Integer Factor Calculation & Math Expressions
Hey guys! Let's dive into some math problems involving integer factor calculations and evaluating expressions. This might sound intimidating, but trust me, we'll break it down and make it super understandable. We're going to tackle these problems step by step, so grab your calculators (or your mental math muscles!) and let's get started!
Calculating Expressions by Converting Factors to Integers
When you first look at expressions with decimals, it might seem a bit messy. But the trick here is to convert those decimals into integers by multiplying. This makes the calculations much smoother and easier to handle. Let's walk through each expression, focusing on making those conversions and simplifying along the way.
1) 2.4 * 9.5 - (-3.5)
Okay, so we've got 2.4 * 9.5 - (-3.5). The key here is to think about how we can get rid of those decimals. We can rewrite 2.4 as 24/10 and 9.5 as 95/10. This might seem like extra work, but it sets us up for cleaner calculations. And don't forget about that - (-3.5), which is the same as + 3.5 or + 35/10.
So, let's rewrite the expression: (24/10) * (95/10) + (35/10). Now, multiply 24 by 95, which gives us 2280. So we have 2280/100 + 35/10. To add these fractions, we need a common denominator, which is 100. Convert 35/10 to 350/100, and we have 2280/100 + 350/100, which simplifies to 2630/100. Finally, divide that and you'll get 26.3. That wasn't so bad, right? Remember, breaking down the decimals is your best friend here.
2) (-9) - 2.4 - (-7.2)
Moving on to the next one: (-9) - 2.4 - (-7.2). Let's rewrite 2.4 as 24/10 and -(-7.2) as +7.2 or +72/10. The expression now looks like this: (-9) - (24/10) + (72/10). To make things easier, let's convert -9 to a fraction with a denominator of 10, which is -90/10. Now we have: (-90/10) - (24/10) + (72/10). Combine the fractions: (-90 - 24 + 72) / 10. This simplifies to (-42) / 10, which is -4.2. See? Another one down!
3) 5 - (-7) * 0.6 / 0.2 - (-8.1) * 2.5
This one's a bit longer, but don't worry, we'll tackle it. We've got 5 - (-7) * 0.6 / 0.2 - (-8.1) * 2.5. Remember our order of operations (PEMDAS/BODMAS)? Multiplication and division come before addition and subtraction. First, let’s convert those decimals: 0.6 becomes 6/10, 0.2 becomes 2/10, 8.1 becomes 81/10, and 2.5 becomes 25/10. Let's also handle those negative signs: -(-7) is +7 and -(-8.1) is +8.1. So, the expression becomes: 5 + 7 * (6/10) / (2/10) + (81/10) * (25/10).
Now, let’s break it down further. 7 * (6/10) is 42/10. Dividing by a fraction is the same as multiplying by its reciprocal, so (42/10) / (2/10) becomes (42/10) * (10/2), which simplifies to 21. Next, (81/10) * (25/10) is 2025/100, which is 20.25. Putting it all together: 5 + 21 + 20.25. Adding these up, we get 46.25. You're doing great!
4) (-2.5) * (-6) * 0.9 * (-0.36) / 12 * (-8.5) * (-0.23) / (-3) * 1.8 * 3.6 / 5 - (-2.3) * (-0.6)
Okay, this one looks like a beast, but don't let it scare you! We'll break it into manageable chunks. Let’s start by converting all the decimals to fractions:
-2.5 = -25/10-6 = -60.9 = 9/10-0.36 = -36/100-8.5 = -85/10-0.23 = -23/1001.8 = 18/103.6 = 36/10-2.3 = -23/10-0.6 = -6/10
The expression is now: (-25/10) * (-6) * (9/10) * (-36/100) / 12 * (-85/10) * (-23/100) / (-3) * (18/10) * (36/10) / 5 - (-23/10) * (-6/10).
First, let's handle the multiplications and divisions step by step. It's going to be a long process, but we'll keep it organized. Remember, a negative times a negative is a positive, and a positive times a negative is a negative.
(-25/10) * (-6) = 150/10 = 1515 * (9/10) = 135/10(135/10) * (-36/100) = -4860/1000
Now we divide by 12: (-4860/1000) / 12 = -4860/12000 = -405/1000.
Next part:
(-405/1000) * (-85/10) = 34425/10000(34425/10000) * (-23/100) = -791775/1000000
Divide by -3:
(-791775/1000000) / (-3) = 263925/1000000
Next part:
(263925/1000000) * (18/10) = 4750650/10000000(4750650/10000000) * (36/10) = 171023400/100000000
Divide by 5:
(171023400/100000000) / 5 = 171023400/500000000 = 3420468/10000000
Now, let's calculate the second term: (-23/10) * (-6/10) = 138/100 = 1.38.
Finally, subtract the second term from the first term:
(3420468/10000000) - (138/100)
This requires finding a common denominator, which is 10,000,000:
(3420468/10000000) - (1380000/10000000) = -10379532/10000000 = -1.0379532
Phew! That was a marathon! So, the final answer is approximately -1.0379532. Great job sticking with it! This shows how breaking down a massive problem into smaller steps makes it solvable. It’s all about being organized and patient.
Calculating Expressions Directly
Sometimes, instead of converting to integers, we just need to work directly with the decimals and integers as they are. This involves understanding order of operations and handling positive and negative numbers carefully. Let's see how this works!
1) (-8.8 + 2) - (-1) + 4 * (-1)
In this expression, (-8.8 + 2) - (-1) + 4 * (-1), we need to follow PEMDAS/BODMAS. First, let's handle the parentheses: (-8.8 + 2) = -6.8. Now the expression looks like: -6.8 - (-1) + 4 * (-1). Next up, multiplication: 4 * (-1) = -4. Now we have: -6.8 - (-1) + (-4). Let's take care of that subtraction of a negative: -6.8 - (-1) is the same as -6.8 + 1, which equals -5.8. Finally, we have -5.8 + (-4), which is -5.8 - 4. Adding those together gives us -9.8. Not too shabby, huh?
Conclusion
So guys, we've walked through some pretty complex calculations today, from converting decimals to fractions to working directly with decimal numbers and integers. The key takeaway is that breaking down the problem into smaller, manageable steps is crucial. Whether you're converting decimals to integers or following the order of operations, a systematic approach will help you get to the right answer.
Remember, practice makes perfect! The more you work with these kinds of problems, the more comfortable you'll become. So keep those calculators handy, keep that mental math sharp, and you'll be a math whiz in no time! You got this!