Question

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

Short answer

\(\frac{5}{8}\)

Step-by-step solution

Multiply the Numerators

To multiply fractions, multiply the numerators (the top numbers) together first. Here, we have the numerators 5 and 2. So, we calculate \(5 \times 2 = 10\).

Multiply the Denominators

Next, multiply the denominators (the bottom numbers) together. Here, the denominators are 4 and 4. So, we calculate \(4 \times 4 = 16\).

Form the New Fraction

Combine the results from Step 1 and Step 2 to form a new fraction. This gives us \(\frac{10}{16}\).

Simplify the Fraction

To simplify \(\frac{10}{16}\), find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 10 and 16 is 2. Divide both the numerator and the denominator by their GCD: \(\frac{10 \div 2}{16 \div 2} = \frac{5}{8}\).

Key concepts

Simplifying Fractions

Simplifying fractions is an essential skill in math that makes working with them much easier. When you simplify a fraction, you reduce it to its simplest form, ensuring the numerator and the denominator are as small as possible while keeping the same value.
This process involves dividing both the numerator and the denominator by their greatest common divisor (GCD), a concept we'll explore further in a moment.

• If a fraction is already in its simplest form, it cannot be simplified any further.
• Simplifying helps to quickly evaluate and compare fractions in calculations.

By simplifying, you make the fraction less cumbersome and more intuitive to work with, especially in further calculations like addition, subtraction, or comparing sizes.

Numerator

The numerator is the number at the top part of a fraction. It represents how many parts we have out of the whole. For example, in the fraction \( \frac{5}{4} \), the numerator is 5, which tells you that you have 5 parts.

When multiplying fractions, the numerators from each fraction must be multiplied together to form the new numerator of the product fraction.

• In our exercise, the numerators 5 and 2 are multiplied to form 10.

Remember, changing the numerator will directly affect the size of the fraction, indicating how large or small your fraction is, compared to one whole unit.

Denominator

The denominator is the bottom part of a fraction, showing the total number of equal parts something is divided into. It sets the "whole" context for the fraction. For instance, in \( \frac{5}{4} \), the denominator is 4, indicating that a whole is divided into 4 parts.

When multiplying fractions, you simply multiply the denominators together to find the denominator of the resulting fraction.

• In our worked-out solution, 4 and 4 are multiplied to give 16.
• This new denominator shows how the whole has been reshaped by the multiplication process.

Understanding how denominators work is crucial for comparing fractions and for any operation involving fractions.

Greatest Common Divisor

The greatest common divisor (GCD) is vital for simplifying fractions. It is the largest number that can exactly divide both the numerator and denominator without leaving a remainder. Finding the GCD allows you to reduce a fraction to its simplest form.

To determine the GCD, you identify all the factors of the numerator and denominator and choose the largest common one.

• In our solution, the GCD of 10 and 16 is 2.
• By dividing the numerator and the denominator by the GCD, we simplify the fraction to \( \frac{5}{8} \).

Finding and using the GCD simplifies the process of working with fractions and ensures your results are as straightforward as possible.