Ordering Expressions By Value: A Math Calculation Guide

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Ordering Expressions by Value: A Math Calculation Guide

Hey guys! Today, we're diving into a fun math problem where we need to calculate the sums of different expressions and then arrange them in order from the largest value to the smallest. Don't worry, it sounds trickier than it is! We'll break it down step by step. Let's get started!

Understanding the Expressions

First, let's take a look at the expressions we're working with:

  1. 215+2252 \frac{1}{5} + 2 \frac{2}{5}

  2. −825+915-8 \frac{2}{5} + 9 \frac{1}{5}

  3. 925+(−535)9 \frac{2}{5} + (-5 \frac{3}{5})

These expressions involve mixed numbers (a whole number and a fraction combined) and addition. Some also include negative numbers, which adds a little twist. The key here is to remember how to add mixed numbers and how to handle negative numbers correctly. We'll go through each expression, calculate its sum, and then compare the results. Think of it like a mini-math puzzle – fun, right?

Converting Mixed Numbers to Improper Fractions

Before we can add these mixed numbers, we need to convert them into improper fractions. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This makes addition and subtraction much easier. To convert a mixed number to an improper fraction, we use a simple formula:

Whole number × Denominator + Numerator / Denominator

Let’s apply this to our expressions. For example, in the first expression, we have $2 \frac{1}{5}$. To convert this, we do:

2 × 5 + 1 = 11

So, $2 \frac{1}{5}$ becomes $\frac{11}{5}$. We'll do this for all the mixed numbers in our expressions to make the calculations smoother. This step is crucial because it transforms the mixed numbers into a format that's easier to work with when adding and subtracting. Trust me, it's a game-changer!

Adding the Fractions

Once we have our improper fractions, we can add them together. When adding fractions with the same denominator, we simply add the numerators and keep the denominator the same. For instance, if we have $\frac{3}{5} + \frac{1}{5}$, we add 3 and 1 to get 4, so the result is $\frac{4}{5}$. This is a fundamental rule of fraction addition, and it's what makes this process relatively straightforward.

However, we also need to pay attention to the signs. If we're adding a negative fraction, it's the same as subtracting. For example, if we have $\frac{7}{5} + (-\frac{2}{5})$, it's the same as $\frac{7}{5} - \frac{2}{5}$. Understanding these rules about signs is super important to avoid mistakes and get the correct answers. We'll make sure to handle the signs carefully in each of our expressions.

Calculating the Sums

Now, let's roll up our sleeves and calculate the sums of the expressions. We'll go through each one step by step, showing all the calculations so you can follow along easily. Remember, the goal is to get a single value for each expression that we can then compare and order. This is where the actual math happens, so let's take our time and make sure we get it right!

Expression 1: $2 \frac{1}{5} + 2 \frac{2}{5}$

First, we convert the mixed numbers to improper fractions:

  • 215=(2imes5)+15=1152 \frac{1}{5} = \frac{(2 imes 5) + 1}{5} = \frac{11}{5}

  • 225=(2imes5)+25=1252 \frac{2}{5} = \frac{(2 imes 5) + 2}{5} = \frac{12}{5}

Now, we add the improper fractions:

115+125=11+125=235\frac{11}{5} + \frac{12}{5} = \frac{11 + 12}{5} = \frac{23}{5}

So, the sum of the first expression is $\frac23}{5}$. We can also convert this back to a mixed number if we want $\frac{23{5} = 4 \frac{3}{5}$. Keep this value in mind as we move on to the next expression. This is our first data point, and we'll use it to compare with the others later.

Expression 2: $-8 \frac{2}{5} + 9 \frac{1}{5}$

Let's convert the mixed numbers to improper fractions, remembering to keep the negative sign for the first term:

  • −825=−(8imes5)+25=−425-8 \frac{2}{5} = -\frac{(8 imes 5) + 2}{5} = -\frac{42}{5}

  • 915=(9imes5)+15=4659 \frac{1}{5} = \frac{(9 imes 5) + 1}{5} = \frac{46}{5}

Now, we add the improper fractions:

−425+465=−42+465=45-\frac{42}{5} + \frac{46}{5} = \frac{-42 + 46}{5} = \frac{4}{5}

So, the sum of the second expression is $\frac{4}{5}$. This one is positive, which is good to note. We'll compare this value with the others to see where it fits in the order. Remember, keeping track of the signs is super important here!

Expression 3: $9 \frac{2}{5} + (-5 \frac{3}{5})$

Convert the mixed numbers to improper fractions:

  • 925=(9imes5)+25=4759 \frac{2}{5} = \frac{(9 imes 5) + 2}{5} = \frac{47}{5}

  • −535=−(5imes5)+35=−285-5 \frac{3}{5} = -\frac{(5 imes 5) + 3}{5} = -\frac{28}{5}

Now, we add the improper fractions:

475+(−285)=47−285=195\frac{47}{5} + (-\frac{28}{5}) = \frac{47 - 28}{5} = \frac{19}{5}

So, the sum of the third expression is $\frac19}{5}$. We can also convert this back to a mixed number $\frac{19{5} = 3 \frac{4}{5}$. Great job! We've now calculated the sums for all three expressions. The next step is to put them in order.

Arranging in Descending Order

Okay, we've done the hard work of calculating the sums. Now comes the fun part: arranging them in order from largest to smallest. This is like putting the pieces of a puzzle together to see the full picture. We have the following sums:

  1. Expression 1: $\frac{23}{5} = 4 \frac{3}{5}$
  2. Expression 2: $\frac{4}{5}$
  3. Expression 3: $\frac{19}{5} = 3 \frac{4}{5}$

To make it easier to compare, let's look at the mixed number forms. We have $4 \frac{3}{5}$, $\frac{4}{5}$, and $3 \frac{4}{5}$. It's pretty clear that $4 \frac{3}{5}$ is the largest, followed by $3 \frac{4}{5}$, and then $\frac{4}{5}$ is the smallest.

Ordering the Expressions

So, the expressions in descending order (from largest to smallest value) are:

  1. 215+225 (=235=435)2 \frac{1}{5} + 2 \frac{2}{5} \space ( = \frac{23}{5} = 4 \frac{3}{5})

  2. 925+(−535) (=195=345)9 \frac{2}{5} + (-5 \frac{3}{5}) \space (= \frac{19}{5} = 3 \frac{4}{5})

  3. −825+915 (=45)-8 \frac{2}{5} + 9 \frac{1}{5} \space (= \frac{4}{5})

And there you have it! We've successfully calculated the sums of the expressions and arranged them in the correct order. This is a fantastic example of how we can break down complex math problems into smaller, manageable steps. Great job, guys!

Conclusion

We made it! We've successfully calculated the sums of the given expressions and arranged them in descending order. This exercise was a great way to practice working with mixed numbers, improper fractions, and negative numbers. Remember, the key to solving math problems is to break them down into smaller steps and tackle each one methodically. Keep practicing, and you'll become a math whiz in no time! If you have any questions or want to try more problems like this, just let me know. Keep up the awesome work!