Algebraic Expressions & Rate Problems Explained

What's up, math enthusiasts! Today, we're diving deep into two juicy problems that'll have your brain doing a happy dance. We're talking algebra and rates, the bread and butter of problem-solving. So grab your pencils, open your minds, and let's get this mathematical party started!

Problem 1: Cracking the Algebraic Code

Alright guys, let's kick things off with our first challenge: Which of the following is equivalent to the expression $4(x+5)+4 x+8$? This is where we flex those algebraic muscles and show this expression who's boss. We've got four potential champions: A. $4(2 x+7)$, B. $8(x+4)$, C. $5 x+17$, and D. $8 x+13$. Your mission, should you choose to accept it, is to find the one that’s a perfect match, a true equivalent. This isn't just about picking an answer; it's about understanding the process of simplification and equivalence. When two expressions are equivalent, it means they will always produce the same result, no matter what value you plug in for the variable, in this case, 'x'. Think of it like having two different outfits that look amazing on you – they're different, but they both achieve the same stylish effect. Our goal here is to simplify the given expression, $4(x+5)+4x+8$, into its most basic form and then compare that simplified version to the options provided. This involves using the distributive property and combining like terms. The distributive property is like a magic wand that lets us multiply a number outside parentheses by each term inside the parentheses. So, $4(x+5)$ becomes $4*x + 4*5$, which is $4x + 20$. Now, our expression looks like this: $4x + 20 + 4x + 8$. See how we've broken down the first part? The next step is combining 'like terms'. Like terms are those that have the same variable raised to the same power. In our expression, we have two 'x' terms ($4x$ and $4x$) and two constant terms ($20$ and $8$). When we combine the 'x' terms, $4x + 4x$, we get $8x$. And when we combine the constants, $20 + 8$, we get $28$. So, the simplified expression is $8x + 28$. Now, we need to see which of our options matches this. Let's break down each option:

• A. $4(2x+7)$: Using the distributive property here gives us $4 * 2x + 4 * 7$, which equals $8x + 28$. Boom! This looks like our winner already!
• B. $8(x+4)$: Distributing the 8 gives us $8 * x + 8 * 4$, which is $8x + 32$. Close, but not quite the same.
• C. $5x+17$: This expression is already simplified, but it clearly doesn't match $8x + 28$ because the coefficients of 'x' and the constant terms are different.
• D. $8x+13$: This expression also doesn't match $8x + 28$ because the constant terms are different.

So, by systematically simplifying the original expression and then simplifying (or evaluating) each option, we found that option A, $4(2x+7)$, is indeed equivalent to the original expression. It's all about patience and applying those fundamental algebraic rules, guys. You got this!

Problem 2: Decoding Rate and Time

Next up, we're shifting gears to a problem involving rates and time. It took Khalid 90 minutes to complete 40 tasks. Which of the following is an equivalent rate? This is super practical, like figuring out how fast you can binge-watch your favorite show or how long it takes to finish a huge project. We're given a rate: 40 tasks in 90 minutes. We need to find another rate that represents the exact same speed or efficiency. The options are: A. 10 tasks in 0.9 minutes, B. 1 task in 2.25 minutes, C. 20 tasks in 45 minutes, D. 4 tasks in 20 minutes. The key word here is equivalent rate. Just like equivalent expressions, equivalent rates mean the same thing, just possibly expressed in different units or with different numbers. To find an equivalent rate, we can simplify the given rate or scale it up or down. Let's calculate Khalid's original rate in tasks per minute. The rate is rac{40 ext{ tasks}}{90 ext{ minutes}}. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10. So, rac{40 ext{ tasks}}{90 ext{ minutes}} = rac{4 ext{ tasks}}{9 ext{ minutes}}. This means Khalid completes 4 tasks every 9 minutes. Now, let's see if any of our options match this simplified rate or can be scaled to match it. We can also express the rate as tasks per one minute by dividing 40 by 90: rac{40}{90} = rac{4}{9} ext{ tasks/minute}. As a decimal, this is approximately $0.444$ tasks per minute. Let's analyze each option:

• A. 10 tasks in 0.9 minutes: The rate here is rac{10 ext{ tasks}}{0.9 ext{ minutes}}. To compare this, let's first make the denominator a whole number by multiplying both top and bottom by 10: rac{10 ext{ tasks} imes 10}{0.9 ext{ minutes} imes 10} = rac{100 ext{ tasks}}{9 ext{ minutes}}. This is definitely not the same as rac{4 ext{ tasks}}{9 ext{ minutes}}. Or, we can calculate tasks per minute: rac{10}{0.9} ext{ tasks/minute} ext{ is } rac{100}{9} ext{ tasks/minute} ext{, which is about } 11.11 ext{ tasks/minute}. Way too fast!
• B. 1 task in 2.25 minutes: The rate is rac{1 ext{ task}}{2.25 ext{ minutes}}. Let's see how many tasks this is per 9 minutes. If 1 task takes 2.25 minutes, then in 9 minutes, Khalid would complete $9 ext{ minutes} / 2.25 ext{ minutes/task} = 4$ tasks. So, this rate is 4 tasks in 9 minutes. Bingo! This matches our simplified rate! Let's double-check by converting 2.25 minutes to a fraction: 2.25 = 2 rac{1}{4} = rac{9}{4} minutes. So the rate is rac{1 ext{ task}}{rac{9}{4} ext{ minutes}} = rac{4}{9} ext{ tasks/minute}. This is exactly our original simplified rate.
• C. 20 tasks in 45 minutes: The rate is rac{20 ext{ tasks}}{45 ext{ minutes}}. We can simplify this by dividing both numbers by 5: rac{20 ext{ tasks} egexdiv 5}{45 ext{ minutes} egexdiv 5} = rac{4 ext{ tasks}}{9 ext{ minutes}}. Look at that! This also matches our simplified rate. So, both B and C are equivalent. In a multiple-choice question, you'd typically pick the one provided as the correct answer. If this were a real test, we'd need to be careful, but mathematically, both B and C represent the same rate.
• D. 4 tasks in 20 minutes: The rate is rac{4 ext{ tasks}}{20 ext{ minutes}}. Simplifying this gives us rac{1 ext{ task}}{5 ext{ minutes}}, or $0.2$ tasks per minute. This is much slower than our original rate.

So, for this problem, both Option B and Option C are equivalent rates to Khalid's performance. It’s all about finding that common ground, that consistent pace. Whether you simplify the fraction or scale it up, the core rate remains the same. Keep practicing these, and you'll be a rate-solving wizard in no time!

Wrapping It Up

There you have it, folks! Two distinct problems, solved with clarity and a dash of fun. We navigated the world of algebraic expressions, simplifying and finding equivalents, and then tackled rate problems, ensuring we understood what 'equivalent' truly means in different contexts. Remember, math isn't just about numbers; it's about logic, patterns, and problem-solving skills that are useful everywhere. Keep exploring, keep questioning, and keep those brains sharp! Catch you in the next one!