Calculating 8 × 2 7/12: A Step-by-Step Guide

by Alex Johnson 45 views

Let's dive into the world of fractions and multiplication! In this guide, we'll break down how to calculate 8 multiplied by the mixed number 2 7/12 and, most importantly, express the final answer as a simplified fraction. So, grab your thinking caps, and let's get started!

Understanding the Problem

The problem we're tackling is 8 × 2 7/12. This involves multiplying a whole number (8) by a mixed number (2 7/12). To solve this effectively, we need to convert the mixed number into an improper fraction. This conversion is crucial because it allows us to multiply fractions in a straightforward manner. Before we jump into the solution, let's understand why this conversion is so important.

Why Convert Mixed Numbers to Improper Fractions?

When dealing with mixed numbers, multiplying directly can become quite cumbersome. A mixed number combines a whole number and a fraction, which doesn't fit neatly into the standard fraction multiplication process. By converting to an improper fraction, we express the entire quantity as a single fraction, making the multiplication process much cleaner and easier to manage. This is because multiplying fractions involves simply multiplying the numerators (the top numbers) and the denominators (the bottom numbers). When you have a mixed number, you're essentially dealing with two separate parts – the whole number and the fraction – which complicates the direct multiplication. Converting to an improper fraction unifies these parts into a single fractional representation, streamlining the calculation.

So, let's proceed with the first step: converting our mixed number into an improper fraction.

Step 1: Converting the Mixed Number to an Improper Fraction

The mixed number we need to convert is 2 7/12. To convert a mixed number to an improper fraction, we follow a simple process:

  1. Multiply the whole number part (2) by the denominator of the fractional part (12).
  2. Add the result to the numerator of the fractional part (7).
  3. Place the sum over the original denominator (12).

Let's apply this to our mixed number:

(2 × 12) + 7 = 24 + 7 = 31

So, 2 7/12 is equal to 31/12 as an improper fraction. Now that we've successfully converted the mixed number into an improper fraction, the next step is to multiply this fraction by the whole number. This is where the process becomes much simpler, as we're now dealing with a standard fraction multiplication problem. Remember, the goal here is to express both numbers as fractions so that we can easily multiply them. We've already converted 2 7/12 into 31/12. Next, we'll address the whole number 8.

Step 2: Multiplying the Fractions

Now that we have converted the mixed number 2 7/12 into the improper fraction 31/12, we can proceed with the multiplication. Our problem now looks like this:

8 × 31/12

To multiply a whole number by a fraction, we can rewrite the whole number as a fraction by placing it over 1. So, 8 becomes 8/1. This gives us:

8/1 × 31/12

Now, we can multiply the numerators (top numbers) and the denominators (bottom numbers) separately:

Numerator: 8 × 31 = 248

Denominator: 1 × 12 = 12

So, the result of the multiplication is 248/12. This fraction represents the product of our original numbers, but it's not yet in its simplest form. Simplifying fractions is a crucial step in these types of calculations because it allows us to express the answer in the most concise and understandable way. A simplified fraction has the smallest possible numbers in the numerator and the denominator while maintaining the fraction's value. To simplify, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. In the next step, we'll focus on simplifying this fraction to its simplest form.

Step 3: Simplifying the Fraction

We have the fraction 248/12. To simplify it, we need to find the greatest common divisor (GCD) of 248 and 12. The GCD is the largest number that divides both numbers without leaving a remainder. Let's find the GCD:

The factors of 12 are: 1, 2, 3, 4, 6, and 12. The factors of 248 are: 1, 2, 4, 8, 31, 62, 124, and 248.

The greatest common divisor of 248 and 12 is 4.

Now, we divide both the numerator and the denominator by 4:

248 ÷ 4 = 62

12 ÷ 4 = 3

So, the simplified fraction is 62/3. However, this is an improper fraction (where the numerator is greater than the denominator). While improper fractions are perfectly valid, it's often preferable to express the final answer as a mixed number for better clarity, especially in contexts where the magnitude of the number is important. Converting an improper fraction back to a mixed number involves dividing the numerator by the denominator and expressing the result as a whole number and a remainder. Let's proceed with this conversion in the next step.

Step 4: Expressing as a Mixed Number (Optional, but Recommended)

Our simplified fraction is 62/3. To express it as a mixed number, we divide the numerator (62) by the denominator (3):

62 ÷ 3 = 20 with a remainder of 2.

This means that 3 goes into 62 twenty times, with 2 left over. So, we can write 62/3 as a mixed number:

20 2/3

This is the simplest form of our answer, expressed as a mixed number. The mixed number representation gives us a clear sense of the quantity: it's 20 whole units and 2/3 of another unit. This form is often preferred in practical applications where understanding the magnitude of the number is important. For instance, if you were measuring ingredients for a recipe, knowing you need 20 2/3 cups gives you a more intuitive sense than just saying 62/3 cups. While 62/3 is mathematically correct, 20 2/3 provides a more tangible understanding of the quantity.

Final Answer

Therefore, 8 × 2 7/12 = 62/3, which can also be expressed as the mixed number 20 2/3.

Key Takeaways

  • Converting mixed numbers to improper fractions simplifies multiplication.
  • Simplifying fractions involves finding the greatest common divisor (GCD).
  • Expressing improper fractions as mixed numbers can provide better clarity.

Understanding these steps is crucial for mastering fraction multiplication and simplifying your results. Remember, practice makes perfect! The more you work with fractions, the more comfortable and confident you'll become in handling these types of calculations.

For further learning and practice, you can explore resources like Khan Academy's fraction lessons.