Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises 47-76, perform the indicated division or state that the expression is undefined. $$ 6 \div\left(-\frac{2}{5}\right) $$

Step-by-Step Solution

Answer

-15

Step 1: Identify the reciprocal of the fraction

The reciprocal of their fraction \(-\frac{2}{5}\) is \(-\frac{5}{2}\). The negative sign is also included in the reciprocal.

Step 2: Perform the multiplication

Now, to perform the division, multiply the whole number 6 with the reciprocal of the fraction from Step 1, which results in \(6 * -\frac{5}{2} = -15\).

Key Concepts

Reciprocal of a Fraction

To understand the concept of a reciprocal, imagine flipping a fraction upside down. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
For instance, the reciprocal of the fraction \(-\frac{2}{5}\) is \(-\frac{5}{2}\).
It's important to note that the negative sign is maintained; it doesn't disappear with the flipping.

• Numerator becomes the denominator.
• Denominator becomes the numerator.
• Sign stays the same.

Understanding reciprocals is crucial, especially in algebraic division, because dividing by a fraction is the same as multiplying by its reciprocal.

Multiplying Fractions

Multiplying fractions is straightforward. You multiply the numerators together and the denominators together. When performing an operation like \(6 * -\frac{5}{2}\), treat 6 as a fraction \(\frac{6}{1}\).

• Multiply numerators: \(6 \cdot -5 = -30\).
• Multiply denominators: \(1 \cdot 2 = 2\).

This gives us \(-\frac{30}{2}\), which simplifies to \(-15\).

Multiplying by the reciprocal is often used to transform a division problem into a multiplication one. This is helpful when simplifying expressions and solving equations.

Undefined Expressions

In arithmetic and algebra, an expression is undefined if it involves a division by zero. This means any situation where the denominator becomes zero creates an undefined expression. This is a fundamental rule: one cannot divide a number by zero.

This concept plays a vital role when working with fractions as part of larger mathematical problems. Before proceeding with division, always check that the denominator won’t become zero to avoid undefined results.

• Be wary of complex expressions where components might seem hidden.
• Solve fractions thoroughly to ensure numerators and denominators are correctly identified.

So, ensure every division makes sense, and you'll avoid entering undefined territories in your calculations.