Simplifying Algebraic Expressions: A Step-by-Step Guide

Jan 17, 2026 by Editorial Team 56 views

Hey there, math enthusiasts! Today, we're diving into the world of algebraic expressions, specifically how to simplify the expression: $\frac{18k + 36}{60}$ . Don't worry, it might seem a bit daunting at first, but trust me, it's like a puzzle, and we're going to break it down step-by-step to make it super easy to understand. Our goal is to find the most simplified version of this expression from the given options: A) $\frac{1}{k-5}$ , B) $\frac{3(k+2)}{10}$ , C) $\frac{3k}{k-3}$ , and D) $\frac{5k-7}{6k^2}$ . Let's get started and see which one fits the bill. Remember, simplifying expressions is a fundamental skill in algebra, and it's super important for more complex problems down the line. We will go through the simplification process in a very detailed manner so everyone can understand it. Let’s get our hands dirty, shall we?

Step-by-Step Simplification

Alright, guys, let's roll up our sleeves and simplify the expression $\frac{18k + 36}{60}$ . The key to simplifying such expressions is to find the greatest common factor (GCF) of the terms in the numerator and the denominator. This process involves identifying the largest number that divides evenly into all terms. Finding the GCF helps us reduce the expression to its simplest form, making it easier to work with. Here's how we'll do it:

Factor the Numerator: Look at the numerator, $18k + 36$ . We need to find the GCF of 18k and 36. The GCF of 18 and 36 is 18. So, we can factor out 18 from both terms. This gives us $18(k + 2)$ . It's like we're reversing the distributive property! So the expression becomes $\frac{18(k + 2)}{60}$ .
Factor the Denominator: The denominator is 60. We can find the prime factorization of 60 to help us. 60 can be factored into $2 \times 2 \times 3 \times 5$ . However, for our purposes here, we just need to see if there is any common factor between the numerator and denominator. We see that the numerator has a 18 and the denominator has a 60, both are divisible by 6, but also by 2, and 3. The common factor is 6. This step is about identifying any factors the numerator and denominator share so we can simplify the fraction further.
Simplify the Fraction: Now that we have the expression as $\frac{18(k + 2)}{60}$ , we can simplify. Both 18 and 60 are divisible by a common factor, specifically 6. Let’s divide both the numerator and the denominator by 6. Dividing 18 by 6 gives us 3, and dividing 60 by 6 gives us 10. This step ensures that we reduce the fraction to its lowest terms. So, our simplified expression is $\frac{3(k + 2)}{10}$ . Isn't that neat?

Therefore, after simplifying, we have found our simplified expression, which we are now going to verify by comparing it with the options.

Matching with the Options

Now that we've simplified the expression to $\frac{3(k + 2)}{10}$ , let's see which of the provided options matches our simplified result. Remember, we are looking for an expression that is equivalent to $\frac{3(k + 2)}{10}$ . Here's how we can match it:

Option A: $\frac{1}{k-5}$ This doesn't look like our simplified expression at all. The numerator is 1, and the denominator involves a variable k. We can immediately eliminate this option as a possibility. Our simplified expression has a numerator of $3(k+2)$ and a denominator of 10, which doesn't match this option.
Option B: $\frac{3(k+2)}{10}$ Hey, this looks familiar! This is exactly what we simplified our original expression to. The numerator is $3(k + 2)$ , and the denominator is 10. This is a perfect match, and it's likely our correct answer. Always remember to check all the possible choices.
Option C: $\frac{3k}{k-3}$ This expression looks different from our simplified form. The numerator is $3k$ , and the denominator is $(k-3)$ . These terms don't align with our simplified result, so we can discard this option.
Option D: $\frac{5k-7}{6k^2}$ This expression is also quite different. The numerator is $(5k - 7)$ , and the denominator is $6k^2$ . This doesn't resemble our simplified expression in any way, so this is not the answer. We have confirmed the correct answer by comparing it to all the options. Isn't that fun?

Therefore, by matching our simplified expression with the options, we can confidently say that Option B, $\frac{3(k+2)}{10}$ , is the correct answer. We have successfully simplified the expression and identified the equivalent option.

Why This Matters

Simplifying algebraic expressions isn't just a math exercise; it's a fundamental skill that builds the foundation for more advanced topics. Knowing how to simplify expressions helps you in several ways:

Solving Equations: Simplified expressions make solving equations much easier. When you have a complex equation, simplifying the terms first can help you isolate the variable and find the solution more efficiently.
Understanding Relationships: Simplification helps you see the underlying relationships between variables and constants. This can give you insights into the problem you're trying to solve.
Reducing Errors: Working with simplified expressions reduces the chance of making computational errors. Fewer terms mean fewer opportunities to make mistakes.
Preparing for Future Math: The skills you gain from simplifying expressions will be vital as you move on to more advanced math topics such as calculus, trigonometry, and linear algebra. It's like learning the alphabet before reading a novel; you need the basics to understand the more complicated stuff.

So, as you can see, simplifying expressions is more than just a step in a math problem; it's a building block for future success in mathematics. Keep practicing, and you'll become a pro in no time! Remember, guys, practice makes perfect. Keep up the good work and don't give up! We are always here to guide you.

Conclusion

Alright, folks, we've successfully simplified the expression $\frac{18k + 36}{60}$ and found that the correct answer is $\frac{3(k + 2)}{10}$ . We did this by first factoring the numerator and then simplifying the fraction by dividing by the greatest common factor. This process is applicable to many similar problems, so remember the steps. Remember the importance of simplifying and how it plays a role in future math concepts. Keep practicing, and you'll become a pro in no time! We've covered the steps in detail to make sure everyone understands the concepts. If you have any more questions, feel free to ask! Happy simplifying!