Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{3}{10} \times \frac{4}{12}=\) ______

Short answer

The answer is \( \frac{1}{10} \).

Step-by-step solution

Convert Fractions to Lowest Terms

First, simplify each fraction by finding the greatest common divisor (GCD) and dividing the numerator and the denominator by this number. For \( \frac{3}{10} \), the GCD is 1, so it remains the same: \( \frac{3}{10} \).For \( \frac{4}{12} \), the GCD is 4. Divide both numerator and the denominator by 4: \( \frac{4}{12} = \frac{1}{3} \).

Multiply the Fractions

To find the product of two fractions, multiply the numerators together and the denominators together. Multiply the simplified fractions: \( \frac{3}{10} \times \frac{1}{3} = \frac{3 \times 1}{10 \times 3} = \frac{3}{30} \).

Simplify the Result

Simplify \( \frac{3}{30} \) by finding the GCD of 3 and 30, which is 3. Divide both the numerator and the denominator by 3: \( \frac{3}{30} = \frac{1}{10} \).

Key concepts

Reducing Fractions

Reducing fractions means making a fraction simpler or more digestible to work with. A fraction is reduced when both its numerator (the top number) and its denominator (the bottom number) are divided by the largest number that can evenly divide both of them. This largest number is called the Greatest Common Divisor (GCD). By reducing fractions, we make calculations easier and findings faster to understand.

• Example: The fraction \( \frac{4}{8} \) can be reduced by dividing both numbers by their GCD, 4, resulting in \( \frac{1}{2} \).
• Remember: Not all fractions need reducing if their numerator and denominator do not share any divisors other than 1.

Greatest Common Divisor

The Greatest Common Divisor, or GCD, is crucial in fraction operations, especially in reducing and simplifying fractions. The GCD is the largest number that can divide two or more integers without leaving a remainder. Finding the GCD allows you to simplify fractions effectively.

• Determine the GCD by listing the common divisors of both the numerator and denominator.
• For example, to find the GCD of 12 and 18, list divisors: 12 (1, 2, 3, 4, 6, 12) and 18 (1, 2, 3, 6, 9, 18). The greatest common one is 6.

The GCD helps maintain accuracy, especially when developing fractions for real-life applications or further mathematical processes.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. After multiplying fractions, always check if the resulting fraction can be simplified.

• Following multiplication, look for any common factors between the resulting fraction's numerator and denominator.
• Example: After multiplying fractions and obtaining \( \frac{6}{18} \), divide both by their GCD, 6, to get \( \frac{1}{3} \).
• Also, ensure you carry out operations on both the numerator and the denominator to maintain the value of the fraction.

Simplified fractions are cleaner, easier to read, and more mathematically sound for further computations or comparisons.

Mixed Numbers

Mixed numbers are an interesting part of fraction multiplication. A mixed number combines a whole number and a fraction. Handling these requires converting a mixed number into an improper fraction before multiplying.

• An improper fraction is where the numerator is larger than the denominator.
• For instance, the mixed number 1\( \frac{1}{2} \) becomes the improper fraction \( \frac{3}{2} \) by multiplying the whole number by the denominator and adding the numerator.
• Once converted, multiply as usual and, if needed, convert back to a mixed number.

Working with mixed numbers in this way ensures accurate results and further simplifies back to meaningful results when necessary.