Problem 91 - Prealgebra

Question

The problems below review material involving fractions and mixed numbers. Perform the indicated operations. Write your answers as whole numbers, proper fractions, or mixed numbers. $$\frac{\frac{3}{8}}{\frac{6}{7}}$$

Step-by-Step Solution

Answer

The simplified result of the division is \(\frac{7}{16}\).

Step 1: Understand Division of Fractions

To divide by a fraction, you need to multiply by its reciprocal. The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This means that instead of dividing, you reverse the numerator and the denominator of the fraction you are dividing by and then multiply.

Step 2: Write the Reciprocal

The fraction you need to divide by is \(\frac{6}{7}\). Therefore, its reciprocal is \(\frac{7}{6}\).

Step 3: Replace Division with Multiplication

Convert the division problem \(\frac{\frac{3}{8}}{\frac{6}{7}}\) into a multiplication problem using the reciprocal: \(\frac{3}{8} \times \frac{7}{6}\).

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{3 \times 7}{8 \times 6} = \frac{21}{48}\).

Step 5: Simplify the Result

To simplify \(\frac{21}{48}\), find the greatest common divisor (GCD) of 21 and 48, which is 3. Divide both the numerator and the denominator by 3: \(\frac{21 \div 3}{48 \div 3} = \frac{7}{16}\).

Step 6: Express in Simplified Form

The fraction \(\frac{7}{16}\) is already in its simplest form as both 7 and 16 have no common divisors other than 1.

Key Concepts

Division of Fractions

Dividing fractions might sound tricky at first, but it's actually simple with the right approach. Instead of dividing directly, you convert the problem using a key math trick involving reciprocal of fractions. This method helps in transforming the division into a multiplication problem, which is typically easier to handle.

• For any division involving fractions, you take the second fraction (or the divisor), find its reciprocal, and multiply it by the first fraction (or the dividend).
• In the given example, you had to divide \( \frac{3}{8} \) by \( \frac{6}{7} \). By converting it to multiplication with the reciprocal, it becomes \( \frac{3}{8} \times \frac{7}{6} \).
• This change simplifies the process and uses basic multiplication of fractions instead of division.

Reciprocal of a Fraction

The concept of a reciprocal is fundamental when working with fractions. The reciprocal of a fraction \( \frac{a}{b} \) is simply flipping the numerator and the denominator, resulting in \( \frac{b}{a} \).

• Reciprocals are crucial because they convert division problems into multiplication ones.
• For example, the reciprocal of \( \frac{6}{7} \) is \( \frac{7}{6} \). By turning the division problem into a multiplication, you make calculations more straightforward.
• This step ensures that you get the same result as dividing would, but with less complexity.

Knowing how to find and use reciprocals quickly can make fraction calculations go much more smoothly.

Simplifying Fractions

Simplifying fractions means making them as concise as possible. A fraction is simplified when the numerator and the denominator have no common factors other than one.

• To simplify \( \frac{21}{48} \), find the greatest common divisor (GCD) of 21 and 48. In this case, it's 3.
• Divide both the numerator and denominator by the GCD to reduce the fraction: \( \frac{21}{48} = \frac{21 \div 3}{48 \div 3} = \frac{7}{16} \).
• Keep checking until no further reduction is possible. Sometimes, a little trial and error helps to ensure that the fraction is in simplest form.

Remember that simplifying fractions makes them easier to understand and use, especially in practical applications and further calculations.