Calculate The Product:$ 10 \frac{2}{3} \times 1 \frac{3}{8} }$Select One Of The Following Options A. ${ 15 \frac{3 {4} }$B. ${ 15 \frac{1}{2} }$C. ${ 14 \frac{2}{3} }$D. ${ 16 \frac{1}{2} }$
Understanding Mixed Numbers
Before we dive into the problem, let's quickly review what mixed numbers are. A mixed number is a combination of a whole number and a fraction. It's written in the form of a b/c, where a is the whole number and b/c is the fraction. For example, 3 1/2 is a mixed number that represents the sum of 3 and 1/2.
The Problem: Multiplying Mixed Numbers
Now, let's tackle the problem at hand: calculating the product of two mixed numbers, 10 2/3 and 1 3/8. To do this, we'll need to follow a few steps.
Step 1: Convert Mixed Numbers to Improper Fractions
To multiply mixed numbers, it's often easier to convert them to improper fractions first. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. Then, we write the result as a fraction with the original denominator.
For 10 2/3, we can convert it to an improper fraction as follows:
10 2/3 = (10 × 3) + 2 = 30 + 2 = 32/3
Similarly, for 1 3/8, we can convert it to an improper fraction as follows:
1 3/8 = (1 × 8) + 3 = 8 + 3 = 11/8
Step 2: Multiply the Improper Fractions
Now that we have both mixed numbers converted to improper fractions, we can multiply them together. To multiply fractions, we simply multiply the numerators and multiply the denominators.
32/3 × 11/8 = (32 × 11) / (3 × 8)
Step 3: Simplify the Result
To simplify the result, we can divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 352 and 24 is 8.
352 ÷ 8 = 44
24 ÷ 8 = 3
So, the simplified result is:
44/3
Step 4: Convert the Result Back to a Mixed Number
Finally, we can convert the improper fraction back to a mixed number. To do this, we divide the numerator by the denominator and write the result as a mixed number.
44 ÷ 3 = 14 with a remainder of 2
So, the result is:
14 2/3
The Answer:
Now that we've walked through the steps, let's look at the answer choices and see which one matches our result.
A. 15 3/4
B. 15 1/2
C. 14 2/3
D. 16 1/2
The correct answer is:
C. 14 2/3
Conclusion
Q: What is the difference between a mixed number and an improper fraction?
A: A mixed number is a combination of a whole number and a fraction, written in the form of a b/c, where a is the whole number and b/c is the fraction. An improper fraction, on the other hand, is a fraction where the numerator is greater than or equal to the denominator.
Q: How do I convert a mixed number to an improper fraction?
A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add the numerator. Then, you write the result as a fraction with the original denominator.
For example, to convert 10 2/3 to an improper fraction, you would:
10 2/3 = (10 × 3) + 2 = 30 + 2 = 32/3
Q: How do I multiply mixed numbers?
A: To multiply mixed numbers, you first convert them to improper fractions, then multiply the fractions together, and finally simplify the result.
Q: What is the greatest common divisor (GCD)?
A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. In the context of multiplying mixed numbers, the GCD is used to simplify the result.
Q: How do I simplify a fraction?
A: To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).
Q: Can I multiply mixed numbers with different denominators?
A: Yes, you can multiply mixed numbers with different denominators. To do this, you first convert each mixed number to an improper fraction, then multiply the fractions together, and finally simplify the result.
Q: What if I get a negative result when multiplying mixed numbers?
A: If you get a negative result when multiplying mixed numbers, it means that the product is negative. This is because the product of two negative numbers is positive, and the product of a negative number and a positive number is negative.
Q: Can I use a calculator to multiply mixed numbers?
A: Yes, you can use a calculator to multiply mixed numbers. However, it's often more helpful to understand the steps involved in multiplying mixed numbers, as this can help you to check your work and understand the underlying math.
Q: Are there any shortcuts for multiplying mixed numbers?
A: While there are no shortcuts for multiplying mixed numbers, you can use the following tips to make the process easier:
- Use a calculator to convert mixed numbers to improper fractions.
- Use a calculator to multiply the fractions together.
- Use a calculator to simplify the result.
By following these tips, you can make the process of multiplying mixed numbers easier and more efficient.