Calculating Maria's Running Time A Step-by-Step Guide

Hey guys! Let's break down this math problem together. It's all about figuring out how long Maria spends running each Saturday. The problem tells us that Maria exercises for a total of $1 \frac{5}{6}$ hours, and she spends $\frac{3}{4}$ of that time running. So, how do we find the exact amount of time she's running? That's what we're going to figure out!

Understanding the Problem

Before we dive into the math, let's make sure we fully grasp what the problem is asking. Maria has a set amount of exercise time, and a fraction of that time is dedicated to running. Think of it like a pie – her total exercise time is the whole pie, and running time is a slice of that pie. Our mission is to calculate the size of that slice. To find a fraction of a quantity, we use multiplication. That's our key tool here.

So, in simpler terms, we need to find $\frac{3}{4}$ of $1 \frac{5}{6}$ hours. This means we'll be multiplying these two values together. But before we can multiply, we need to make sure our numbers are in the right format. Mixed numbers (like $1 \frac{5}{6}$) can be a bit tricky to work with directly, so we'll convert them into improper fractions. This will make the multiplication process much smoother and easier to manage. Once we've got our numbers in the proper form, it's just a matter of multiplying the numerators (the top numbers) and the denominators (the bottom numbers). Then, we simplify the result, and boom! We've got our answer.

Step-by-Step Solution

Let's get down to the nitty-gritty and solve this problem step-by-step. We'll take it slow and break down each part so it's super clear. First up, converting that mixed number into an improper fraction.

Converting Mixed Numbers to Improper Fractions

Remember that a mixed number is a combination of a whole number and a fraction. In our case, we have $1 \frac{5}{6}$. To convert this to an improper fraction, we multiply the whole number (1) by the denominator of the fraction (6) and then add the numerator (5). This gives us the new numerator, and we keep the same denominator. So, let's do the math: (1 * 6) + 5 = 11. This means $1 \frac{5}{6}$ is equal to $\frac{11}{6}$.

Now that we've got our mixed number converted, we're ready to multiply! We've transformed $1 \frac{5}{6}$ hours into the equivalent improper fraction, $\frac{11}{6}$ hours. This conversion is crucial because it allows us to perform multiplication much more easily. Think of it like swapping out a tool for a better one – improper fractions are the better tool for multiplying fractions.

Multiplying the Fractions

Now, we need to multiply the fraction of time Maria spends running (rac{3}{4}) by the total exercise time (now in improper fraction form, $\frac{11}{6}$). Remember, to multiply fractions, we simply multiply the numerators together and the denominators together. So, we have: $\frac{3}{4} * \frac{11}{6} = \frac{(3 * 11)}{(4 * 6)} = \frac{33}{24}$.

Simplifying the Result

We've multiplied our fractions and got $\frac{33}{24}$. But this fraction looks a bit clunky, right? We want to simplify it to its simplest form. This means reducing the fraction by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it. In this case, the GCF of 33 and 24 is 3. So, let's divide both the numerator and the denominator by 3: $\frac{33 ÷ 3}{24 ÷ 3} = \frac{11}{8}$.

We've successfully simplified our fraction to $\frac{11}{8}$. This is an improper fraction, which means the numerator is larger than the denominator. While this is a perfectly valid answer, it's often more helpful to convert it back into a mixed number to better understand the quantity. Think of it like this: imagine you're measuring ingredients for a recipe – you'd probably rather know you need "one and a half cups" rather than "three halves of a cup." Converting to a mixed number gives us a more intuitive sense of the amount of time Maria spends running.

Converting Back to a Mixed Number

To convert the improper fraction $\frac{11}{8}$ back to a mixed number, we divide the numerator (11) by the denominator (8). The quotient (the whole number result of the division) becomes the whole number part of our mixed number, the remainder becomes the new numerator, and we keep the same denominator. 11 divided by 8 is 1 with a remainder of 3. So, $\frac{11}{8}$ is equal to $1 \frac{3}{8}$.

The Final Answer

So, after all that math, we've found that Maria runs for $1 \frac{3}{8}$ hours each Saturday. This matches option C. We did it!

Why This Answer Makes Sense

Let's take a step back and think about why this answer makes sense in the context of the problem. Maria exercises for a little less than 2 hours ($1 \frac{5}{6}$ hours is close to 2 hours), and she runs for a little less than $\frac{3}{4}$ of that time. So, we'd expect her running time to be less than $\frac{3}{4}$ of 2 hours, which is 1.5 hours. Our answer, $1 \frac{3}{8}$ hours, is indeed less than 1.5 hours, which gives us confidence that we're on the right track. Estimating the answer before doing the calculations can be a helpful way to catch any major errors along the way. It's like having a built-in safety check for your math!

Common Mistakes to Avoid

Math problems like this can sometimes trip us up if we're not careful. Here are a few common mistakes to watch out for:

• Forgetting to convert mixed numbers: This is a big one! Multiplying fractions is much easier when they're in improper fraction form. Trying to multiply mixed numbers directly can lead to errors.
• Incorrectly converting mixed numbers: Make sure you follow the correct procedure: multiply the whole number by the denominator, add the numerator, and keep the same denominator.
• Not simplifying the fraction: Always simplify your final answer to its simplest form. This makes the answer clearer and easier to understand.
• Flipping the fractions when multiplying: Remember, we only flip fractions when dividing. When multiplying, we multiply straight across.

By being aware of these potential pitfalls, you can increase your chances of acing similar problems in the future.

Practice Makes Perfect

Alright, guys, we've tackled this problem together, but the best way to truly master these skills is through practice. Try working through similar problems on your own. Change the numbers, change the context, and really challenge yourself. The more you practice, the more confident you'll become in your ability to solve fraction problems. And remember, math can be fun! It's like a puzzle waiting to be solved. So, keep practicing, keep exploring, and keep those math muscles strong!

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• Multiplying Fractions
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• Calculating Time
• Exercise Time

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