Problem 28 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ -9 \div\left(-\frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The quotient of \(-9\) divided by \(-\frac{3}{4}\) is \(12\).

Step 1 - Setup

Firstly, express the problem in appropriate terms: Multiply \(-9\) by the reciprocal of \(-\frac{3}{4}\).

Step 2 - Find Reciprocal

To find the reciprocal of a fraction, interchange the numerator and the denominator. Hence, the reciprocal of \(-\frac{3}{4}\) is \(-\frac{4}{3}\).

Step 3 - Multiply

Now, multiply \(-9\) with \(-\frac{4}{3}\). This results in \(12\).

Step 4 - Simplify

With two negative numbers, the multiplication operation gives a positive solution. Therefore, \(-9\) divided by \(-\frac{3}{4}\) equals \(12\).

Key Concepts

Reciprocal

Understanding the concept of a reciprocal is crucial when working with division problems involving fractions. In mathematics, the reciprocal of a number is simply achieved by flipping it. For fractions, this means swapping the numerator (the top number) and the denominator (the bottom number).
For example:

• The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
• If we have a negative fraction, like \(-\frac{3}{4}\), its reciprocal becomes \(-\frac{4}{3}\). The sign of the fraction does not change. Only the numerator and denominator switch places.

Knowing how to find the reciprocal is especially important when you divide by a fraction because dividing by a fraction is equivalent to multiplying by its reciprocal.

Negative Numbers

When you encounter negative numbers in division or multiplication, it's important to understand how the signs interact. Negative numbers have a few key rules:

• Multiplying or dividing two negative numbers results in a positive number. This happens because the two negatives cancel each other out.
• Multiplying or dividing a negative number by a positive number results in a negative number.

In the given exercise, we had both numbers as negative: \(-9\) and \(-\frac{3}{4}\). According to the rule mentioned, their product is positive. Therefore, the result of multiplying \(-9\) by \(-\frac{4}{3}\) gives us a positive \(12\).
Understanding these rules is essential for correctly solving problems involving negative numbers in arithmetic operations.

Multiplication of Fractions

Multiplication of fractions is a simple process that involves multiplying the numerators together and the denominators together.
For instance, if you have two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the result of multiplying them will be: \( \frac{a \times c}{b \times d} \).

To multiply a whole number by a fraction, consider the whole number as a fraction with a denominator of 1. For example, \(-9\) can be written as \( \frac{-9}{1} \).

In the exercise given, we multiplied \(-9\) by the reciprocal of \(-\frac{3}{4}\) which is \(-\frac{4}{3}\). This operation looks like:
\[\frac{-9}{1} \times \frac{-4}{3} = \frac{-9 \times -4}{1 \times 3} = \frac{36}{3} \]
Since the product of the two negatives gives a positive number, the final step is simplifying \( \frac{36}{3} \) to get \(12\).
The method of multiplying the numerators and denominators separately makes it straightforward to arrive at these solutions.