Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(2 \frac{2}{5} \times 4 \frac{1}{6}=\) ______

Short answer

The product is 10.

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, transform the mixed numbers into improper fractions. For \(2 \frac{2}{5}\), multiply the whole number 2 by the denominator 5, then add the numerator 2, resulting in an improper fraction of \(\frac{12}{5}\). Similarly, for \(4 \frac{1}{6}\), multiply the whole number 4 by the denominator 6, then add the numerator 1, yielding \(\frac{25}{6}\).

Multiply the Fractions

Next, multiply the improper fractions obtained: \(\frac{12}{5} \times \frac{25}{6}\). Multiply the numerators (12 and 25) to get 300, and the denominators (5 and 6) to get 30. This results in the fraction \(\frac{300}{30}\).

Reduce to Lowest Terms

Finally, reduce the fraction \(\frac{300}{30}\) to its simplest form. Divide both the numerator and the denominator by their greatest common divisor, which is 30: \(\frac{300 \div 30}{30 \div 30} = \frac{10}{1}\). Thus, the fraction simplifies to 10.

Key concepts

Improper Fractions

Have you ever encountered fractions where the numerator is larger than the denominator? If so, you've likely seen improper fractions. These fractions are essential in many mathematical operations, especially when dealing with mixed numbers.

To convert a mixed number to an improper fraction:

• Multiply the whole number part by the fraction's denominator.
• Add the numerator of the fraction to the result from the previous step.
• The result becomes the new numerator, while the denominator remains the same.

For example, converting the mixed number \(2 \frac{2}{5}\) involves multiplying 2 (the whole number) by 5, which gives 10. Add the numerator 2 to get a new numerator of 12. So, \(2 \frac{2}{5}\) becomes \(\frac{12}{5}\). Converting improper fractions is a straightforward process that simplifies calculations in the multiplication of fractions.

Mixed Numbers

Mixed numbers are numbers that combine whole numbers and fractions. They often appear in real-life situations, like recipes or measurements, because they allow for precise communication of quantities. However, they're often not so convenient for mathematical operations such as multiplication or division.

When you have mixed numbers like \(4 \frac{1}{6}\), the most practical step is to first convert them into improper fractions before any operations. Follow these steps:

• Multiply the whole number by the fraction's denominator.
• Add the original fraction's numerator to this product to get the new numerator.
• Write the result as an improper fraction with the same denominator.

Hence, for \(4 \frac{1}{6}\), multiply 4 by 6 to get 24, then add 1 to achieve a new numerator of 25, resulting in \(\frac{25}{6}\). Understanding how to handle mixed numbers this way streamlines the problem-solving process.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means the numerator and the denominator share no common factors other than 1. It makes fractions easier to understand and use, especially when performing operations like multiplication or division.

The simplification process follows a few key steps:

• Identify the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator of the fraction by the GCD.
• The result is the fraction in its simplest form.

Take the fraction \(\frac{300}{30}\) from our exercise, for instance. The GCD of 300 and 30 is 30. By dividing both terms by this number, you reduce \(\frac{300}{30}\) to \(\frac{10}{1}\), or simply 10. Simplifying is crucial for getting the most accurate and understandable results.