Problem 59 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer as a fraction or as a mixed number in simplest form. $$\frac{3}{5} \div \frac{6}{7}$$

Step-by-Step Solution

Answer

The result of dividing \(\frac{3}{5}\) by \(\frac{6}{7}\) is \(\frac{7}{10}\).

Step 1: Rewrite the division as multiplication

To divide fractions, we must multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor), which is essentially flipping the second fraction upside down. \[ \frac{3}{5} \div \frac{6}{7} = \frac{3}{5} \cdot \frac{7}{6} \]

Step 2: Multiply the fractions

Perform the multiplication operation on the fractions. This means multiplying the numerators together (the numbers on the 'top' of each fraction) to find the numerator of the solution, and similarly, multiplying the denominators together (the numbers on the 'bottom' of each fraction) for the denominator of the solution. \[ \frac{3}{5} \cdot \frac{7}{6} = \frac{3 \cdot 7}{5 \cdot 6} = \frac{21}{30} \]

Step 3: Simplify the result

The next move is to reduce the resulting fraction into its simplest or reduced form, which is achieved by dividing both the numerator and denominator by their greatest common divisor (GCD). Both 21 and 30 are divisible by 3, so we divide both by 3. \[ \frac{21}{30} = \frac{21 \div 3}{30 \div 3} = \frac{7}{10} \]

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This means breaking the fraction down so that the numerator and denominator have no common factors other than 1. By simplifying fractions, calculations become easier and results clearer.

To simplify a fraction:

• Find the common factors of the numerator and the denominator.
• Identify the greatest common divisor (GCD).
• Divide both the numerator and the denominator by the GCD.

Doing this ensures that the fraction is in its simplest form, such as reducing \(\frac{21}{30}\) to \(\frac{7}{10}\), simplifying each by dividing by their GCD, which is 3.

Simple fractions make division and multiplication more straightforward, helping to maintain accuracy in further calculations.

Reciprocal

The reciprocal of a fraction is created by flipping the fraction upside down. This means swapping the numerator and the denominator. The reciprocal is essential when dividing fractions because it changes division into multiplication.

For instance, if you have \(\frac{6}{7}\), its reciprocal is \(\frac{7}{6}\). Converting division into multiplication with reciprocals makes calculations more manageable.

To divide fractions:

• Take the reciprocal of the divisor.
• Multiply it by the dividend.

This method simplifies the division task, as seen in the example: \(\frac{3}{5} \div \frac{6}{7} = \frac{3}{5} \cdot \frac{7}{6}\). Now it's just a straightforward multiplication.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. Knowing how to find the GCD is crucial for simplifying fractions effectively.

Here's how you can find the GCD:

• List the factors of both numbers.
• Identify the largest factor that appears in both lists.

For example, for the fraction \(\frac{21}{30}\), the common factors of 21 are \([1, 3, 7, 21]\) and for 30 are \([1, 2, 3, 5, 6, 10, 15, 30]\). The largest common factor is 3, which we divide both numerator and denominator by to simplify the fraction to \(\frac{7}{10}\).

By regularly using the GCD, students ensure their fractions are always in the simplest form, facilitating easy computation in complex mathematical problems.