Problem 45 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{a}{3 b} \div \frac{a}{6 c} $$

Step-by-Step Solution

Answer

\( \frac{2c}{b} \)

Step 1: Rewrite the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. Rewrite the given expression \[ \frac{a}{3b} \times \frac{6c}{a} \]

Step 2: Simplify the Multiplication

Multiply the numerators together and the denominators together: \[ \frac{a \times 6c}{3b \times a} \]

Step 3: Cancel Common Factors

Since both the numerator and the denominator contain the factor \( a \), cancel them out: \[ \frac{6c}{3b} \]

Step 4: Reduce the Fraction

Divide both the numerator and the denominator by their greatest common divisor (3): \[ \frac{6c \/ 3}{3b \/ 3} = \frac{2c}{b} \]

Key Concepts

Fraction Division

When dealing with fraction division, it's essential to remember one key rule—divide by multiplying by the reciprocal of the fraction. This means, instead of dividing by a fraction, you flip the fraction (swap the numerator and denominator), and then multiply. For example, if you're dividing \(\frac{a}{3b}\) by \(\frac{a}{6c}\), you can rewrite the operation as a multiplication problem: \(\frac{a}{3b} \times \frac{6c}{a}\). By converting division to multiplication, the problem becomes simpler to solve.

Multiplication of Fractions

Once you've rewritten a fraction division problem as multiplication, the next step is multiplying fractions. This involves a straightforward process. Multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example: \(\frac{a}{3b} \times \frac{6c}{a}\) becomes \(\frac{a \times 6c}{3b \times a}\). Simplifying this product is easier once multiplication has taken place.

Canceling Common Factors

After multiplying the fractions, you often end up with common factors in both the numerator and the denominator. Canceling these common factors simplifies the fraction. In our example \(\frac{a \times 6c}{3b \times a}\), both the numerator and the denominator contain the factor \(a\). By canceling out \(a\), we get \(\frac{6c}{3b}\). Removing these common elements keeps the fraction simpler and easier to understand.

Reducing Fractions

The final step in simplifying an algebraic fraction is reducing it to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For \(\frac{6c}{3b}\), the GCD of 6 and 3 is 3. Dividing both by 3, we obtain \(\frac{2c}{b}\). The final simplified form of any fraction is achieved when no further reduction is possible, ensuring simplicity and clarity in the solution.