Problem 81 - Beginning and Intermediate Algebra

Question

Divide. $$ -\frac{5}{9} \div\left(-\frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The result is \(\frac{20}{27}\).

Step 1: Identify the Problem

We need to divide two negative fractions: \(-\frac{5}{9}\) by \(-\frac{3}{4}\).

Step 2: Use the Division Rule for Fractions

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we convert the division to multiplication: \[-\frac{5}{9} \div \left(-\frac{3}{4}\right) = -\frac{5}{9} \times \left(-\frac{4}{3}\right)\].

Step 3: Multiply the Fractions

Multiply the numerators and the denominators. This gives: \(-5 \times -4 = 20\) and \(9 \times 3 = 27\). Therefore, the product is \(\frac{20}{27}\).

Step 4: Simplify the Result

Check if the fraction \(\frac{20}{27}\) can be simplified. Since 20 and 27 have no common factors other than 1, \(\frac{20}{27}\) is already in its simplest form.

Key Concepts

Negative Fractions

When dealing with negative fractions, the negative sign can appear on either the numerator, the denominator, or out front of the entire fraction. This placement makes no difference on the value of the fraction as long as only one negative sign is present. For example,

• \(-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}\)

This means that when you're dividing two negative fractions, like in the exercise \(\left(-\frac{5}{9}\right) \div \left(-\frac{3}{4}\right)\)the negative signs effectively cancel each other out. This is because dividing a negative by a negative results in a positive, thanks to the rules of division. It's like multiplying two negative numbers in cases of regular arithmetic, where two negatives make a positive.
A helpful insight is to simplify handling the signs first to avoid confusion, then proceed with the arithmetic.

Multiplication and Division

An essential skill in working with fractions is understanding how to switch between multiplication and division. When dividing fractions, the operation can be transformed into a multiplication problem by using the reciprocal of the divisor. In mathematical terms, dividing by a fraction \(\left(\frac{a}{b}\right)\) is the same as multiplying by its reciprocal \(\left(\frac{b}{a}\right)\).
In our given problem,

• \(-\frac{5}{9} \div \left(-\frac{3}{4}\right)\)

converts to

• \(-\frac{5}{9} \times \frac{-4}{3}\)

This method keeps the calculation streamlined and ensures the final manipulation relates directly to multiplication, a generally simpler operation. Once converted, simply multiply the numerators and denominators. This process is fundamental to transforming division problems into operations you might find easier to handle.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. After multiplying fractions, as in the example

• \(-\frac{5}{9} \times \frac{-4}{3} = \frac{20}{27}\)

the resulting fraction \(\frac{20}{27}\) should be checked for common factors.
To identify common factors, it's useful to list out the divisors of both numbers. For 20, the divisors are 1, 2, 4, 5, 10, and 20, while for 27 they are 1, 3, 9, and 27. Since the only common divisor is 1, this fraction is already in its simplest form.
Simplification helps in presenting a more intuitive and accessible form of the fraction, making it easier to interpret and utilize in further calculations.