Question 13 Step-by-Step Solution

Let's tackle this math problem step by step, guys! We've got a combination of mixed fractions, division, multiplication, and absolute values, but don't worry, we'll break it down so it's super easy to understand. Our main goal is to solve the expression: $2 \frac{1}{2} \div(5 \times 4 \div|-12+5|)=$

Step 1: Convert the Mixed Fraction to an Improper Fraction

First, we need to convert the mixed fraction $2 \frac{1}{2}$ into an improper fraction. This makes it easier to work with in calculations. To do this, we multiply the whole number (2) by the denominator (2) and add the numerator (1). Then, we put the result over the original denominator. So, $2 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$.

Why do we do this? Converting to an improper fraction simplifies multiplication and division. Trust me; it’s a neat trick!

Step 2: Simplify the Absolute Value

Next up, let's simplify the expression inside the absolute value bars: $|-12 + 5|$. Remember, the absolute value of a number is its distance from zero, so it's always positive. $-12 + 5 = -7$, and the absolute value of -7 is $|-7| = 7$. So, we've simplified the absolute value part of our problem.

What's the deal with absolute values? Think of them as a way to make negative numbers positive. It’s like a math superhero that always makes things positive!

Step 3: Perform the Operations Inside the Parentheses

Now, let's focus on the parentheses: $(5 \times 4 \div |-12+5|)$. We already know that $|-12 + 5| = 7$, so we can rewrite the expression as $(5 \times 4 \div 7)$. Following the order of operations (PEMDAS/BODMAS), we perform multiplication and division from left to right. First, $5 \times 4 = 20$. Then, we divide 20 by 7: $20 \div 7 = \frac{20}{7}$.

PEMDAS/BODMAS, huh? It’s just a fancy acronym to remember the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). It's like the math rulebook!

Step 4: Divide the Fractions

Now we have $\frac{5}{2} \div \frac{20}{7}$. Dividing fractions can seem tricky, but it's actually pretty straightforward. To divide fractions, we multiply by the reciprocal of the second fraction. The reciprocal of $\frac{20}{7}$ is $\frac{7}{20}$. So, we rewrite the division as multiplication: $\frac{5}{2} \times \frac{7}{20}$.

Reciprocal? What’s that? The reciprocal of a fraction is simply flipping it over. So, $\frac{a}{b}$ becomes $\frac{b}{a}$. It’s like doing a fraction flip!

Step 5: Multiply the Fractions

To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, $\frac{5}{2} \times \frac{7}{20} = \frac{5 \times 7}{2 \times 20} = \frac{35}{40}$.

Multiplying fractions is a breeze! Just multiply across the top and across the bottom. Easy peasy!

Step 6: Simplify the Resulting Fraction

Finally, we simplify the fraction $\frac{35}{40}$. Both 35 and 40 are divisible by 5. Dividing both the numerator and the denominator by 5, we get $\frac{35 \div 5}{40 \div 5} = \frac{7}{8}$.

Simplifying fractions is key! Always reduce your fraction to its simplest form. It’s like tidying up your math!

Step 7: Identify the Correct Answer

Let's look at our options:

A. $-\frac{1}{6}$ B. $\frac{1}{22}$ C. $\frac{5}{54}$ D. $\frac{7}{8}$

Oops! It seems there was a mistake in the provided options. The correct answer should be $\frac{7}{8}$, which is not listed among the choices. However, let's proceed with the correct steps and arrive at the correct solution.

Double-checking is crucial! Always make sure your answer matches one of the options (if provided). If not, it's time to double-check your work.

Alternative Approach: Step-by-Step Breakdown with Explanations

To make sure we’ve got this down pat, let's go through the problem again, but this time with even more detailed explanations. This will help solidify your understanding and give you the confidence to tackle similar problems.

Rewriting the Original Expression

We start with: $2 \frac{1}{2} \div(5 \times 4 \div|-12+5|)=$

Our mission is to simplify this step by step, so let’s get started!

Step 1 Revisited: Converting the Mixed Fraction

As we discussed, $2 \frac{1}{2}$ needs to become an improper fraction. We do this by multiplying the whole number (2) by the denominator (2) and then adding the numerator (1), all over the original denominator:

$2 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$

Why is this important? Improper fractions are much easier to handle when we're dealing with multiplication and division. It avoids confusion and keeps our calculations smooth.

Step 2 Revisited: Tackling the Absolute Value

Next, we look at the absolute value part: $|-12 + 5|$.

First, we simplify inside the absolute value bars: $-12 + 5 = -7$.

Then, we take the absolute value, which means we take the distance from zero, making it positive: $|-7| = 7$.

Why absolute value? Absolute value gives us the magnitude of a number, regardless of its sign. Think of it as the “size” of the number, without worrying about whether it’s positive or negative.

Step 3 Revisited: Parentheses Power!

Now we dive into the parentheses: $(5 \times 4 \div |-12+5|)$. We know $|-12 + 5| = 7$, so we have:

$(5 \times 4 \div 7)$

Following the order of operations, we multiply and divide from left to right:

$5 \times 4 = 20$

Then:

$20 \div 7 = \frac{20}{7}$

Order of operations is our friend! PEMDAS/BODMAS ensures we all get the same answer. It’s like the universal language of math!

Step 4 Revisited: Division of Fractions Demystified

Now we have: $\frac{5}{2} \div \frac{20}{7}$.

To divide fractions, we multiply by the reciprocal of the second fraction. The reciprocal of $\frac{20}{7}$ is $\frac{7}{20}$.

So, we rewrite the division as multiplication:

$\frac{5}{2} \div \frac{20}{7} = \frac{5}{2} \times \frac{7}{20}$

Dividing fractions? Just flip and multiply! It’s a handy trick to remember.

Step 5 Revisited: Multiplying Fractions Made Easy

Multiply the numerators and the denominators:

$\frac{5}{2} \times \frac{7}{20} = \frac{5 \times 7}{2 \times 20} = \frac{35}{40}$

Multiplying fractions is a breeze! Top times top, bottom times bottom.

Step 6 Revisited: Simplifying to the Max

Simplify the fraction $\frac{35}{40}$. Both 35 and 40 are divisible by 5.

$\frac{35 \div 5}{40 \div 5} = \frac{7}{8}$

Always simplify! It makes the fraction cleaner and easier to understand.

Step 7 Revisited: The Grand Finale

Our final simplified answer is $\frac{7}{8}$.

Wrapping it up! We’ve successfully navigated through the problem, step by step, and arrived at the correct answer. High five!

Conclusion: Mastering Math with Confidence

So, there you have it! We've successfully solved Question 13 by breaking it down into manageable steps. Remember, the key to tackling complex math problems is to take your time, follow the order of operations, and double-check your work. Keep practicing, and you'll become a math whiz in no time! Even though the provided options didn't include the correct answer, we learned the importance of understanding the process and arriving at the correct solution ourselves. Keep up the great work, guys! Remember, math can be fun when you break it down and understand the underlying concepts. You've got this!

This detailed walkthrough should help you not only solve this particular problem but also understand the underlying concepts and strategies that can be applied to many other math problems. Keep practicing and exploring, and you'll become more confident and skilled in math. Math isn't just about numbers; it's about problem-solving, logical thinking, and building a foundation for future learning. So, embrace the challenge and enjoy the journey of learning math!