Math Expression: Solve -[-1 1/5 ÷ 0.6(1.3)] - 5/6

by Andrew McMorgan 50 views

Evaluate the Expression: -[-1 1/5 ÷ 0.6(1.3)] - 5/6

Hey math enthusiasts! Today, we've got a doozy of an expression to tackle: $-[-\left(1 \frac{1}{5}\right) \div 0.6(1.3)\right] - \frac{5}{6}$

Don't let those brackets and fractions scare you, guys. We're going to break this down step-by-step, just like dissecting a cool piece of art or understanding a complex beat. The goal here is to find the single, definitive value of the expression. We'll be using the order of operations (PEMDAS/BODMAS) like our trusty compass to navigate through this mathematical landscape. So, grab your calculators, your notebooks, or just your brilliant brains, and let's dive in!

Decoding the Expression: A First Look

Alright, let's get down to business with our expression: $-[-\left(1 \frac{1}{5}\right) \div 0.6(1.3)\right] - \frac{5}{6}$. Before we even start crunching numbers, it's super important to understand what we're dealing with. We've got subtraction, division, multiplication, a mixed number, and a decimal, all nested within brackets and parentheses. This means we need to be extra careful and follow the order of operations religiously. Remember PEMDAS? Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is our roadmap, and deviating from it is like trying to remix a classic track and messing up the tempo – it just won't sound right!

Our first mission is to simplify what's inside the innermost parentheses and brackets. Notice the mixed number $1 \frac1}{5}$. To make things easier, we'll convert this into an improper fraction. $1 \frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{6}{5}$. And that decimal, $0.6$? Let's convert that to a fraction too $0.6 = \frac{610} = \frac{3}{5}$. Now, our expression starts looking a little less intimidating. The part inside the brackets becomes $-\left(\frac{6}{5}\right) \div \frac{3}{5}(1.3)$. We still have that $1.3$ to deal with. Let's convert that to a fraction as well $1.3 = \frac{13{10}$. So, the inner part is now $-\left(\frac{6}{5}\right) \div \frac{3}{5}\left(\frac{13}{10}\right)$ . Phew! This careful conversion is the foundation for evaluating the expression accurately.

Tackling the Innermost Operations

Now that we've got our expression prepped with fractions, let's get into the nitty-gritty of solving what's inside those brackets. We're looking at $-\left(\frac6}{5}\right) \div \frac{3}{5}\left(\frac{13}{10}\right)$. Remember, multiplication and division have the same priority, so we work from left to right. First up is the division $-\left(\frac{6{5}\right) \div \frac{3}{5}$. To divide by a fraction, we multiply by its reciprocal. So, this becomes $-\left(\frac{6}{5}\right) \times \frac{5}{3}$. See how the 5s cancel out? That leaves us with $-\frac{6}{3}$, which simplifies to $-2$. So, the expression inside the brackets now looks like $-2 \left(\frac{13}{10}\right)$.

Next, we handle the multiplication: $-2 \times \frac13}{10}$. This equals $-\frac{2 \times 13}{10} = -\frac{26}{10}$. We can simplify this fraction to $-\frac{13}{5}$. Now, remember the original expression had a negative sign outside the brackets. So, we have $-\left(-\frac{13}{5}\right)$. Double negatives, guys! That turns into a positive $+\frac{13{5}$. We're making serious progress towards finding the value of the expression!

Final Steps: Bringing It All Together

We've successfully navigated the tricky parts inside the brackets and simplified that whole section to $+\frac13}{5}$. Now, let's bring back the rest of the original expression $+\frac{13{5} - \frac{5}{6}$. We're almost there! To subtract these fractions, we need a common denominator. The least common multiple of 5 and 6 is 30. So, we'll convert both fractions:

135=13×65×6=7830 \frac{13}{5} = \frac{13 \times 6}{5 \times 6} = \frac{78}{30}

56=5×56×5=2530 \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}

Now, the subtraction is straightforward: $ \frac78}{30} - \frac{25}{30} = \frac{78 - 25}{30} = \frac{53}{30} $. And there you have it, folks! The value of the expression is $ \frac{53}{30} $. You can also express this as a mixed number $1 \frac{23{30}$. Awesome job sticking with it and evaluating the expression correctly!

Key Takeaways for Solving Math Expressions

So, what did we learn from wrestling with this beast of an expression? First and foremost, the order of operations (PEMDAS/BODMAS) is your absolute best friend. Always start with parentheses, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. Don't rush it! Breaking down the problem into smaller, manageable steps is crucial, especially when dealing with nested brackets and various number types like fractions and decimals.

Converting all numbers to a consistent format, usually fractions, can significantly simplify the process. Remember that $1 \frac{1}{5}$ is $ \frac{6}{5}$, $0.6$ is $ \frac{3}{5}$, and $1.3$ is $ \frac{13}{10}$. These conversions make operations like division and multiplication much cleaner. Also, don't shy away from simplifying fractions as you go – it keeps the numbers smaller and easier to manage. The trickiest part for many is handling negative signs, especially when they appear in succession. Remember that two negatives make a positive! Finally, when adding or subtracting fractions, always find a common denominator. This ensures you're comparing apples to apples, or in this case, thirty-seconds to thirty-seconds.

By applying these techniques consistently, you can confidently evaluate any complex mathematical expression. It’s all about patience, precision, and a solid understanding of the fundamental rules. Keep practicing, and you'll be a math expression ninja in no time! High fives all around for conquering this challenge and understanding how to find the value of the expression!