Problem 49 - Prealgebra

Question

Evaluate each expression. $$m \div n \text { if } m=-\frac{8}{9} \text { and } n=\frac{7}{18}$$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{-16}{7} \).

Step 1: Identify the Expression

We need to evaluate the expression given by \( m \div n \), where \( m = -\frac{8}{9} \) and \( n = \frac{7}{18} \).

Step 2: Reciprocal of the Divisor

To divide by a fraction, we multiply by its reciprocal. The reciprocal of \( n = \frac{7}{18} \) is \( \frac{18}{7} \).

Step 3: Multiply by Reciprocal

The division \( m \div n \) can be rewritten as a multiplication: \( -\frac{8}{9} \times \frac{18}{7} \).

Step 4: Multiply the Numerators

Multiply the numerators of the fractions: \(-8 \times 18 = -144\).

Step 5: Multiply the Denominators

Multiply the denominators of the fractions: \(9 \times 7 = 63\).

Step 6: Write the Fraction

Combine the results from above into a single fraction: \( \frac{-144}{63} \).

Step 7: Simplify the Fraction

Simplify \( \frac{-144}{63} \) by finding the greatest common divisor of 144 and 63, which is 9. This gives us \( \frac{-16}{7} \) after simplification.

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is the first step in dividing fractions. A reciprocal simply flips a fraction upside down. In a fraction, such as \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). To find the reciprocal, swap the numerator (top number) and the denominator (bottom number).

• For example, if you have the fraction \( \frac{7}{18} \), its reciprocal is \( \frac{18}{7} \).
• Similarly, the reciprocal of \( \frac{-8}{9} \) would be \( \frac{-9}{8} \), although, here, we are working with \( \frac{7}{18} \).

Reciprocals are crucial because they transform division into multiplication, an operation many find easier to compute. You just need to remember: when dividing by a fraction, multiply by its reciprocal!

Multiplication of Fractions

Multiplying fractions is typically more straightforward than dividing them. This process involves two main steps: multiplying the numerators together and then multiplying the denominators.
Here’s a simple way to calculate:

• First, multiply the numerators. For instance, with \( -\frac{8}{9} \) and \( \frac{18}{7} \), this step involves calculating \(-8 \times 18\), which equals \(-144\).
• Next, multiply the denominators. Here, \(9 \times 7\), which equals \(63\).
The result of these two calculations forms the new fraction: \( \frac{-144}{63} \).
This operation is consistent no matter how complex the fractions appear. Remember, multiplication of fractions is always simplified by multiplying across the numerators and denominators.

Simplifying Fractions

Simplifying, or reducing, fractions is about making them as simple as possible. This means dividing both the numerator and the denominator by their greatest common divisor (GCD).
To simplify \( \frac{-144}{63} \), follow these steps:

• Find the GCD of both numbers. Here, the GCD of 144 and 63 is 9.
• Divide both the numerator and denominator by the GCD. So, \(-144 \div 9 = -16\) and \(63 \div 9 = 7\).

This simplifies \( \frac{-144}{63} \) to \( \frac{-16}{7} \).
Simplifying helps us understand fractions more clearly and can make later calculations easier. It's always a useful step to keep in mind when working with fractions.