Problem 54 - Prealgebra

Question

Multiply and divide as indicated. $$\frac{7}{8} \cdot 32$$

Step-by-Step Solution

Answer

The result is 28.

Step 1: Convert to Fraction Multiplication

Convert the whole number 32 into a fraction. This is done by writing it as \( \frac{32}{1} \). Now we have \( \frac{7}{8} \cdot \frac{32}{1} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions. So, multiply 7 by 32: \( 7 \times 32 = 224 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions. So, multiply 8 by 1: \( 8 \times 1 = 8 \).

Step 4: Form the New Fraction

Combine the results from the previous steps into a single fraction: \( \frac{224}{8} \).

Step 5: Simplify the Fraction

Divide the numerator by the denominator to simplify the fraction. \( 224 \div 8 = 28 \). So, the simplified result is 28.

Key Concepts

Numerators and Denominators

When working with fractions, it's important to understand the roles of numerators and denominators. The numerator is the top number in a fraction, representing how many parts of a whole are being considered. The denominator is the bottom number, showing into how many equal parts the whole is divided. For example, in the fraction \( \frac{7}{8} \), 7 is the numerator and 8 is the denominator. This fraction means you have 7 out of 8 equal parts of something.
Understanding these components is crucial when multiplying or dividing fractions because they dictate how the operations are performed. For multiplication, you multiply the numerators to find the new numerator, and similarly, multiply the denominators to find the new denominator. In division, though, you would multiply by the reciprocal, which involves flipping the second fraction. Knowing this helps simplify and solve fraction problems effectively.

Simplifying Fractions

Simplifying fractions is an essential skill in making numbers easier to understand and use. When simplifying, you aim to reduce the fraction to its simplest form, where the numerator and denominator have no common divisors other than 1.
To simplify \( \frac{224}{8} \), you divide both the numerator and the denominator by their greatest common divisor (GCD). Here, both 224 and 8 can be divided by 8. Performing this division, \( 224 \div 8 = 28 \) and \( 8 \div 8 = 1 \), simplifies the fraction to 28 since \( \frac{28}{1} = 28 \).

• Find the GCD of the numerator and the denominator.
• Divide both by the GCD to simplify.

Simplifying makes calculations easier and results more interpretable, helping you understand the amount more clearly in real-life problems and mathematical expressions.

Whole Numbers to Fractions

Sometimes it's necessary to convert whole numbers into fractions, especially during fraction multiplication or division operations. A whole number can be easily converted into a fraction by placing it over 1. For example, the whole number 32 can be represented as \( \frac{32}{1} \).
This process makes it possible to perform operations with fractions and whole numbers seamlessly. In fraction multiplication, treating 32 as \( \frac{32}{1} \) allows it to be multiplied by another fraction, like \( \frac{7}{8} \), with ease. Just follow the same multiplication rules: multiply the numerators and multiply the denominators.

• Write the whole number over 1, e.g., \( \frac{32}{1} \).
• Makes operations with fractions straightforward by maintaining consistency in fraction format.

This concept not only simplifies calculations but also helps visualize the problem in consistent terms, aiding in managing complex mathematical scenarios.