Age Calculation: Daria, Rares, Maria, And Mihai's Ages

Let's dive into this intriguing age puzzle featuring Daria, Rares, Maria, and Mihai! We're tasked with figuring out each person's age based on the clues provided. This type of problem often requires us to translate the word problem into algebraic equations, which we can then solve to find the unknowns. So, grab your thinking caps, guys, and let's break it down step by step.

Understanding the Problem

The core of solving any mathematical problem, especially word problems, lies in understanding what the problem is asking and what information we already have. In this case, we're given relationships between the ages of Daria, Rares, and Maria. We also know that subtracting 60 from the sum of their ages reveals Mihai's age. The ultimate goal is to determine the age of each person involved.

Let's recap the key information:

• Daria is one year older than Rares.
• Maria is one year less than 29 years older than Rares.
• If we subtract 60 from the sum of the four ages, we get Mihai's age.

This information seems a bit tangled right now, but don't worry! We'll untangle it by assigning variables and forming equations. This is a classic strategy in algebra to handle such problems, and it makes things much clearer. Now, let's move on to the next step, which involves setting up the equations that will help us crack this age conundrum.

Setting Up the Equations

Alright, let's get mathematical! The best way to solve this age problem is to translate the given information into algebraic equations. This involves assigning variables to the unknown ages and then writing equations based on the relationships described in the problem. This might sound a bit intimidating, but trust me, it's a super effective way to organize the information and find the solution. Let's break it down:

• Let R be Rares's age.
• Since Daria is one year older than Rares, Daria's age can be represented as R + 1.
• Maria is one year less than 29 years older than Rares, so Maria's age is (R + 29) - 1, which simplifies to R + 28.
• Let M be Mihai's age.

Now, let's consider the information about the sum of their ages. We know that if we add all their ages together and subtract 60, we get Mihai's age. This gives us the equation:

R + (R + 1) + (R + 28) + M - 60 = M

Notice that M appears on both sides of the equation. This might seem a bit strange, but it's actually a useful setup! It hints that we can simplify the equation to find a direct relationship between Rares's age and a constant value. This is a crucial step towards solving the problem. By carefully organizing the information into equations, we've created a framework that allows us to use the power of algebra to find the ages of these individuals. Next, we'll simplify the equation and see what we can uncover about their ages.

Simplifying the Equation

Okay, guys, now for the fun part – simplifying the equation! We've got a bit of an algebraic puzzle here, and simplifying it will bring us closer to finding the solution. Remember the equation we set up in the last section? It was:

R + (R + 1) + (R + 28) + M - 60 = M

The goal here is to combine like terms and see if we can isolate R, which represents Rares's age. This will give us a starting point to figure out the ages of the others.

Let's start by combining the R terms and the constants on the left side of the equation:

R + R + R + 1 + 28 + M - 60 = M
3R + 29 + M - 60 = M
3R - 31 + M = M

Now, we have M on both sides of the equation, which is super convenient! We can subtract M from both sides, and it will cancel out:

3R - 31 + M - M = M - M
3R - 31 = 0

Look at that! We've managed to eliminate M and we're left with a much simpler equation involving only R. This is a huge step forward. By simplifying the equation, we've essentially distilled the information into its most essential form, making it easier to solve for Rares's age. Next up, we'll solve this simplified equation and find out exactly how old Rares is!

Solving for Rares's Age

Alright, let's solve for Rares's age! We've simplified the equation down to a very manageable form: 3R - 31 = 0. This is a simple linear equation, and we can solve it using basic algebraic techniques. The key is to isolate R on one side of the equation. This means we need to get rid of the -31 and the 3 that's multiplying R. Let's do it step by step.

First, we'll add 31 to both sides of the equation to get rid of the -31 on the left side:

3R - 31 + 31 = 0 + 31
3R = 31

Now, we have 3R = 31. To isolate R, we need to divide both sides of the equation by 3:

3R / 3 = 31 / 3
R = 31 / 3

Okay, we've got R = 31/3. This is an improper fraction, which means the numerator is larger than the denominator. To make it easier to understand, let's convert it to a mixed number:

31 / 3 = 10 with a remainder of 1
So, R = 10 1/3

However, in the context of age, we typically deal with whole numbers. It seems there might be a slight issue in the problem statement or the information provided, as Rares's age is not a whole number. This is a good reminder that in real-world math problems, it's important to check if the answer makes sense in the given context.

However, for the sake of continuing with the problem-solving process, let's assume there was a minor error in the original problem, and the equation should have resulted in a whole number for Rares's age. If we were to round R to the nearest whole number, we would get 10.

Let's assume Rares is 10 years old and proceed to find the ages of the others based on this assumption. In the next section, we'll calculate the ages of Daria and Maria using Rares's age as our foundation.

Calculating Daria and Maria's Ages

Now that we've (hypothetically) found Rares's age, let's move on to calculating Daria and Maria's ages. Remember, the problem gave us clear relationships between their ages and Rares's age. This makes our task relatively straightforward. We just need to plug in Rares's age (which we're assuming is 10 for the sake of this example) into the equations we set up earlier.

• Daria's age: We know Daria is one year older than Rares. So, Daria's age is R + 1. If Rares is 10, then Daria is 10 + 1 = 11 years old.
• Maria's age: Maria is one year less than 29 years older than Rares. This means Maria's age is (R + 29) - 1. If Rares is 10, then Maria is (10 + 29) - 1 = 39 - 1 = 38 years old.

So, based on our assumption that Rares is 10 years old, we've calculated that Daria is 11 years old and Maria is 38 years old. This is a great example of how solving for one variable can unlock the values of other related variables. By using Rares's age as a reference point, we were able to easily determine the ages of Daria and Maria.

However, remember that Mihai's age is still a bit of a mystery. We haven't used all the information given in the problem yet. We know that subtracting 60 from the sum of all their ages gives us Mihai's age. In the next section, we'll use this information to calculate Mihai's age and complete the age puzzle!

Finding Mihai's Age

Alright, guys, time to complete the puzzle and find Mihai's age! We've already (hypothetically) determined the ages of Rares, Daria, and Maria. Now we need to use the final piece of information – the relationship between the sum of their ages and Mihai's age – to crack this last part of the problem. Remember, the problem stated that if we subtract 60 from the sum of the four ages, we get Mihai's age. Let's put that into action.

First, we need to calculate the sum of the ages of Rares, Daria, and Maria. We're working with the assumption that Rares is 10, Daria is 11, and Maria is 38. So, the sum of their ages is:

10 (Rares) + 11 (Daria) + 38 (Maria) = 59 years

Now, let's call Mihai's age M. The problem states that if we add Mihai's age to the sum of the other three ages and subtract 60, we still get Mihai's age. This might seem a bit circular, but it's the key to solving for M. We can write this as an equation:

59 + M - 60 = M

Notice that the M on both sides will cancel each other out, similar to what we saw earlier when simplifying the equation for Rares's age. This indicates that there may be an inconsistency in the problem's information, as it doesn't allow us to directly solve for Mihai's age using this equation. The equation simplifies to:

-1 = 0

This is clearly not a valid statement, which confirms our suspicion that there's an issue with the problem's setup. In a real-world scenario, this would be a good point to double-check the original problem statement for any errors or missing information.

Conclusion and Reflection

We've journeyed through an interesting age problem involving Daria, Rares, Maria, and Mihai. We started by understanding the problem and identifying the key information. Then, we translated the word problem into algebraic equations, which allowed us to use mathematical tools to find a solution. We simplified equations, solved for Rares's age (with a hypothetical assumption), and calculated Daria and Maria's ages based on that assumption.

However, we encountered a snag when trying to determine Mihai's age. The information provided led to a contradiction, indicating a potential issue with the problem statement itself. This is a valuable lesson in problem-solving: it's not just about finding an answer, but also about critically evaluating whether the answer makes sense and whether the given information is consistent.

In a real-world scenario, encountering such an inconsistency would prompt us to go back and review the original information, look for any errors, or seek clarification. It highlights the importance of attention to detail and logical reasoning in mathematical problem-solving. While we couldn't definitively determine everyone's age due to the inconsistency, we successfully demonstrated the process of setting up equations and solving for unknowns, which are crucial skills in mathematics and beyond.