Problem 19 - Fundamentals of Mathematics

Question

Find each value. (Section 4.6) \(\frac{4}{15} \div 8\)

Step-by-Step Solution

Answer

The value of \( \frac{4}{15} \div 8 \) is \( \frac{1}{30} \).

Step 1: Rewrite the Division as Multiplication

To simplify the expression \( \frac{4}{15} \div 8 \), first rewrite the division as a multiplication by the reciprocal. This means turning \(8\) into a fraction and then finding its reciprocal. The reciprocal of \(8\) is \(\frac{1}{8}\). Therefore, the expression becomes: \( \frac{4}{15} \times \frac{1}{8} \).

Step 2: Multiply the Numerators

Now, we need to multiply the numerators of the two fractions together. The numerators are \(4\) and \(1\) respectively. So, \(4 \times 1 = 4\).

Step 3: Multiply the Denominators

Next, multiply the denominators of the two fractions. The denominators are \(15\) and \(8\), so \(15 \times 8 = 120\).

Step 4: Simplify the Fraction

The product of the numerators and denominators gives us the fraction \( \frac{4}{120} \). To simplify, find the greatest common divisor (GCD) of 4 and 120, which is 4. Divide both the numerator and the denominator by 4: \( \frac{4 \div 4}{120 \div 4} = \frac{1}{30} \).

Key Concepts

Reciprocal

In mathematics, the reciprocal of a number is simply its inverse with reference to multiplication. For instance, if you have a number like 8, the reciprocal would be the fraction \( \frac{1}{8} \). The main idea is that when you multiply a number by its reciprocal, the product is always 1.
This is particularly helpful in fraction division. When dividing by a number, say \( 8 \), you can multiply by its reciprocal instead, turning the operation into a multiplication problem.
This simplifies the division process because multiplying fractions is often more straightforward than dividing them.

Multiplication of Fractions

When multiplying fractions, the process involves multiplying the numerators (the top numbers) together and then the denominators (the bottom numbers) together.
Let's say we have two fractions: \( \frac{a}{b} \) and \( \frac{c}{d} \). To multiply these, you follow the formula:

• Numerator: \( a \times c \)
• Denominator: \( b \times d \)

This will give you a fraction \( \frac{a \times c}{b \times d} \).
Using our exercise example, \( \frac{4}{15} \times \frac{1}{8} \), you multiply 4 by 1 to get 4 (the new numerator) and 15 by 8 to get 120 (the new denominator). This results in the fraction \( \frac{4}{120} \).

Simplifying Fractions

Simplifying a fraction means making it as basic as possible, while still keeping the same value.
This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, we ended up with \( \frac{4}{120} \) after our multiplication step. The GCD of 4 and 120 is 4.

• Divide the numerator by 4: \( 4 \div 4 = 1 \)
• Divide the denominator by 4: \( 120 \div 4 = 30 \)

Hence, \( \frac{4}{120} \) simplifies to \( \frac{1}{30} \).
This simplified fraction gives the same value, but in a cleaner and more understandable form.