Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{10}{25} \times \frac{5}{3}=\) ______

Short answer

The simplified product is \( \frac{2}{3} \).

Step-by-step solution

Convert Fractions to Lowest Terms

First, simplify each fraction to its lowest terms. To simplify \( \frac{10}{25} \), divide the numerator and denominator by their greatest common divisor (GCD), which is 5:\[ \frac{10}{25} = \frac{10 \div 5}{25 \div 5} = \frac{2}{5} \] The fraction \( \frac{5}{3} \) is already in its simplest form.

Multiply the Fractions

Now multiply the simplified fractions:\[ \frac{2}{5} \times \frac{5}{3} \]Multiply the numerators together: \( 2 \times 5 = 10 \) Multiply the denominators together: \( 5 \times 3 = 15 \)So the product is: \[ \frac{10}{15} \]

Simplify the Product

Simplify the resulting fraction by dividing both the numerator and the denominator by their GCD, which is 5:\[ \frac{10}{15} = \frac{10 \div 5}{15 \div 5} = \frac{2}{3} \]

Key concepts

Simplifying Fractions

Understanding how to simplify fractions is an essential part of working with them, especially in multiplication. Simplification involves reducing a fraction to its simplest form, where the numerator and denominator are as small as possible while still maintaining the same value of the fraction. For example, when simplifying the fraction \( \frac{10}{25} \), we identify the greatest common divisor (GCD) of 10 and 25, which is 5. By dividing both the numerator and the denominator by this GCD, we get \( \frac{2}{5} \). A fraction is considered fully simplified when no whole number except 1 can divide both the numerator and the denominator. When simplifications are applied correctly, the process of working with fractions becomes much easier.

Greatest Common Divisor

The greatest common divisor (GCD) is crucial in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD helps in reducing fractions to their lowest terms. For instance, in the fraction \( \frac{10}{25} \), the GCD is 5, as it is the highest number that evenly divides both 10 and 25. To find the GCD, you can list the factors of each number and identify the largest one common to both lists. Alternatively, the Euclidean algorithm is an efficient method for larger numbers. Using the GCD is a straightforward way to ensure fractions are simplified properly during mathematical operations.

Mixed Numbers

Mixed numbers are numbers that consist of an integer and a fraction, for example, \(2 \frac{1}{3}\). When it comes to multiplication, it's often necessary to convert mixed numbers into improper fractions first. This means expressing the whole number and fraction as one combined fraction. For instance, \(2 \frac{1}{3}\) can be converted by multiplying the whole number by the denominator (3), adding the numerator (1), and placing the result over the original denominator: \(\frac{7}{3}\). This conversion to improper fractions facilitates direct multiplication with other fractions, streamlining the process.

Numerator and Denominator

The terms numerator and denominator are fundamental when dealing with fractions. The numerator is the top part of the fraction, representing how many parts out of the whole are considered. The denominator, on the other hand, is the bottom part, indicating the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{2}{5} \), 2 is the numerator, showing two parts are taken, and 5 is the denominator, showing the whole is divided into five parts. When multiplying fractions, the numerators are multiplied together and the denominators are multiplied together, like in the operation \( \frac{2}{5} \times \frac{5}{3} = \frac{10}{15} \). Correct understanding of these components is essential when simplifying or performing operations with fractions.

Reducing to Lowest Terms

Reducing fractions to their lowest terms makes them easier to understand and work with. This process involves using the greatest common divisor to simplify the fraction as much as possible, ensuring there are no common divisors left between the numerator and denominator except 1. For example, the result \( \frac{10}{15} \) can be further simplified by dividing both the numerator and the denominator by their GCD, which is 5, resulting in \( \frac{2}{3} \). This not only simplifies calculations but also provides a clearer representation of the fraction. Regular practice of reducing fractions to their lowest terms will enhance fluency in fraction arithmetic.