Quebra-cabeça De Idade Da Mãe E Da Filha Resolva O Problema Matemático 30 E 10 Anos

Let's dive into a classic age problem! We're going to figure out the current ages of a mother and her daughter, given some clues about their ages now and in the future. This is a fun little math puzzle that uses algebra to crack the code. So, grab your thinking caps, guys, and let's get started!

Decoding the Age Problem

In this mathematical challenge, we're presented with a scenario involving a mother and her daughter's ages. The key to solving this puzzle lies in translating the word problem into algebraic equations. We'll use variables to represent their current ages and then set up equations based on the information given. This way, we can use the power of algebra to find the solution. Algebra allows us to represent unknown quantities with letters and form equations that describe the relationships between these quantities. By solving these equations, we can uncover the hidden values, in this case, the ages of the mother and daughter.

Setting up the Equations

• Defining the Variables: Let's start by assigning variables to the unknowns. We'll let 'M' represent the mother's current age and 'D' represent the daughter's current age. This is a crucial first step in translating the word problem into mathematical language. By using variables, we can manipulate the unknowns and form equations that capture the relationships described in the problem.
• Equation 1: Current Ages: The first piece of information we have is that the mother's age is three times the daughter's age. We can write this as an equation: M = 3D. This equation captures the direct relationship between their ages right now. It tells us that whatever the daughter's age is, the mother's age is three times that amount. This is a fundamental equation for solving the problem.
• Equation 2: Ages in 10 Years: The next clue is about their ages in 10 years. In 10 years, the mother's age will be M + 10, and the daughter's age will be D + 10. We're told that in 10 years, the mother's age will be twice the daughter's age. This translates to the equation: M + 10 = 2(D + 10). This equation describes their ages in the future and introduces a new relationship that we can use to solve for the unknowns.

With these two equations, we have a system of equations that we can solve to find the values of M and D. The beauty of algebra is that it provides us with the tools to manipulate these equations and isolate the variables, ultimately leading us to the solution. It's like having a secret code that we can decipher using mathematical techniques.

Solving the System of Equations

Now that we have our equations, it's time to solve them. There are a couple of ways we can do this, but one common method is substitution. In this method, we'll use the first equation to substitute for one variable in the second equation. This will give us an equation with only one variable, which we can then solve.

• Substitution Method:
  • We know from the first equation that M = 3D. Let's substitute this into the second equation:
    • 3D + 10 = 2(D + 10)
  • Now we have an equation with only D, the daughter's age. Let's simplify and solve for D:
    • 3D + 10 = 2D + 20
    • 3D - 2D = 20 - 10
    • D = 10

So, we've found that the daughter's current age is 10 years old. Awesome! Now we can use this information to find the mother's age.

• Finding the Mother's Age:
  • We can use the equation M = 3D to find the mother's age.
  • Substitute D = 10 into the equation:
    • M = 3 * 10
    • M = 30

So, the mother's current age is 30 years old. We've cracked the code! By using algebra, we've successfully found the ages of both the mother and the daughter. It's like being a mathematical detective, following the clues and solving the mystery.

Checking the Solution

It's always a good idea to check our solution to make sure it fits the original problem. This is like double-checking our work to ensure we haven't made any mistakes along the way. It gives us confidence that our answer is correct and that we've truly solved the puzzle.

• Current Ages: The mother is 30, and the daughter is 10. Is the mother's age three times the daughter's age? Yes, 30 = 3 * 10. This checks out.
• Ages in 10 Years: In 10 years, the mother will be 40 (30 + 10), and the daughter will be 20 (10 + 10). Will the mother's age be twice the daughter's age? Yes, 40 = 2 * 20. This also checks out.

Since our solution satisfies both conditions of the problem, we can be confident that we've found the correct ages. It's like fitting the last piece of the puzzle into place and seeing the whole picture come together. We've successfully used our algebraic skills to solve a real-world problem!

Analyzing the Answer Choices

Now that we've solved the problem, let's take a look at the answer choices provided. We have two options:

• A) Mãe: 30 anos, Filha: 10 anos
• B) Mãe: 36 anos, Filha: (This option is incomplete in the original problem statement)

Identifying the Correct Solution

By solving the equations, we found that the mother is 30 years old, and the daughter is 10 years old. This perfectly matches option A. So, the correct answer is:

• A) Mãe: 30 anos, Filha: 10 anos

Option B is incomplete, so we can't evaluate it fully, but our solution definitively matches option A. This reinforces the idea that our algebraic approach has led us to the correct answer. It's like having a map that guides us directly to the treasure!

Why This Matters: The Power of Algebraic Thinking

This problem might seem like just a simple age puzzle, but it highlights the power of algebraic thinking. Algebraic thinking isn't just about manipulating equations; it's about translating real-world situations into mathematical models. It's about identifying relationships, representing unknowns, and using logical steps to find solutions. It's a skill that's valuable in many areas of life, not just in math class.

• Problem-Solving: By breaking down the problem into smaller parts and using variables to represent the unknowns, we were able to create a clear path to the solution. This is a key skill in problem-solving, whether it's in math, science, or everyday life.
• Logical Reasoning: The process of setting up and solving equations requires logical reasoning. We had to understand the relationships between the ages and use mathematical rules to manipulate the equations. This strengthens our ability to think critically and logically.
• Real-World Applications: Age problems are just one example of how algebra can be used to model real-world situations. From calculating distances and speeds to managing finances and predicting trends, algebra is a powerful tool for understanding the world around us.

So, the next time you encounter a problem, remember the power of algebraic thinking. It's like having a superpower that allows you to unlock hidden solutions and make sense of complex situations.

Key Takeaways from Our Age-Solving Adventure

Let's recap the key concepts we've covered in this age-solving adventure. This will help solidify our understanding and make sure we're ready to tackle similar problems in the future. It's like packing our toolkit with the essential skills and knowledge we'll need for the next challenge.

• Translating Words into Equations: The first and perhaps most crucial step is translating the word problem into mathematical equations. This involves identifying the unknowns, assigning variables, and expressing the relationships between them in the form of equations. This skill is fundamental to applying algebra to real-world problems.
• Setting up a System of Equations: In this case, we had two unknowns (the mother's age and the daughter's age), so we needed two equations to solve the problem. A system of equations is a set of two or more equations that share the same variables. Solving a system of equations allows us to find the values of all the unknowns.
• Solving by Substitution: We used the substitution method to solve the system of equations. This involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable and allows us to solve for the remaining one.
• Checking Your Solution: Always check your solution by plugging the values back into the original equations to make sure they hold true. This is a crucial step in ensuring the accuracy of your answer and catching any potential errors.

By mastering these key takeaways, you'll be well-equipped to solve a wide range of algebraic problems. It's like having a set of keys that unlock the doors to mathematical understanding and problem-solving success.

Final Thoughts: Embrace the Challenge of Math!

So, guys, we've successfully solved this age problem using the power of algebra. It's a testament to the fact that math isn't just about numbers and symbols; it's about logical thinking, problem-solving, and unlocking the mysteries of the world around us. It's like embarking on an adventure where every problem is a new challenge to conquer.

Don't be afraid to embrace the challenge of math. It's a journey of discovery, and with each problem you solve, you'll gain new skills, build confidence, and expand your understanding of the world. It's like climbing a mountain – the view from the top is always worth the effort.

So, keep practicing, keep exploring, and keep challenging yourself. The world of math is full of exciting possibilities, and you have the potential to unlock them all! And remember, if you ever get stuck, there are always resources and people who can help you along the way. It's like having a team of guides to support you on your mathematical journey.

If a mother's age is triple her daughter's age, and in 10 years the mother will be twice the daughter's age, what are their current ages?

Mother and Daughter Age Puzzle Solve the Math Problem 30 and 10 Years Old