Problem 24 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\frac{2}{3} \div\left(-\frac{1}{6}\right)$$

Step-by-Step Solution

Answer

The result is \(-4\).

Step 1: Understand Division of Fractions

When you divide fractions, you actually multiply by the reciprocal of the divisor. The reciprocal of a fraction is obtained by exchanging the numerator and the denominator.

Step 2: Find the Reciprocal of the Divisor

The divisor in this problem is \(-\frac{1}{6}\). To find the reciprocal, we exchange the numerator and denominator, getting \(-6\).

Step 3: Multiply the Fractions

Now multiply \(\frac{2}{3}\) by \(-6\): \[\frac{2}{3} \times (-6) = \frac{2 \times (-6)}{3}\].

Step 4: Perform the Multiplication

Calculate the multiplication in the numerator:\[2 \times (-6) = -12\].

Step 5: Simplify the Fraction

Now, the problem becomes \(\frac{-12}{3}\). Simplify this by dividing the numerator by the denominator:\[\frac{-12}{3} = -4\].

Key Concepts

Understanding Division of Fractions

Dividing fractions might initially seem tricky, but it's just a matter of turning division into a simple multiplication problem. The key is to use the reciprocal. A reciprocal of a fraction is created by swapping its numerator and denominator. For instance, the reciprocal of \( \frac{1}{4} \) is \( 4 \). By multiplying a fraction by its reciprocal, you get 1.

In any division involving fractions, convert the division problem into multiplication by multiplying with the reciprocal of the fraction you are dividing by. This turns the complex task of division into something more straightforward: multiplication.

Real Numbers in Fraction Operations

Real numbers encompass all the numbers represented on the number line, including positive numbers, negative numbers, whole numbers, fractions, and decimals. When performing operations with fractions, you deal with real numbers. It's essential to handle these operations accurately since mixing operations might involve negative and positive real numbers.

In our original exercise, understanding that \( -\frac{1}{6} \) is a real number helps in realizing that when you work with it in fraction operations, it obeys all standard arithmetic rules. Remember, sign changes and basic arithmetic operations work the same way with real numbers, whether they're whole numbers or fractions.

The Role of Reciprocals

A reciprocal plays a pivotal role in dividing fractions. As earlier mentioned, it involves flipping the fraction. It offers a handy solution by transforming division into multiplication.

For example, to divide \( \frac{2}{3} \) by \( -\frac{1}{6} \), you first find the reciprocal of \( -\frac{1}{6} \) which is \( -6 \). This changes the division into a multiplication of \( \frac{2}{3} \times -6 \). Understanding how the reciprocal alters the operation allows you to easily solve division problems involving fractions.

Simplifying Fractions Effectively

Simplifying fractions serves to condense them into their most responsive form. It is done by dividing both the numerator and the denominator by their greatest common divisor. When fractions are simplified, they are easier to work with and understand.

In our example, after multiplying \( \frac{2}{3} \times -6 \), you get \( \frac{-12}{3} \). Simplifying this fraction involves dividing both the numerator and the denominator by 3, the greatest common divisor, resulting in \( -4 \). This final simplification step ensures the fraction represents the easiest and most intuitive numeric value.