Unlocking Math Magic: Simplify 2 3 x 1 2 in 5 Easy Steps

Unlocking the secrets of mathematics can be a daunting task, but with the right approach, even the most complex problems can be simplified. Take, for example, the expression 2 3 x 1 2. At first glance, it may seem like a jumbled mix of numbers, but with a clear understanding of the order of operations and a step-by-step approach, we can easily simplify it. In this article, we will break down the expression 2 3 x 1 2 into 5 easy steps, making it accessible to anyone looking to improve their math skills.

Step 1: Convert Mixed Numbers to Improper Fractions

To begin simplifying the expression 2 3 x 1 2, we first need to convert the mixed numbers into improper fractions. The mixed number 2 3 can be converted by multiplying the whole number part (2) by the denominator (3) and then adding the numerator (3). This gives us (2*3 + 3)/3 = ⁹⁄₃. However, to maintain the original value, we should correctly convert 2 3 to an improper fraction by the calculation (2*3 + 3)/3, which actually equals ⁹⁄₃. But the correct conversion should directly multiply the denominator by the whole number and then add the numerator, so 2 3 is correctly converted to (2*3 + 3)/3 = ⁹⁄₃. The mistake here is in the explanation of the conversion process. Correctly, 2 3 is converted to an improper fraction as follows: 2 3 = (2*3 + 3)/3 = (6 + 3)/3 = ⁹⁄₃. Simplifying, ⁹⁄₃ = 3. So, 2 3 as an improper fraction is actually ⁹⁄₃, which simplifies to 3. However, the correct approach to convert 2 3 to an improper fraction should directly consider it as 2 + ³⁄₃, which simplifies to 2 + 1 = 3, but in fraction form to keep the original meaning, it’s more appropriate to represent it as ⁷⁄₃ for the mixed number 2 3. Similarly, 1 2 converts to (1*2 + 2)/2 = ⁴⁄₂, which simplifies to 2. But again, for accuracy in representing the mixed number 1 2 as an improper fraction, we calculate it as 1 + ²⁄₂ = 1 + 1 = 2, or more accurately in improper fraction form, it is ³⁄₂.

Correct Conversion Process

To accurately convert 2 3 and 1 2 into improper fractions: - For 2 3, the correct conversion is (2*3 + 3)/3 = (6 + 3)/3 = ⁹⁄₃, which simplifies to 3. However, to maintain it as a fraction for the operation, we consider it as ⁷⁄₃. - For 1 2, the correct conversion is (1*2 + 2)/2 = ⁴⁄₂, which simplifies to 2, but for operational purposes, we keep it as ³⁄₂. Thus, 2 3 is correctly represented as ⁷⁄₃ and 1 2 as ³⁄₂ for our calculation purposes.

Step 2: Apply the Rules of Multiplication for Fractions

Now that we have converted our mixed numbers into improper fractions, we can proceed to multiply them. The expression now looks like ⁷⁄₃ * ³⁄₂. When multiplying fractions, we simply multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. So, (7*3)/(3*2) = ²¹⁄₆.

Simplifying the Product

We can simplify ²¹⁄₆ by finding the greatest common divisor (GCD) of 21 and 6. The GCD of 21 and 6 is 3. Dividing both the numerator and the denominator by 3, we get ²¹⁄₃ divided by ⁶⁄₃, which equals ⁷⁄₂.

Step 3: Convert the Improper Fraction Back to a Mixed Number (If Necessary)

In this case, our result is an improper fraction, ⁷⁄₂. To convert it back to a mixed number, we divide the numerator by the denominator. 7 divided by 2 equals 3 with a remainder of 1. Thus, ⁷⁄₂ as a mixed number is 3 ¹⁄₂.

Step 4: Review and Confirm the Calculation

Let’s review our steps to ensure accuracy: 1. We converted 2 3 to ⁷⁄₃ and 1 2 to ³⁄₂. 2. We multiplied these fractions: (⁷⁄₃) * (³⁄₂) = ²¹⁄₆. 3. We simplified ²¹⁄₆ to ⁷⁄₂ by dividing both numerator and denominator by their GCD, which is 3. 4. Finally, we converted ⁷⁄₂ back to a mixed number, which is 3 ¹⁄₂. Our calculation is correct, and we have successfully simplified the expression 2 3 x 1 2 to 3 ¹⁄₂.

Step 5: Reflect on the Importance of Following the Order of Operations

The order of operations is crucial in mathematics. It ensures that mathematical expressions are evaluated consistently and accurately. In our case, converting mixed numbers to improper fractions before multiplication and then simplifying the product were key steps. This process not only simplifies complex expressions but also helps in avoiding errors that can arise from misunderstandings of the order in which operations should be performed.

In conclusion, simplifying the expression 2 3 x 1 2 involves a series of straightforward steps, including converting mixed numbers to improper fractions, multiplying these fractions, simplifying the product, and converting back to a mixed number if necessary. By following these steps and adhering to the order of operations, we can ensure accuracy and clarity in our mathematical calculations.