Solving The Mathematical Expression 48 5/6 - 29 7/15 - 16 - 2 13/15 + 4/5 + 96 7/10 - 95 1/2 A Step-by-Step Guide

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Hey guys! Let's break down this mathematical expression together. It might look intimidating at first, but don't worry, we'll take it one step at a time. This guide will walk you through each part of the problem, so you can understand exactly how to solve it. We'll focus on clarity and making sure you grasp the underlying concepts. So, let’s get started and make math a little less scary!

Understanding the Problem

First, let's rewrite the mathematical expression to make it clearer:

48 5/6 - 29 7/15 - 16 - 2 13/15 + 4/5 + 96 7/10 - 95 1/2

This expression involves a combination of mixed numbers and fractions. To solve it effectively, we need to convert all mixed numbers into improper fractions. This will make it easier to perform the addition and subtraction operations. Remember, a mixed number is a whole number combined with a fraction, like 48 5/6. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number), like 293/6. Converting to improper fractions is a crucial first step, as it allows us to work with uniform values and apply the rules of fraction arithmetic more easily.

Converting mixed numbers to improper fractions involves multiplying the whole number by the denominator and adding the numerator. This result becomes the new numerator, and we keep the original denominator. For instance, to convert 48 5/6, we multiply 48 by 6, which gives us 288, and then add the numerator 5, resulting in 293. So, 48 5/6 becomes 293/6. Similarly, we'll convert all other mixed numbers in the expression to their improper fraction equivalents. This conversion process is fundamental to simplifying the expression and setting the stage for accurate calculations. So, let’s dive into the conversion process and get one step closer to solving the problem!

Converting Mixed Numbers to Improper Fractions

Okay, guys, let's dive into converting these mixed numbers into improper fractions. This is a crucial step, so pay close attention! We'll go through each one methodically to make sure we get it right. Remember, the goal is to transform each mixed number into a fraction where the numerator is larger than the denominator, which makes calculations much easier.

  • 48 5/6: As we discussed earlier, we multiply the whole number (48) by the denominator (6) and add the numerator (5). So, (48 * 6) + 5 = 288 + 5 = 293. The new numerator is 293, and we keep the original denominator, which is 6. Therefore, 48 5/6 becomes 293/6.
  • 29 7/15: Next up, we have 29 7/15. We multiply 29 by 15, which gives us 435, and then add the numerator 7, resulting in 442. Keeping the original denominator of 15, we get 442/15.
  • 2 13/15: Now, let's tackle 2 13/15. Multiply 2 by 15 to get 30, and add the numerator 13, which gives us 43. So, 2 13/15 becomes 43/15.
  • 96 7/10: For 96 7/10, we multiply 96 by 10, giving us 960, and then add the numerator 7, resulting in 967. Keeping the original denominator of 10, we get 967/10.
  • 95 1/2: Finally, let's convert 95 1/2. Multiply 95 by 2 to get 190, and add the numerator 1, which gives us 191. The new fraction is 191/2.

Now that we've converted all the mixed numbers into improper fractions, our mathematical expression looks like this:

293/6 - 442/15 - 16 - 43/15 + 4/5 + 967/10 - 191/2

See how much simpler it looks already? The next step is to deal with the whole number (16) and then find a common denominator for all these fractions. Stick with me, and we'll make this problem a piece of cake!

Dealing with the Whole Number and Finding a Common Denominator

Alright, let’s keep this momentum going! Our next step is to handle that whole number, 16, and then find a common denominator for all our fractions. Dealing with the whole number is straightforward – we can simply turn it into a fraction. Finding a common denominator might sound a bit tricky, but trust me, it’s just about finding a number that all our denominators can divide into evenly. Let's break it down:

Converting the Whole Number

First, we need to convert the whole number 16 into a fraction. To do this, we can simply write it as 16/1. This doesn't change its value, but it allows us to treat it like any other fraction in our expression. So now, our expression looks like this:

293/6 - 442/15 - 16/1 - 43/15 + 4/5 + 967/10 - 191/2

Finding the Least Common Denominator (LCD)

Now comes the crucial part: finding the least common denominator (LCD). The LCD is the smallest number that each of our denominators (6, 15, 1, 5, 10, and 2) can divide into without leaving a remainder. To find the LCD, we can list the multiples of each denominator and identify the smallest multiple they all share. Alternatively, we can use the prime factorization method, which is often more efficient for larger numbers.

Let's use the prime factorization method:

  • 6 = 2 * 3
  • 15 = 3 * 5
  • 1 = 1
  • 5 = 5
  • 10 = 2 * 5
  • 2 = 2

To find the LCD, we take the highest power of each prime number that appears in any of the factorizations: 2, 3, and 5. So, the LCD is 2 * 3 * 5 = 30.

Converting Fractions to Equivalent Fractions with the LCD

Now that we have our LCD, which is 30, we need to convert each fraction in our expression into an equivalent fraction with a denominator of 30. We do this by multiplying both the numerator and the denominator of each fraction by the same number, which will give us an equivalent fraction with the desired denominator. Let's convert each fraction:

  • 293/6: To get the denominator to 30, we multiply 6 by 5. So, we also multiply the numerator by 5: (293 * 5) / (6 * 5) = 1465/30
  • 442/15: To get the denominator to 30, we multiply 15 by 2. So, we also multiply the numerator by 2: (442 * 2) / (15 * 2) = 884/30
  • 16/1: To get the denominator to 30, we multiply 1 by 30. So, we also multiply the numerator by 30: (16 * 30) / (1 * 30) = 480/30
  • 43/15: To get the denominator to 30, we multiply 15 by 2. So, we also multiply the numerator by 2: (43 * 2) / (15 * 2) = 86/30
  • 4/5: To get the denominator to 30, we multiply 5 by 6. So, we also multiply the numerator by 6: (4 * 6) / (5 * 6) = 24/30
  • 967/10: To get the denominator to 30, we multiply 10 by 3. So, we also multiply the numerator by 3: (967 * 3) / (10 * 3) = 2901/30
  • 191/2: To get the denominator to 30, we multiply 2 by 15. So, we also multiply the numerator by 15: (191 * 15) / (2 * 15) = 2865/30

Now our mathematical expression looks like this:

1465/30 - 884/30 - 480/30 - 86/30 + 24/30 + 2901/30 - 2865/30

We’ve done the hard work of converting everything to a common denominator. The next step is the fun part: adding and subtracting the fractions! Let’s keep pushing forward!

Adding and Subtracting the Fractions

Okay, guys, we're in the home stretch now! We've got all our fractions with the same denominator, which means we can finally add and subtract them. This is where all our hard work pays off. Remember, when fractions have the same denominator, we simply add or subtract the numerators and keep the denominator the same. Let's dive in!

Combining the Numerators

Our expression is:

1465/30 - 884/30 - 480/30 - 86/30 + 24/30 + 2901/30 - 2865/30

Now, let's combine the numerators:

1465 - 884 - 480 - 86 + 24 + 2901 - 2865

Let’s do this step by step:

  • 1465 - 884 = 581
  • 581 - 480 = 101
  • 101 - 86 = 15
  • 15 + 24 = 39
  • 39 + 2901 = 2940
  • 2940 - 2865 = 75

So, our combined numerator is 75.

Writing the Resulting Fraction

Now that we have the combined numerator, we can write our resulting fraction. We keep the common denominator, which is 30. So, our fraction is:

75/30

Simplifying the Fraction

Our final step is to simplify this fraction. We need to find the greatest common divisor (GCD) of 75 and 30 and divide both the numerator and the denominator by it. The GCD of 75 and 30 is 15. So, let's divide both by 15:

  • 75 ÷ 15 = 5
  • 30 ÷ 15 = 2

Therefore, the simplified fraction is 5/2.

Converting the Improper Fraction to a Mixed Number (Final Answer)

We're almost there, guys! Our final step is to convert the improper fraction 5/2 back into a mixed number. This will give us our final answer in a more readable format. Remember, an improper fraction has a numerator that is greater than or equal to the denominator, and a mixed number combines a whole number and a proper fraction.

Dividing the Numerator by the Denominator

To convert 5/2 to a mixed number, we divide the numerator (5) by the denominator (2):

5 ÷ 2 = 2 with a remainder of 1

Forming the Mixed Number

The quotient (2) becomes the whole number part of our mixed number. The remainder (1) becomes the numerator of the fractional part, and we keep the original denominator (2). So, the mixed number is:

2 1/2

The Final Answer

So, after all our hard work, we've solved the mathematical expression!

48 5/6 - 29 7/15 - 16 - 2 13/15 + 4/5 + 96 7/10 - 95 1/2 = 2 1/2

Woo-hoo! You did it! I know it looked tough at the beginning, but by breaking it down step by step, we made it manageable. You've conquered a complex mathematical expression today. Great job, everyone!