Problem 14 - Prealgebra

Question

Find each quotient. Write in simplest form. $$\frac{a b}{4} \div \frac{b}{6}$$

Step-by-Step Solution

Answer

The quotient in simplest form is \( \frac{3a}{2} \).

Step 1: Rewrite the Division as Multiplication

To divide by a fraction, you multiply by its reciprocal. Thus, \( \frac{a b}{4} \div \frac{b}{6} \) becomes \( \frac{a b}{4} \times \frac{6}{b} \).

Step 2: Multiply the Fractions

Multiply the numerators and the denominators: \( \frac{a b \times 6}{4 \times b} \). This simplifies to \( \frac{6ab}{4b} \).

Step 3: Simplify the Fraction

Cancel the common factor \( b \) in the numerator and denominator: \( \frac{6a}{4} \).

Step 4: Reduce the Fraction to Simplest Form

Simplify \( \frac{6a}{4} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{3a}{2} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in math that helps make calculations easier and answers more understandable. When you simplify a fraction, you are reducing it to its simplest form, meaning it cannot be reduced any further while remaining an equivalent value.

• Identify the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can evenly divide both the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.
• The resulting fraction is the simplest form.

For example, in the final steps of the original problem, the fraction \( \frac{6a}{4} \) is simplified by dividing both 6 and 4 by their GCD, which is 2, resulting in \( \frac{3a}{2} \). Simplifying fractions helps keep your calculations neat and your answers clear.

Dividing Fractions

Dividing fractions may seem tricky at first, but it becomes straightforward once you learn the steps. Division of fractions involves flipping the second fraction and then performing multiplication.

• Take the reciprocal of the second fraction. The reciprocal is what you get when you swap the numerator and the denominator.
• Change the division sign to a multiplication sign.
• Multiply the fractions as you normally would.

In our original exercise, when dividing \( \frac{a b}{4} \) by \( \frac{b}{6} \), we take the reciprocal of \( \frac{b}{6} \) to get \( \frac{6}{b} \) and then multiply the two fractions. This turns the problem into a simpler multiplication task.

Reciprocal of a Fraction

The reciprocal of a fraction is fundamental when dealing with both multiplication and division of fractions. To find the reciprocal, you simply switch the fraction's numerator and denominator. This process transforms the fraction without changing its relationship in division.

• A fraction \( \frac{x}{y} \) becomes \( \frac{y}{x} \) when taking the reciprocal.
• The product of a fraction and its reciprocal will always be 1, because \( x \times \frac{1}{x} = 1 \).
• Reciprocals simplify the process of dividing fractions, effectively transforming it into a multiplication task.

When you are given a problem like \( \frac{a b}{4} \div \frac{b}{6} \), you find the reciprocal of \( \frac{b}{6} \), which is \( \frac{6}{b} \), and then multiply instead of divide.

Multiplying Fractions

Multiplying fractions is a straightforward process that involves multiplying the numerators together and the denominators together. You can simplify before or after multiplying, whichever is more convenient.

• Multiply across the numerators to find the new numerator.
• Multiply across the denominators to find the new denominator.
• Simplify the resulting fraction by dividing the numerator and denominator by their GCD if necessary.

In our exercise, after rewriting the division as multiplication, we multiply \( \frac{a b}{4} \) by \( \frac{6}{b} \), which gives \( \frac{6ab}{4b} \). After canceling the common \( b \) terms, the simplified fraction is \( \frac{6a}{4} \), which we then further simplify to \( \frac{3a}{2} \). Multiplication is often simpler than division and is made easier by looking for opportunities to simplify the fractions early.