Problem 29 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{3}{4} \div \frac{1}{2} \cdot 6$$

Step-by-Step Solution

Answer

The quotient is 9.

Step 1: Understand the Division Operation

The expression involves division of a fraction by another fraction. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). So, we replace \( \frac{1}{2} \) with its reciprocal, \( 2 \).

Step 2: Rewrite the Expression

Using the reciprocal of \( \frac{1}{2} \), rewrite the division as a multiplication: \( \frac{3}{4} \div \frac{1}{2} \) becomes \( \frac{3}{4} \times 2 \). So, the entire expression becomes \( \frac{3}{4} \times 2 \times 6 \).

Step 3: Multiply the Fractions and Whole Numbers

Multiply the fractions and whole numbers together. First, calculate \( \frac{3}{4} \times 2 \), which simplifies to \( \frac{3 \times 2}{4} = \frac{6}{4} \). Then simplify \( \frac{6}{4} \) to \( \frac{3}{2} \). Now multiply by 6: \( \frac{3}{2} \times 6 \).

Step 4: Final Calculation

Multiply \( \frac{3}{2} \times 6 \). We can write 6 as \( \frac{6}{1} \), so the multiplication becomes: \( \frac{3}{2} \times \frac{6}{1} = \frac{3 \times 6}{2} = \frac{18}{2} \). This simplifies to 9.

Key Concepts

Reciprocal Fractions

When you divide one fraction by another, you are not performing usual division as you might with whole numbers. Instead, you use reciprocal fractions. The reciprocal of a fraction is simply flipping the numerator and the denominator. For example, if you have the fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \).
This means if you want to divide by a fraction, you multiply by its reciprocal instead. In the original exercise, we replaced \( \frac{1}{2} \) with its reciprocal, \( 2 \), so division turned into multiplication. Using reciprocals makes fraction division simpler and more straightforward. Understanding this process provides clarity on how division of fractions works effectively.

Multiplying Fractions

Once the division is turned into a multiplication problem, you are set to multiply fractions. Multiplying fractions involves simple steps:

• Multiply the numerators together


• Multiply the denominators together

After handling the reciprocal conversion in our example, multiplying \( \frac{3}{4} \times 2 \) might look like: - First, \( 3 \times 2 = 6 \)- And the denominator remains \( 4 \) Thus, the result is \( \frac{6}{4} \). Multiplying doesn't change the fractions' main components but positions them resulting in new numerators and denominators. It's as easy as counting to 3 if you follow these simple steps!

Simplifying Fractions

Once you have multiplied the fractions together, you might end up with a larger fraction. In our example, the result was \( \frac{6}{4} \). Simplifying is about making fractions easier to understand and work with. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator


• Divide both by their GCD

For \( \frac{6}{4} \), we find that the GCD is 2, leading to the simplified form \( \frac{3}{2} \). This smaller fraction \( \frac{3}{2} \) is much easier to use. When you simplify fractions correctly, you maintain the value, but in a form that’s quicker to handle in further calculations.