Multiplying Improper Fractions

Oct 13, 2025 by wbrum.org 34 views

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Understanding the Problem

When multiplying mixed numbers, it's essential to first convert them into improper fractions. This process involves multiplying the whole number part by the denominator and then adding the numerator to get the new numerator. The denominator remains the same.

Converting Mixed Numbers to Improper Fractions

To convert the mixed numbers $6 \frac{3}{5}$ and $2 \frac{1}{3}$ into improper fractions, we follow these steps:

For $6 \frac{3}{5}$ $6 \frac{3}{5}$ :
- Multiply the whole number part (6) by the denominator (5): $6 \times 5 = 30$
- Add the numerator (3) to the result: $30 + 3 = 33$
- The new numerator is 33, and the denominator remains 5. So, the improper fraction is $\frac{33}{5}$
For $2 \frac{1}{3}$ $2 \frac{1}{3}$ :
- Multiply the whole number part (2) by the denominator (3): $2 \times 3 = 6$
- Add the numerator (1) to the result: $6 + 1 = 7$
- The new numerator is 7, and the denominator remains 3. So, the improper fraction is $\frac{7}{3}$

Multiplying Improper Fractions

Now that we have the mixed numbers converted into improper fractions, we can multiply them together.

Multiplying the Numerators and Denominators

To multiply two improper fractions, we multiply the numerators together and the denominators together.

Multiply the numerators: $33 \times 7 = 231$
Multiply the denominators: $5 \times 3 = 15$

Writing the Product as a Fraction

The product of the two improper fractions is $\frac{231}{15}$ . However, this fraction can be simplified further.

Simplifying the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

The GCD of 231 and 15 is 3
Divide both the numerator and the denominator by the GCD: $\frac{231 \div 3}{15 \div 3} = \frac{77}{5}$

Writing the Product as a Mixed Number

The product of the two mixed numbers can be written as a mixed number by converting the improper fraction back into a mixed number.

Converting the Improper Fraction to a Mixed Number

To convert the improper fraction $\frac{77}{5}$ into a mixed number, we follow these steps:

Divide the numerator (77) by the denominator (5): $77 \div 5 = 15$ with a remainder of 2
The whole number part is 15, and the remainder is 2. So, the mixed number is $15 \frac{2}{5}$

Conclusion

In conclusion, the product of the two mixed numbers $6 \frac{3}{5}$ and $2 \frac{1}{3}$ is $15 \frac{2}{5}$ . This result can be obtained by first converting the mixed numbers into improper fractions, multiplying them together, and then simplifying the resulting fraction.

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Frequently Asked Questions

Q: What is the product of $6 \frac{3}{5}$ and $2 \frac{1}{3}$ ?

A: The product of $6 \frac{3}{5}$ and $2 \frac{1}{3}$ is $15 \frac{2}{5}$ .

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator. The new numerator is the result, and the denominator remains the same.

Q: How do I multiply two improper fractions?

A: To multiply two improper fractions, multiply the numerators together and the denominators together.

Q: How do I simplify a fraction?

A: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator, and divide both numbers by the GCD.

Q: What is the greatest common divisor (GCD) of 231 and 15?

A: The GCD of 231 and 15 is 3.

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, divide the numerator by the denominator and write the result as a whole number and a remainder. The whole number part is the quotient, and the remainder is the new numerator.

Q: What is the product of $6 \frac{3}{5}$ and $2 \frac{1}{3}$ in decimal form?

A: To convert the mixed number $15 \frac{2}{5}$ to a decimal, divide the numerator (2) by the denominator (5) and add the result to the whole number part (15). The result is 15.4.

Q: Can I use a calculator to multiply mixed numbers?

A: Yes, you can use a calculator to multiply mixed numbers. However, it's essential to first convert the mixed numbers to improper fractions or decimals before performing the multiplication.

Q: What are some real-world applications of multiplying mixed numbers?

A: Multiplying mixed numbers has various real-world applications, such as calculating the area of a rectangle with mixed number dimensions, finding the volume of a rectangular prism with mixed number dimensions, and determining the cost of a product with mixed number prices.

Q: Can I multiply mixed numbers with different denominators?

A: Yes, you can multiply mixed numbers with different denominators. However, it's essential to first convert the mixed numbers to improper fractions with the same denominator before performing the multiplication.

Q: How do I divide mixed numbers?

A: To divide mixed numbers, convert them to improper fractions, invert the second fraction, and multiply the fractions together.

Additional Resources

For more information on multiplying mixed numbers, visit the Khan Academy website.
For a step-by-step guide on converting mixed numbers to improper fractions, visit the Mathway website.
For a list of real-world applications of multiplying mixed numbers, visit the Math Open Reference website.

Conclusion

In conclusion, multiplying mixed numbers involves converting them to improper fractions, multiplying the fractions together, and simplifying the resulting fraction. This process has various real-world applications and can be performed using a calculator or by hand.