Decoding The Value Of 2 1/4 × (3/2 - 3/4) + (2/8 ÷ 1/2) A Step-by-Step Guide

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Hey there, math enthusiasts! Ever stumbled upon a seemingly complex mathematical expression and felt a bit intimidated? Well, fear not! Today, we're going to break down a particular expression step-by-step, making sure you understand each operation along the way. Our mission? To find the value of the following expression:

214×(32−34)+(28÷12)2 \frac{1}{4} \times\left(\frac{3}{2}-\frac{3}{4}\right)+\left(\frac{2}{8} \div \frac{1}{2}\right)

So, grab your calculators (or your mental math muscles!), and let's dive in!

Unraveling the Expression: A Step-by-Step Guide

1. Converting Mixed Fractions to Improper Fractions

Our journey begins with the mixed fraction, $2 \frac{1}{4}$. To make things easier to work with, we'll convert it into an improper fraction. Remember, an improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Here’s how we do it:

  • Multiply the whole number (2) by the denominator (4): 2 * 4 = 8
  • Add the numerator (1) to the result: 8 + 1 = 9
  • Keep the original denominator (4).

So, $2 \frac{1}{4}$ becomes $\frac{9}{4}$. This conversion is crucial because it allows us to perform multiplication and division operations with fractions more smoothly. By converting, we transform a mixed number into a single fraction, making it easier to combine with other fractions in the expression. This is just like changing different currencies into one common currency before making a big purchase – it simplifies the overall calculation.

2. Tackling the Parentheses: Subtraction

Next up, we focus on the expression within the first set of parentheses: $\left(\frac{3}{2}-\frac{3}{4}\right)$. To subtract these fractions, they need a common denominator. The least common multiple of 2 and 4 is 4. So, we'll rewrite $\frac{3}{2}$ with a denominator of 4:

32=3×22×2=64\frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}

Now we can subtract:

64−34=6−34=34\frac{6}{4} - \frac{3}{4} = \frac{6-3}{4} = \frac{3}{4}

Why is finding a common denominator so important? Think of fractions as slices of a pie. You can't directly compare or subtract slices if they're cut into different sizes. A common denominator ensures that all the fractions are expressed in terms of the same “slice size,” allowing for accurate subtraction. This step is not just about following rules; it's about making the fractions speak the same language so we can understand their relationship clearly. It's like translating different units (inches and feet) into a common unit (inches) before measuring the length of an object.

3. Diving into Division: The Second Set of Parentheses

Moving on, we encounter the second set of parentheses: $\left(\frac2}{8} \div \frac{1}{2}\right)$. Dividing fractions might seem tricky, but there’s a neat trick we multiply by the reciprocal of the second fraction. The reciprocal of $\frac{1{2}$ is $\frac{2}{1}$. So, the division becomes:

28÷12=28×21=2×28×1=48\frac{2}{8} \div \frac{1}{2} = \frac{2}{8} \times \frac{2}{1} = \frac{2 \times 2}{8 \times 1} = \frac{4}{8}

We can simplify $\frac{4}{8}$ by dividing both the numerator and denominator by their greatest common divisor, which is 4:

48=4÷48÷4=12\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}

Why do we flip the second fraction and multiply when dividing fractions? This might seem like a magic trick, but there's a solid mathematical reason behind it. Dividing by a fraction is the same as asking how many times that fraction fits into the number you're dividing. Instead of trying to figure out how many $ rac{1}{2}$s are in $ rac{2}{8}$, we can flip $\frac{1}{2}$ to $\frac{2}{1}$ and multiply. This effectively answers the question in reverse – it finds how many times $ rac{2}{1}$ needs to be added to itself to reach $\frac{2}{8}$, which is the same as the original division problem. This approach transforms a division problem into a more manageable multiplication problem.

4. Multiplication: Bringing it Together

Now, let's tackle the multiplication part of the expression: $\frac{9}{4} \times \frac{3}{4}$. Multiplying fractions is straightforward: multiply the numerators and multiply the denominators:

94×34=9×34×4=2716\frac{9}{4} \times \frac{3}{4} = \frac{9 \times 3}{4 \times 4} = \frac{27}{16}

Multiplying fractions is a fundamental operation in many mathematical contexts. It's used in scenarios ranging from scaling recipes to calculating probabilities. The simplicity of multiplying numerators and denominators directly makes it a cornerstone of fractional arithmetic. Understanding this operation is key to unlocking more complex mathematical concepts later on. It's like learning the basic chords on a guitar before trying to play a complicated song.

5. The Grand Finale: Addition

We're in the home stretch! Our expression now looks like this:

2716+12\frac{27}{16} + \frac{1}{2}

To add these fractions, we need a common denominator again. The least common multiple of 16 and 2 is 16. So, we'll rewrite $\frac{1}{2}$ with a denominator of 16:

12=1×82×8=816\frac{1}{2} = \frac{1 \times 8}{2 \times 8} = \frac{8}{16}

Now we can add:

2716+816=27+816=3516\frac{27}{16} + \frac{8}{16} = \frac{27+8}{16} = \frac{35}{16}

Think of adding fractions with a common denominator like combining parts of a whole. If you have 27 slices of a pie that's cut into 16 pieces, and someone gives you another 8 slices of the same size, you simply add the number of slices together. This intuitive understanding of addition makes it easier to grasp why a common denominator is crucial. It ensures that you're adding comparable units – slices of the same size – rather than trying to add apples and oranges. This concept extends beyond fractions to various areas of mathematics and real-life scenarios involving proportions and ratios.

6. Expressing the Result: Improper Fraction or Mixed Number?

The final result is $\frac{35}{16}$. This is an improper fraction, which is perfectly valid. However, we can also express it as a mixed number to get a better sense of its value. To do this, we divide 35 by 16:

35 ÷ 16 = 2 with a remainder of 3

So, $\frac{35}{16}$ is equal to $2 \frac{3}{16}$.

Why both improper fractions and mixed numbers are useful? Improper fractions are essential for calculations. They make it easier to multiply, divide, add, and subtract fractions because you're dealing with a single, straightforward fraction. Mixed numbers, on the other hand, give a clearer sense of the quantity. When you say $2 \frac{3}{16}$, it's immediately evident that you have a little more than 2 wholes. The choice between using an improper fraction and a mixed number often depends on the context and what you want to emphasize: computational convenience or a quick grasp of the quantity.

The Final Answer

Therefore, the value of the expression $2 \frac{1}{4} \times\left(\frac{3}{2}-\frac{3}{4}\right)+\left(\frac{2}{8} \div \frac{1}{2}\right)$ is $\frac{35}{16}$ or $2 \frac{3}{16}$.

Wrapping it Up

Wow, we made it! We've successfully navigated through the expression, breaking it down into manageable steps. Remember, the key to tackling complex mathematical problems is to take it one step at a time. By understanding the order of operations (PEMDAS/BODMAS) and the principles behind each operation, you can confidently solve even the most daunting expressions. So, keep practicing, and you'll become a math whiz in no time!

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  • Improper Fractions: Related to the conversion process, attracting those who need help with this skill.
  • Order of Operations: Essential for solving mathematical expressions correctly, a key search term for math students.
  • PEMDAS/BODMAS: The acronyms for the order of operations, directly targeting users familiar with these terms.
  • Common Denominator: A crucial concept for fraction addition and subtraction.
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  • Step-by-Step Guide: Indicates the article's approach, appealing to users who prefer detailed explanations.
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By strategically incorporating these keywords, we increase the article's visibility in search engine results, making it more accessible to students, educators, and anyone seeking help with similar mathematical problems.

Conclusion

I hope this detailed breakdown has been helpful in understanding how to solve this mathematical expression. Remember, math is like a puzzle – each piece fits together logically. By understanding the rules and practicing regularly, you can master any mathematical challenge. Keep exploring, keep learning, and most importantly, keep having fun with math! If you have any questions or would like to explore other mathematical concepts, feel free to ask. Happy calculating, guys!