Solving Math Expressions: 42/75 - (-22/30) And 85/4 + 25/-5

Hey guys! Today, we're diving into some mathematical expressions. We've got two problems to tackle: 42/75 - (-22/30) and 85/4 + 25/-5. Don't worry, we'll break it down step by step so it's super easy to follow. Math can be fun, and I promise we'll make it enjoyable! Let's get started and solve these expressions together.

Breaking Down the First Expression: 42/75 - (-22/30)

Okay, let's jump into our first problem: 42/75 - (-22/30). This looks a bit intimidating at first, but trust me, it's totally manageable. The key here is to remember our order of operations and how to handle fractions. We're going to take this one step at a time, so you can see exactly how it's done. First, we need to address the subtraction of a negative number. Remember, subtracting a negative is the same as adding a positive. This is a crucial first step, guys, so let’s make sure we nail it!

Step 1: Dealing with the Negative Sign

So, what happens when we subtract a negative? It turns into a positive! Our expression 42/75 - (-22/30) becomes 42/75 + 22/30. See? Not so scary anymore. This is a fundamental rule in math, and it’s super important to remember. Whenever you see a double negative like this, just think of it as addition. Now, we've simplified our problem a bit, but we're not done yet. We still need to add these fractions, and to do that, we need a common denominator.

Step 2: Finding the Common Denominator

To add fractions, we need a common denominator. This means we need to find a number that both 75 and 30 can divide into evenly. The easiest way to find this is to list the multiples of each number and see where they overlap. But, we're going to use a more efficient method: finding the Least Common Multiple (LCM). The LCM of 75 and 30 is 150. This might sound a bit complicated, but it's just the smallest number that both 75 and 30 can divide into without any remainders. So, our goal now is to rewrite both fractions with a denominator of 150.

Step 3: Converting the Fractions

Now that we know our common denominator is 150, let's convert our fractions. For 42/75, we need to multiply both the numerator and the denominator by 2 because 75 * 2 = 150. So, (42 * 2) / (75 * 2) = 84/150. Next up, we have 22/30. We need to multiply both the numerator and the denominator by 5 because 30 * 5 = 150. This gives us (22 * 5) / (30 * 5) = 110/150. Now we have two fractions with the same denominator: 84/150 and 110/150. We're getting closer to the solution, guys!

Step 4: Adding the Fractions

Alright, we've got our fractions with a common denominator, so now we can finally add them! We have 84/150 + 110/150. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same. So, (84 + 110) / 150 = 194/150. We've done the addition, but our work isn't quite over yet. We need to simplify this fraction to its lowest terms.

Step 5: Simplifying the Fraction

Our fraction is currently 194/150. To simplify it, we need to find the greatest common divisor (GCD) of 194 and 150 and divide both the numerator and the denominator by that number. The GCD of 194 and 150 is 2. So, we divide both by 2: (194 / 2) / (150 / 2) = 97/75. Now our fraction is in its simplest form. We can also express this as a mixed number if we want to. To do that, we divide 97 by 75, which gives us 1 with a remainder of 22. So, the mixed number is 1 22/75. And there you have it! We've solved the first expression. That wasn't too bad, right?

Tackling the Second Expression: 85/4 + 25/-5

Now, let's move on to our second expression: 85/4 + 25/-5. This one has a mix of fractions and a negative number, so we'll need to be careful with our steps. The first thing we should address is the fraction with the negative denominator. Remember, we want to make our lives as easy as possible, and dealing with negative signs correctly from the start is key. So, let's dive right in and see how we can simplify this expression.

Step 1: Simplifying the Second Term

We have 25/-5 as part of our expression. A fraction with a negative denominator can be a bit confusing, so let’s simplify it. We can rewrite 25/-5 as -25/5. This is because dividing by a negative number is the same as having a negative fraction. Now, we can further simplify -25/5 by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, (-25 / 5) / (5 / 5) = -5/1 = -5. Now our expression looks much simpler: 85/4 + (-5). This is a great start, guys! We've taken a potentially tricky fraction and turned it into a simple integer.

Step 2: Converting the Integer to a Fraction

To add 85/4 and -5, we need to express -5 as a fraction with the same denominator as 85/4. The denominator here is 4, so we need to rewrite -5 as a fraction with a denominator of 4. To do this, we multiply -5 by 4/4, which is just -5 * (4/4) = -20/4. Remember, multiplying by 1 (in this case, 4/4) doesn't change the value of the number, just its form. Now our expression is 85/4 + (-20/4). We're almost there!

Step 3: Adding the Fractions

Now that we have two fractions with the same denominator, we can add them. We have 85/4 + (-20/4). To add fractions with the same denominator, we simply add the numerators and keep the denominator the same. So, (85 + (-20)) / 4 = 65/4. We've done the addition, and we have a fraction as our result. But let's see if we can simplify it further, or express it as a mixed number.

Step 4: Simplifying and Expressing as a Mixed Number

Our fraction is currently 65/4. We need to check if we can simplify it. In this case, 65 and 4 do not have any common factors other than 1, so the fraction is already in its simplest form. However, we can express it as a mixed number, which can sometimes be more intuitive. To do this, we divide 65 by 4. 65 divided by 4 is 16 with a remainder of 1. So, the mixed number is 16 1/4. And that's it! We've solved the second expression and expressed the result in both fraction and mixed number forms.

Final Thoughts

So, guys, we've tackled two mathematical expressions today: 42/75 - (-22/30) and 85/4 + 25/-5. We broke down each problem step by step, from dealing with negative signs to finding common denominators and simplifying fractions. Remember, the key to math is to take it one step at a time and not be afraid to make mistakes. Every mistake is a learning opportunity! Keep practicing, and you'll become a math whiz in no time. If you have any questions or want to try more problems, let me know. Happy calculating!