Problem 92 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer in simplest form. $$ \frac{7}{8} \div \frac{5}{2} $$

Step-by-Step Solution

Answer

\(\frac{7}{20}\)

Step 1: Reciprocal of the divisor

Since division of fractions is the same as multiplying by the reciprocal of the second fraction, we rewrite \(\frac{5}{2}\) as its reciprocal which is \(\frac{2}{5}\) . So, the task becomes \(\frac{7}{8} \times \frac{2}{5}\) .

Step 2: Multiplication of fractions

To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). For the numerators, \(7 \times 2 = 14\) and for the denominators, \(8 \times 5 = 40\), making the answer \(\frac{14}{40}\).

Step 3: Simplify the result

To present the answer in the simplest form, we need to simplify \(\frac{14}{40}\). The greatest common factor (GCF) of 14 and 40 is 2. So, we divide both the numerator and the denominator by 2, reducing the fraction to \(\frac{7}{20}\), which is the answer in simplest form.

Key Concepts

Reciprocal

When dividing fractions, a crucial concept to grasp is the reciprocal. The reciprocal of a fraction is achieved by swapping its numerator and denominator.
For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This swap is vital, especially in division operations involving fractions.

• To divide \(\frac{7}{8}\) by \(\frac{5}{2}\), we need the reciprocal of \(\frac{5}{2}\).
• The reciprocal is \(\frac{2}{5}\).

Once transformed into a multiplication problem, dividing by \(\frac{5}{2}\) becomes multiplying \(\frac{7}{8}\) by \(\frac{2}{5}\).
Understanding how to find and use the reciprocal paves the way for solving division problems involving fractions easily.

Multiplying Fractions

Multiplying fractions might seem complex, but with the right steps, it's quite easy. When multiplying fractions like \(\frac{7}{8}\) and \(\frac{2}{5}\), you follow a straightforward process.

• Multiply the numerators: You take the top numbers of the fractions, which are 7 and 2, and multiply them to get 14.
• Multiply the denominators: Similarly, multiply the bottom numbers, 8 and 5, which result in 40.

Therefore, the product of \(\frac{7}{8}\) and \(\frac{2}{5}\) is \(\frac{14}{40}\).
This process is consistent regardless of the fractions involved and forms a foundation for working with fractions in different mathematical contexts.

Simplifying Fractions

Simplifying fractions is all about reducing them to their simplest form. A fraction is in its simplest form when you can no longer divide the numerator and the denominator by any common number other than 1.

• In \(\frac{14}{40}\), the greatest common factor (GCF) of 14 and 40 is 2.
• Divide both the numerator and the denominator by their GCF: \(\frac{14 \div 2}{40 \div 2} = \frac{7}{20}\).

After simplification, \(\frac{7}{20}\) becomes the final answer in its simplest form.
Simplifying is crucial as it makes fractions easier to understand and work with, particularly in more complex calculations. Understanding how to identify common factors helps in arriving at the cleanest and most concise solution.