Problem 33 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{9}{16} \div \frac{15}{8}$$

Step-by-Step Solution

Answer

The value is \( \frac{3}{10} \).

Step 1: Write Down the Problem

The problem given is to divide the fraction \( \frac{9}{16} \) by \( \frac{15}{8} \).

Step 2: Change Division to Multiplication

When you divide by a fraction, you multiply by its reciprocal. Therefore, \( \frac{9}{16} \div \frac{15}{8} \) becomes \( \frac{9}{16} \times \frac{8}{15} \).

Step 3: Multiply the Numerators

Multiply the numerators of the fractions: \( 9 \times 8 = 72 \).

Step 4: Multiply the Denominators

Multiply the denominators of the fractions: \( 16 \times 15 = 240 \).

Step 5: Simplify the Fraction

Simplify the fraction \( \frac{72}{240} \). The greatest common divisor of 72 and 240 is 24, so divide both by 24: \( \frac{72 \div 24}{240 \div 24} = \frac{3}{10} \).

Key Concepts

What is a Reciprocal?

To understand fraction division, you first need to know what a reciprocal is. The reciprocal of a fraction is simply swapping its numerator and denominator. This means, if you have a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
This concept is key in fraction division. Instead of dividing by a fraction, you multiply by its reciprocal.
For example, if you need to divide \( \frac{9}{16} \) by \( \frac{15}{8} \), you multiply \( \frac{9}{16} \) by the reciprocal of \( \frac{15}{8} \), which is \( \frac{8}{15} \).

• Original: \( \frac{15}{8} \)
• Reciprocal: \( \frac{8}{15} \)

This change from division to multiplication makes it much easier to work with the fractions.

Simplifying Fractions

After performing operations such as multiplication, it's important to simplify the resulting fraction. Simplifying a fraction means making the numerator and denominator as small as possible, while still keeping the same value.
This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD), which we'll discuss more in the next section.
As an example, after multiplying \( \frac{9}{16} \) by \( \frac{8}{15} \), you get \( \frac{72}{240} \).

• First, find a number that both 72 and 240 can be divided by.
• In this case, both are divisible by 24.
• Divide both 72 and 240 by 24 to simplify the fraction to \( \frac{3}{10} \).

Simplifying fractions helps reduce them to their simplest form, making calculations easier and results clearer.

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is the largest number that can evenly divide both the numerator and denominator of a fraction. Finding the GCD is essential to simplifying fractions.
To find the GCD, list the factors of each number and choose the largest common one.

• For 72: Factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
• For 240: Factors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240.

The largest common factor between both lists is 24.
By dividing both 72 and 240 by their GCD of 24, you get a simplified fraction that is easier to understand and work with, like going from \( \frac{72}{240} \) to \( \frac{3}{10} \).
Simplification using the GCD ensures that the fraction is reduced to its most simplified form.