Problem 82 - Beginning and Intermediate Algebra

Question

Divide. $$ -\frac{1}{10} \div\left(-\frac{8}{11}\right) $$

Step-by-Step Solution

Answer

The result of the division is \(\frac{11}{80}\).

Step 1: Understand the Problem

The exercise asks us to divide two fractions: \(-\frac{1}{10}\) and \(-\frac{8}{11}\). Dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Identify the Reciprocal of the Divisor

The divisor in this problem is \(-\frac{8}{11}\). The reciprocal of \(-\frac{8}{11}\) is \(-\frac{11}{8}\).

Step 3: Change Division to Multiplication

Convert the division problem into a multiplication problem using the reciprocal. So, \(-\frac{1}{10} \div (-\frac{8}{11})\) becomes \(-\frac{1}{10} \times (-\frac{11}{8})\).

Step 4: Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together: \[(-1) \times (-11) = 11 \]\[10 \times 8 = 80\]So, the result of \(-\frac{1}{10} \times (-\frac{11}{8})\) is \(\frac{11}{80}\).

Step 5: Simplify if Necessary

Check if the resulting fraction \(\frac{11}{80}\) can be simplified. Since 11 is a prime number and doesn't divide 80, it is already in its simplest form.

Key Concepts

Reciprocal of Fractions

When working with fractions, knowing how to find and use the reciprocal is crucial. The "reciprocal" of a fraction is simply created by swapping its numerator and its denominator. This effectively flips the fraction upside down. For example, the reciprocal of \(-\frac{8}{11}\) is \(-\frac{11}{8}\).

Utilizing the reciprocal comes into play when we are dealing with division of fractions. Instead of dividing by a fraction, multiplying by its reciprocal allows us to solve the problem much more easily. Think of it like a shortcut that turns the division into a multiplication problem.

Remember these points:

• The reciprocal of a positive fraction is positive.
• The reciprocal of a negative fraction remains negative, because flipping the numerator and denominator doesn't change the sign.
• Finding the reciprocal is an essential step in dividing fractions.

Multiplying Fractions

Once we have converted a division problem into a multiplication problem using the reciprocal, the next step is to multiply fractions. Multiplying fractions is straightforward and involves two simple steps:

1. Multiply the numerators (the top numbers) to get the new numerator.
2. Multiply the denominators (the bottom numbers) to get the new denominator.

Let's look at an example: to multiply \(-\frac{1}{10}\) by \(-\frac{11}{8}\), you multiply the numerators \(-1\) and \(-11\) which equals \(11\), and the denominators \(10\) and \(8\) which equals \(80\).

The product is \(\frac{11}{80}\). A handy tip is to keep track of signs:

• Multiplying two negative numbers results in a positive product.
• Multiplying a positive and a negative number results in a negative product.

So, be sure to simplify if possible, but in our example, \(\frac{11}{80}\) is already in its simplest form.

Simplifying Fractions

After multiplying fractions, the final step is often to simplify the fraction, which means making it as simple as possible. Simplifying a fraction involves reducing it to its smallest possible terms.

This can be done if the numerator and denominator share a common factor. In the example \(\frac{11}{80}\), both the numerator and the denominator do not have any common factors other than 1. Therefore, the fraction is already in its simplest form.

Here are some steps to follow for simplifying fractions:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and denominator by this GCF.
• If the GCF is 1, then the fraction is already as simple as it can get.

In situations where no further reduction is possible, your work on simplifying is complete.