Solving The Math Expression: A Step-by-Step Guide

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Solving the Math Expression: A Step-by-Step Guide

Hey guys! Today, we're diving into a fascinating mathematical expression and breaking it down step by step. Math can seem daunting, but with a clear approach, it becomes super manageable. We're going to tackle this expression together: 439 * 13/35 + (23/35 - 2/7) * 7/16 - (3/4 - 5/16). Grab your calculators (or your mental math gears!) and let's get started!

Understanding the Order of Operations

Before we jump into the numbers, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS:

  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

This order ensures we solve the expression in the correct sequence, leading to the accurate answer. Ignoring this order can throw off the entire calculation, so always keep PEMDAS in mind. In our expression, we have parentheses, multiplication, addition, and subtraction. This means we'll start by simplifying the expressions within the parentheses first. Remember, math is like building with LEGOs; each step is a brick that supports the next one!

Step 1: Simplifying the Parentheses

Our expression has two sets of parentheses: (23/35 - 2/7) and (3/4 - 5/16). Let's tackle them one at a time.

Parenthesis 1: (23/35 - 2/7)

To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 35 and 7 is 35. So, we'll convert 2/7 to an equivalent fraction with a denominator of 35. To do this, we multiply both the numerator and the denominator of 2/7 by 5:

(2/7) * (5/5) = 10/35

Now we can subtract:

23/35 - 10/35 = 13/35

So, the first set of parentheses simplifies to 13/35. See? We're making progress already! Math is all about breaking big problems into smaller, manageable pieces. It's like solving a puzzle – each piece you fit gives you a clearer picture of the solution.

Parenthesis 2: (3/4 - 5/16)

Similarly, we need a common denominator for 3/4 and 5/16. The LCM of 4 and 16 is 16. Convert 3/4 to an equivalent fraction with a denominator of 16 by multiplying both the numerator and the denominator by 4:

(3/4) * (4/4) = 12/16

Now subtract:

12/16 - 5/16 = 7/16

The second set of parentheses simplifies to 7/16. We've cleared the first hurdle! Taking it step-by-step makes it less intimidating, right? Think of each set of parentheses as a mini-mission accomplished.

Step 2: Performing the Multiplication

Now that we've simplified the parentheses, let's rewrite the expression:

439 * 13/35 + (13/35) * 7/16 - 7/16

We have two multiplication operations to perform. Let's tackle them from left to right.

Multiplication 1: 439 * 13/35

Multiplying 439 by 13/35 gives us:

(439 * 13) / 35 = 5707 / 35

This fraction doesn't simplify nicely to a whole number, so we'll keep it as an improper fraction for now. We can convert it to a mixed number or decimal later if needed. Remember, sometimes in math, we work with fractions because they give us the most accurate representation of a number.

Multiplication 2: (13/35) * 7/16

Multiplying 13/35 by 7/16 gives us:

(13 * 7) / (35 * 16) = 91 / 560

We can simplify this fraction. Both 91 and 560 are divisible by 7:

91 / 7 = 13 560 / 7 = 80

So, the simplified fraction is 13/80. Nice! We're simplifying as we go, making the numbers easier to work with in the next steps. It's like decluttering your workspace before starting a new task.

Step 3: Rewriting and Performing Addition and Subtraction

Our expression now looks like this:

5707/35 + 13/80 - 7/16

Now, we need to perform the addition and subtraction from left to right. This means we'll add 5707/35 and 13/80 first, and then subtract 7/16 from the result.

Addition: 5707/35 + 13/80

To add these fractions, we need a common denominator. The LCM of 35 and 80 is 560. Let's convert each fraction to an equivalent fraction with a denominator of 560.

For 5707/35, we multiply both the numerator and the denominator by 16:

(5707/35) * (16/16) = 91312/560

For 13/80, we multiply both the numerator and the denominator by 7:

(13/80) * (7/7) = 91/560

Now we can add:

91312/560 + 91/560 = 91403/560

Subtraction: 91403/560 - 7/16

Next, we subtract 7/16 from 91403/560. We need a common denominator again. Luckily, 560 is a multiple of 16, so we just need to convert 7/16 to an equivalent fraction with a denominator of 560.

To do this, we multiply both the numerator and the denominator of 7/16 by 35:

(7/16) * (35/35) = 245/560

Now subtract:

91403/560 - 245/560 = 91158/560

Step 4: Simplifying the Final Fraction

Our final result is 91158/560. This is a large fraction, so let's try to simplify it. Both the numerator and the denominator are even, so we can divide both by 2:

91158 / 2 = 45579 560 / 2 = 280

So, our simplified fraction is 45579/280. We can leave it as an improper fraction, or convert it to a mixed number.

Step 5: Converting to a Mixed Number (Optional)

To convert 45579/280 to a mixed number, we divide 45579 by 280:

45579 Ă· 280 = 162 with a remainder of 219

So, the mixed number is 162 219/280.

Final Answer

The solution to the expression 439 * 13/35 + (23/35 - 2/7) * 7/16 - (3/4 - 5/16) is 45579/280 or, as a mixed number, 162 219/280.

Conclusion

There you have it! We've successfully navigated a complex mathematical expression by breaking it down into manageable steps. Remember, the key to solving math problems is understanding the order of operations and taking it one step at a time. Don't get overwhelmed by the size of the problem; focus on solving each piece, and the solution will reveal itself. Keep practicing, and you'll become a math whiz in no time! You've got this!