Simplify: (1/3 + 2/5) * 2/3

Welcome, math enthusiasts! Today, we're diving into a fun arithmetic problem that will test your skills in fraction manipulation. We're going to evaluate the expression $\left(\frac{1}{3}+\frac{2}{5}\right) \cdot \frac{2}{3}$ and present our answer in the simplest possible form. This isn't just about getting the right number; it's about understanding the process of working with fractions, which is a fundamental building block in mathematics. Whether you're a student tackling homework, a curious learner, or just someone who enjoys a good mental workout, this guide will walk you through each step with clarity and ease. We'll break down the operations, explain the reasoning behind each move, and ensure you feel confident in your ability to simplify complex fraction expressions. So, grab your metaphorical pencil and paper, and let's embark on this mathematical journey together!

Understanding the Order of Operations

Before we even touch the numbers, it's crucial to remember the order of operations, often remembered by the acronym PEMDAS or BODMAS. This governs the sequence in which we perform calculations to ensure a consistent and correct result. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our expression, $\left(\frac{1}{3}+\frac{2}{5}\right) \cdot \frac{2}{3}$, the parentheses are our first priority. This means we must add the two fractions inside the parentheses before we multiply by $\frac{2}{3}$. Ignoring this rule would lead to an entirely different, and incorrect, answer. Think of the parentheses as a way of grouping operations, signaling that this specific calculation needs to be completed first. This principle is vital not just for this problem, but for virtually all mathematical expressions you'll encounter. Master the order of operations, and you've unlocked a key to solving a vast array of problems accurately. It’s the bedrock upon which complex calculations are built, ensuring that everyone, everywhere, arrives at the same correct solution when presented with the same mathematical puzzle. So, always look for those parentheses first – they're your green light to proceed!

Step 1: Adding Fractions Inside the Parentheses

Our first major step is to tackle the addition within the parentheses: $\frac{1}{3}+\frac{2}{5}$. To add fractions, they must have a common denominator. This means the bottom number (the denominator) of both fractions needs to be the same. We can't simply add the numerators (top numbers) and the denominators together, as that would lead to an incorrect result. The easiest way to find a common denominator is often to multiply the denominators of the two fractions together. In this case, the denominators are 3 and 5. So, $3 \times 5 = 15$. This 15 will be our common denominator. Now, we need to adjust each fraction so that it has a denominator of 15, without changing its value. To change $\frac{1}{3}$ to an equivalent fraction with a denominator of 15, we ask ourselves: 'What do we multiply 3 by to get 15?' The answer is 5. So, we multiply both the numerator and the denominator of $\frac{1}{3}$ by 5: $\frac{1 \times 5}{3 \times 5} = \frac{5}{15}$. Similarly, for $\frac{2}{5}$, we ask: 'What do we multiply 5 by to get 15?' The answer is 3. So, we multiply both the numerator and the denominator of $\frac{2}{5}$ by 3: $\frac{2 \times 3}{5 \times 3} = \frac{6}{15}$. Now that both fractions have the same denominator, we can add their numerators: $\frac{5}{15} + \frac{6}{15} = \frac{5+6}{15} = \frac{11}{15}$. This is a critical step; getting this right ensures the rest of the calculation flows smoothly. The result of our operation inside the parentheses is $\frac{11}{15}$. Remember, finding a common denominator is like finding a shared language for your numbers, allowing them to communicate (be added or subtracted) effectively. It's a fundamental concept that opens the door to more complex fraction arithmetic.

Step 2: Multiplying the Result by the External Fraction

Now that we've successfully added the fractions inside the parentheses, our expression has been simplified to $\frac{11}{15} \cdot \frac{2}{3}$. The next step, according to our order of operations, is multiplication. To multiply fractions, the process is quite straightforward: you multiply the numerators together and multiply the denominators together. So, we take the numerator of the first fraction (11) and multiply it by the numerator of the second fraction (2): $11 \times 2 = 22$. Then, we take the denominator of the first fraction (15) and multiply it by the denominator of the second fraction (3): $15 \times 3 = 45$. Putting these together, our new fraction is $\frac{22}{45}$. This is the product of our multiplication. It's important to note that unlike addition and subtraction, multiplication of fractions does not require a common denominator. You simply multiply across. This makes multiplication a bit simpler in that regard. Before moving on, it's always a good habit to check if this fraction can be simplified. We look for a common factor (a number that divides evenly into both the numerator and the denominator) for 22 and 45. The factors of 22 are 1, 2, 11, and 22. The factors of 45 are 1, 3, 5, 9, 15, and 45. The only common factor they share is 1. When the only common factor is 1, the fraction is already in its simplest form. Therefore, $\frac{22}{45}$ is our answer at this stage.

Step 3: Ensuring the Answer is in Simplest Form

We've reached the final stage of our calculation, and we need to make sure our answer, $\frac{22}{45}$, is in its simplest form. A fraction is considered to be in its simplest form (or lowest terms) when the greatest common divisor (GCD) of its numerator and denominator is 1. This means there are no common factors between the top number and the bottom number other than 1. To check this, we list the factors of the numerator (22) and the denominator (45).

• Factors of 22: 1, 2, 11, 22.
• Factors of 45: 1, 3, 5, 9, 15, 45.

By comparing these lists, we can see that the only number that appears in both lists is 1. This confirms that the greatest common divisor of 22 and 45 is indeed 1. Therefore, the fraction $\frac{22}{45}$ cannot be simplified any further. It is already in its simplest form. Sometimes, after multiplying fractions, you might end up with a fraction that can be simplified. For instance, if we had gotten $\frac{20}{40}$, we would see that both 20 and 40 are divisible by 20, so we would divide both the numerator and denominator by 20 to get $\frac{1}{2}$. Always perform this simplification step at the end to ensure your final answer meets the requirement of being in the simplest form. It's like cleaning up your work – making sure everything is neat and tidy. This attention to detail ensures your mathematical answers are not only correct but also presented in the most concise and standard way possible.

Conclusion: The Final Simplified Answer

Through a systematic approach, following the order of operations, and applying the rules for fraction addition and multiplication, we have successfully evaluated the expression $\left(\frac{1}{3}+\frac{2}{5}\right) \cdot \frac{2}{3}$. We first addressed the parentheses, finding a common denominator to add $\frac{1}{3}$ and $\frac{2}{5}$, which resulted in $\frac{11}{15}$. Subsequently, we multiplied this sum by $\frac{2}{3}$, yielding $\frac{22}{45}$. Finally, we confirmed that $\frac{22}{45}$ is indeed in its simplest form, as 22 and 45 share no common factors other than 1. Therefore, the final answer, written in its simplest form, is $\frac{22}{45}$.

This problem highlights the importance of fundamental arithmetic skills and the consistent application of mathematical rules. Mastering these steps will undoubtedly help you tackle more complex algebraic and calculus problems in the future. Keep practicing, and don't hesitate to revisit these concepts whenever needed!

For further exploration and practice with fractions, you might find the resources at Khan Academy incredibly helpful. They offer a wide range of lessons and exercises on all aspects of mathematics, from basic arithmetic to advanced topics. You can visit them at https://www.khanacademy.org/math/arithmetic/arith-review-fractions.