Problem 37 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{24}{75} \div \frac{8}{15}$$

Step-by-Step Solution

Answer

The solution is \( \frac{3}{5} \).

Step 1: Understand the Division of Fractions

When dividing by a fraction, you can multiply by its reciprocal. So, to solve \( \frac{24}{75} \div \frac{8}{15} \), we will multiply \( \frac{24}{75} \) by the reciprocal of \( \frac{8}{15} \), which is \( \frac{15}{8} \).

Step 2: Perform the Multiplication

Multiply the numerators and denominators: \[(\frac{24}{75}) \times (\frac{15}{8}) = \frac{24 \times 15}{75 \times 8}\]. Simplifying, we have \( \frac{360}{600} \).

Step 3: Simplify the Fraction

To simplify \( \frac{360}{600} \), find the greatest common divisor (GCD) of both numerators and denominators, which is 120. Therefore, \( \frac{360}{600} = \frac{360 \div 120}{600 \div 120} = \frac{3}{5} \).

Key Concepts

Understanding the Reciprocal of a Fraction

When we talk about the reciprocal of a fraction, we are referring to flipping the fraction's numerator and denominator. This concept is essential when dividing fractions.
For example, if you have the fraction \( \frac{8}{15} \), its reciprocal is \( \frac{15}{8} \).

• The original fraction is expressed as \( \frac{a}{b} \).
• Its reciprocal becomes \( \frac{b}{a} \).

This reciprocal function is crucial because to divide one fraction by another, you multiply the first fraction by the reciprocal of the second.
This trick allows you to turn a division problem into a multiplication one, making it easier to solve.For instance, in our original problem \( \frac{24}{75} \div \frac{8}{15} \), we can reinterpret it as a multiplication problem involving the reciprocals as follows: \( \frac{24}{75} \times \frac{15}{8} \). With this step, you effectively handle the division by fractions.

Multiplying Fractions

Multiplying fractions is simpler than it may first seem. All you need to do is multiply the numerators (the top numbers) together and multiply the denominators (the bottom numbers) together. This will give you the product of the fractions.For example, when you multiply \( \frac{24}{75} \) by \( \frac{15}{8} \), you follow these steps:

• Multiply the numerators: \( 24 \times 15 = 360 \).
• Multiply the denominators: \( 75 \times 8 = 600 \).

Now your multiplied fraction is \( \frac{360}{600} \). This product then becomes the focus for our next step, which is to simplify the fraction.
Remember, multiplying fractions helps avoid the complexities often involved in adding or subtracting them.
You don't need a common denominator, as you multiply directly across.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means we want to express the fraction so that the numerator and the denominator are as small as possible, yet still maintain the same value.
To do this, find the greatest common divisor (GCD) of both the numerator and the denominator.In our problem with \( \frac{360}{600} \), the GCD is 120.

• Divide the numerator by the GCD: \( 360 \div 120 = 3 \).
• Divide the denominator by the GCD: \( 600 \div 120 = 5 \).

So, \( \frac{360}{600} \) simplifies to \( \frac{3}{5} \). Simplifying fractions not only makes them easier to understand but also emphasizes their inherent relationships.
It's a critical skill because fractions in simplest form are often easier to use in further calculations and comparisons.