Problem 78 - Algebra 1: Concepts and Skills

Question

Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{7}{8} \cdot \frac{1}{2} $$

Step-by-Step Solution

Answer

The simplified form of the fraction is \(\frac{7}{16}\)

Step 1: Multiplication of Numerators and Denominators

Multiply the numerators \(7 * 1 = 7\) and the denominators \(8 * 2 = 16\). The result is the fraction \(\frac{7}{16}\).

Step 2: Simplification

Check if the fraction \(\frac{7}{16}\) can be reduced further. In this case, as 7 and 16 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions is a key skill in math that helps make numbers more manageable. The process involves reducing a fraction to its simplest form. This means making the numerator (the top number) and the denominator (the bottom number) as small as possible while still having the same value. To do this, you'll need to find the greatest common divisor (GCD) of both numbers. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder.

For instance, if you have a fraction like \(\frac{8}{12}\), you would find the GCD of 8 and 12, which is 4. You can then divide both the numerator and the denominator by their GCD to simplify the fraction to \(\frac{2}{3}\).

• Make sure to always check if the fraction can be simplified further by finding any common factors.
• Sometimes a fraction is already in its simplest form, like \(\frac{7}{16}\), where no number except 1 divides both the top and bottom evenly.

Numerator and Denominator

Understanding the numerator and denominator is crucial when working with fractions. A fraction represents a part of a whole and is written with two numbers separated by a line.

The **numerator** is the top part of the fraction. It tells you how many equal parts you have. In our example \(\frac{7}{8}\), 7 is the numerator, indicating that we're dealing with seven parts.

On the other hand, the **denominator** is the bottom part. It shows into how many parts the whole is divided. Using \(\frac{7}{8}\) again, 8 serves as the denominator, meaning the whole is divided into eight parts.

• Remember, the fraction tells you "how many of which size piece." The numerator tells you "how many," and the denominator tells you "which size."
• Always keep track of these numbers to ensure you’re performing calculations correctly, especially in operations like addition, subtraction, multiplication, or division of fractions.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it's quite straightforward. To multiply fractions, you multiply the numerators together and then the denominators together. This creates a new fraction.

For example, if you multiply \(\frac{7}{8}\) by \(\frac{1}{2}\), you follow these simple steps:

• Multiply the numerators: \(7 \times 1 = 7\).
• Multiply the denominators: \(8 \times 2 = 16\).

This gives you the new fraction \(\frac{7}{16}\). After finding the product, you should always check if it can be simplified, even though sometimes, like in this case, it is already in the simplest form.

Multiplying fractions helps in scaling numbers and is useful in many mathematical contexts like probability and ratios. Make sure to simplify whenever possible as it makes further calculations easier and cleaner.