Problem 49 - Prealgebra

Question

Find each product \(\frac{1}{4} \times 32\)

Step-by-Step Solution

Answer

The product of \( \frac{1}{4} \times 32 \) is 8.

Step 1: Understand the Multiplication

The problem asks us to find the product of \( \frac{1}{4} \times 32 \). This means we need to multiply a fraction by a whole number.

Step 2: Express Whole Number as a Fraction

To simplify multiplication, express the whole number 32 as a fraction by putting it over 1. So, 32 can be written as \( \frac{32}{1} \).

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together. The multiplication looks like this: \( \frac{1}{4} \times \frac{32}{1} = \frac{1 \times 32}{4 \times 1} = \frac{32}{4} \).

Step 4: Simplify the Fraction

To simplify \( \frac{32}{4} \), divide 32 by 4, which equals 8. Thus, \( \frac{32}{4} = 8 \).

Key Concepts

Simplifying Fractions

Fractions can appear intimidating at first, but they become much easier to handle after learning how to simplify them. To simplify a fraction, you want to make it as basic as possible. This means reducing the fraction to its smallest equivalent form. Consider this scenario from the original exercise \[\frac{32}{4}\]. Here's how you simplify a fraction:

• **Identify the Greatest Common Divisor (GCD):** This is the largest number that can divide both the numerator (the top number) and the denominator (the bottom number) without leaving a remainder. In this case, the GCD of 32 and 4 is 4.
• **Divide Both by the GCD:** Once we've found the GCD, we divide both the numerator and denominator by 4. So, \( \frac{32}{4} \) becomes 8 because \( 32 \div 4 = 8 \) and \( 4 \div 4 = 1 \), giving us \( \frac{8}{1} \) which is equal to 8.

Simplifying fractions not only makes calculations easier but also provides a clearer understanding of the quantities involved. Remember, the key here is to find the biggest number that can evenly divide both parts of the fraction.

Whole Numbers

Understanding how to work with whole numbers is essential, especially in operations involving fractions. In mathematics, whole numbers are the numbers we typically learn first. They include all positive integers, starting from zero. Zero itself is a whole number, too.In the exercise, we see a whole number, 32. Here's how to handle it:

• **Express as a Fraction:** Even though a whole number doesn't seem like a fraction, you can express it as one by writing it over 1. For the number 32, you can write it as \( \frac{32}{1} \).
• **Keeps its Value:** This representation does not change its value. 32 is still 32, but writing it as a fraction allows you to perform operations like multiplication with other fractions.

Expressing whole numbers as fractions is especially useful when multiplying because it aligns with the format of the operation and makes the process seamless.

Numerator and Denominator Manipulation

When multiplying fractions, the key operations happen with the numerators and denominators. A fraction consists of two parts: the numerator on top and the denominator on the bottom.Here is how you manipulate them:

• **Multiply the Numerators:** When multiplying fractions, you start by multiplying the top numbers. In the exercise, \( \frac{1}{4} \times \frac{32}{1} \) becomes \( 1 \times 32 \).
• **Multiply the Denominators:** Next, you multiply the bottom numbers. For our example, \( 4 \times 1 \) is performed.
• **Form the New Fraction:** After multiplying, you create a new fraction composed of the products. It looks like \( \frac{32}{4} \).
• **Simplify if Necessary:** Finally, check if the new fraction can be simplified to its lowest form by dividing both the numerator and the denominator by their GCD. In this case, \( \frac{32}{4} \) simplifies to 8.

This step-by-step process of manipulating numerators and denominators is crucial for accurately performing fraction multiplication.