Problem 57 - Fundamentals of Mathematics

Question

(Section 4.4) Find the value of \(\frac{3}{8} \div \frac{9}{16} \cdot \frac{6}{5}\).

Step-by-Step Solution

Answer

The value is \( \frac{4}{5} \).

Step 1: Convert Division to Multiplication

When dividing by a fraction, we multiply by its reciprocal. Thus, \( \frac{3}{8} \div \frac{9}{16} \) becomes \( \frac{3}{8} \times \frac{16}{9} \).

Step 2: Simplify and Multiply Fractions

First, multiply \( \frac{3}{8} \times \frac{16}{9} \). Simplify by dividing 16 and 8 by 8: \( \frac{3}{1} \times \frac{2}{9} = \frac{6}{9} \). Next, multiply by \( \frac{6}{5} \): \( \frac{6}{9} \times \frac{6}{5} = \frac{36}{45} \).

Step 3: Simplify the Resulting Fraction

The fraction \( \frac{36}{45} \) can be simplified. Both numerator and denominator can be divided by their greatest common divisor, 9: \( \frac{36 \div 9}{45 \div 9} = \frac{4}{5} \).

Key Concepts

Division of Fractions

Dividing fractions might seem tricky at first, but it becomes straightforward once you know the secret trick: Instead of dividing, you multiply by the reciprocal. The reciprocal of a fraction is simply flipping its numerator and denominator.
For example, if you need to divide \( \frac{3}{8} \) by \( \frac{9}{16} \), you actually multiply \( \frac{3}{8} \) by the reciprocal of \( \frac{9}{16} \). This means you flip \( \frac{9}{16} \) to get \( \frac{16}{9} \), and then multiply: \( \frac{3}{8} \times \frac{16}{9} \).
This approach makes division much simpler, and you avoid complex division operations by merely multiplying! It's essential to remember that every division of fractions will convert into a multiplication by using reciprocals.

Simplification of Fractions

Simplification of fractions is necessary as it makes the numbers easier to work with and understand. When you simplify, you look for common factors in the numerator and denominator and divide both by their greatest common divisor.

In our exercise, after multiplying the fractions \( \frac{3}{8} \times \frac{16}{9} \), the result is \( \frac{6}{9} \). It can be simplified further because both 6 and 9 can be divided by 3. So, \( \frac{6}{9} = \frac{2}{3} \).
It's an important step to make sure your final answer is in its simplest form, showing the most reduced and concise version possible. This ensures accuracy and clarity in communication, especially when the result is a part of other calculations.

Reciprocal of a Fraction

The reciprocal of a fraction is one of the simplest yet most powerful concepts in math. It just means flipping a fraction's numerator and denominator. This simple swap changes how we perceive and use the fraction entirely.
For instance, the reciprocal of \( \frac{9}{16} \) is \( \frac{16}{9} \). This operation is pivotal in converting division into multiplication, providing a much simpler method to handle fraction division.

• To find the reciprocal, just swap the top and bottom numbers.
• Always use it in fraction division to turn the process into multiplication.

Using reciprocals this way not only makes math easier but also provides a foundation for a deeper understanding of rational numbers and their properties.