
Setting Up the Equations: A Step-by-Step Guide
Okay, first things first, let's assign some variables. Let's say A's present age is 'a' and B's present age is 'b'. This is a crucial first step in translating the word problem into a mathematical form that we can actually work with. By representing the unknown quantities (the present ages of A and B) with variables, we create a foundation for building equations based on the given information. It's like laying the groundwork for solving a puzzle – you need the right pieces in place before you can start fitting them together.
Now, let's look at the first part of the problem: "The ratio of the ages of A and B, four years ago, was 4:5." Four years ago, A's age would have been a - 4 and B's age would have been b - 4. We can then express the ratio of their ages four years ago as a fraction, setting it equal to the given ratio of 4:5. This gives us our first equation, which mathematically represents the relationship between their ages in the past. By expressing this relationship as an equation, we can use algebraic methods to solve for the unknowns, 'a' and 'b'. This translation from words to mathematics is a fundamental skill in problem-solving, and it's the key to unlocking the solution to this age-related puzzle. So, our first equation is:
(a - 4) / (b - 4) = 4/5
Next, let's consider the second part of the problem: "Eight years from now, the ratio of the ages of A and B will be 11:13." Eight years from now, A's age will be a + 8 and B's age will be b + 8. Similar to before, we can express the ratio of their ages eight years from now as a fraction, setting it equal to the provided ratio of 11:13. This forms our second equation, capturing the future relationship between A and B's ages. By having two equations, each representing a different time frame (past and future), we create a system of equations that we can solve simultaneously. This is a powerful technique in algebra, allowing us to find the values of multiple unknowns when we have multiple pieces of information relating them. Thus, our second equation is:
(a + 8) / (b + 8) = 11/13
Solving the System of Equations: Unveiling the Ages
We now have two equations with two unknowns, 'a' and 'b'. This means we can solve for their values! There are a couple of ways to do this, such as using substitution or elimination. Let's use the cross-multiplication method, which is particularly handy for dealing with fractions. This technique involves multiplying both sides of the equation by the denominators to eliminate the fractions, making the equations easier to manipulate and solve.
Starting with the first equation:
(a - 4) / (b - 4) = 4/5
Cross-multiplying gives us:
5(a - 4) = 4(b - 4)
Expanding both sides, we get:
5a - 20 = 4b - 16
Let's rearrange this to get a and b on one side and the constant on the other:
**5a - 4b = 4 **(Equation 1)
Now, let's do the same for the second equation:
(a + 8) / (b + 8) = 11/13
Cross-multiplying gives us:
13(a + 8) = 11(b + 8)
Expanding both sides, we get:
13a + 104 = 11b + 88
Rearranging, we get:
**13a - 11b = -16 **(Equation 2)
Now we have a system of two linear equations:
- 5a - 4b = 4
- 13a - 11b = -16
To solve this system, we can use the elimination method. The goal here is to eliminate one of the variables (either a or b) by manipulating the equations so that their coefficients become opposites. This allows us to add the equations together, canceling out one variable and leaving us with a single equation in one unknown, which we can then easily solve. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. This systematic approach helps us unravel the relationships between the variables and find their specific values, ultimately leading us to the solution of the problem.
To eliminate b, we can multiply Equation 1 by 11 and Equation 2 by 4. This will give us the same coefficient for b but with opposite signs:
Equation 1 multiplied by 11:
55a - 44b = 44
Equation 2 multiplied by 4:
52a - 44b = -64
Now, subtract the second modified equation from the first:
(55a - 44b) - (52a - 44b) = 44 - (-64)
This simplifies to:
3a = 108
Divide both sides by 3 to solve for a:
a = 36
Great! We've found A's present age. Now, let's substitute the value of a back into Equation 1 to find b:
5(36) - 4b = 4
180 - 4b = 4
Subtract 180 from both sides:
-4b = -176
Divide both sides by -4:
b = 44
So, A's present age is 36 years, and B's present age is 44 years.
Finding the Sum and Choosing the Right Answer
We're almost there! The question asks for the sum of their present ages. So, we simply add a and b:
Sum = a + b = 36 + 44 = 80
Therefore, the sum of their present ages is 80 years. Looking at the options, the correct answer is:
(a) 80 years
Key Takeaways and Strategies for Age-Related Problems
- Translate Words into Equations: The most crucial step is converting the word problem into mathematical equations. Use variables to represent unknown ages and form equations based on the given ratios and timeframes.
- System of Equations: Age problems often lead to a system of equations. Master techniques like substitution or elimination to solve these systems effectively.
- Cross-Multiplication: This is a handy trick for equations involving fractions. It simplifies the equations and makes them easier to work with.
- Double-Check: Always double-check your answer against the original problem to ensure it makes sense in the context.
Practice Makes Perfect: Test Your Skills
To really nail these types of problems, practice is key! Try solving similar age-related questions. You can find plenty of examples online or in math textbooks. The more you practice, the more comfortable you'll become with setting up the equations and solving them. Remember, the goal is to understand the underlying concepts and apply them confidently.
Conclusion: You've Conquered the Age Puzzle!
Awesome! We've successfully navigated this age problem together. Remember, breaking down the problem into smaller steps, translating the words into equations, and using algebraic techniques are the key strategies. With practice, you'll be solving these problems like a pro. Keep up the great work, guys, and happy problem-solving!