Problem 61 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{3}{8} \div \frac{9}{16} \cdot \frac{6}{5}$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Division as Multiplication

To solve this problem, we first need to rewrite the division of fractions as a multiplication. Dividing by a fraction means multiplying by its reciprocal. So, replace \( \frac{3}{8} \div \frac{9}{16} \) with \( \frac{3}{8} \times \frac{16}{9} \).

Step 2: Multiply the First Two Fractions

Now, multiply the first fraction \( \frac{3}{8} \) by the reciprocal of the second, \( \frac{16}{9} \): \[\frac{3}{8} \times \frac{16}{9} = \frac{3 \times 16}{8 \times 9} = \frac{48}{72}\]

Step 3: Simplify the Resulting Fraction

To simplify \( \frac{48}{72} \), find the greatest common divisor (GCD) of 48 and 72, which is 24. Divide both the numerator and denominator by 24: \[\frac{48 \div 24}{72 \div 24} = \frac{2}{3}\]

Step 4: Multiply by the Third Fraction

Next, multiply the simplified result \( \frac{2}{3} \) by the third fraction \( \frac{6}{5} \): \[\frac{2}{3} \times \frac{6}{5} = \frac{2 \times 6}{3 \times 5} = \frac{12}{15}\]

Step 5: Simplify the Final Result

Finally, simplify the fraction \( \frac{12}{15} \). The greatest common divisor of 12 and 15 is 3. So we divide the numerator and denominator by 3: \[\frac{12 \div 3}{15 \div 3} = \frac{4}{5}\]

Key Concepts

Reciprocal of a Fraction

When we discuss fractions in mathematics, the reciprocal is a key concept, especially in fraction division. The reciprocal of a fraction is simply a way to "flip" a fraction so that the numerator becomes the denominator and the denominator becomes the numerator.

• For example, the reciprocal of \( \frac{3}{8} \) is \( \frac{8}{3} \).
• Similarly, the reciprocal of \( \frac{9}{16} \) is \( \frac{16}{9} \).

In the context of our problem, converting a division of fractions into multiplication involves using the reciprocal of the divisor. So, to divide by \( \frac{9}{16} \), we multiply by its reciprocal, \( \frac{16}{9} \). This simplification turns the problem from division into a more straightforward multiplication, making it easier to solve.

Simplifying Fractions

Simplifying fractions is a necessary skill in mathematics that makes working with numbers cleaner and more manageable. To simplify a fraction, we reduce it to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).

• For instance, with the fraction \( \frac{48}{72} \), the GCD is 24.
• Hence, dividing the numerator and the denominator by 24 gives us \( \frac{2}{3} \).

This process not only makes the numbers smaller and more comfortable to work with but also ensures the fraction is in its simplest form, removing any common factors within the fraction itself.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept when simplifying fractions. The GCD of two numbers is the largest number that can evenly divide both numbers without leaving a remainder.

• For instance, to find the GCD of 48 and 72, we look for the highest number that can divide both evenly, which is 24.
• Similarly, the GCD of 12 and 15 is 3.

Using the GCD to reduce fractions is a crucial step in simplifying them. By dividing both the numerator and denominator of a fraction by their GCD, we can reduce the fraction to its simplest, most compact form. This step facilitates easier calculations in subsequent steps of any mathematical problem.

Multiplication of Fractions

Multiplying fractions is generally straightforward once you have a clear approach. When multiplying two fractions, you multiply the numerators together for a new numerator, and the denominators together for a new denominator.

• For example, multiplying \( \frac{3}{8} \times \frac{16}{9} \) results in \( \frac{3 \times 16}{8 \times 9} = \frac{48}{72} \).
• Furthermore, if we multiply \( \frac{2}{3} \) by \( \frac{6}{5} \), it becomes \( \frac{2 \times 6}{3 \times 5} = \frac{12}{15} \).

It is important to simplify the result wherever possible. After performing the multiplication, simplifying ensures the fraction is in its most manageable form for further mathematical operations or final answers. This step-by-step approach to multiplication takes away complexity and provides clarity in mathematical problems involving fractions.