Problem 102 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -1 \frac{1}{8}\left(-\frac{3}{8}\right) $$

Step-by-Step Solution

Answer

The result of the operation is \(\frac{27}{64}\).

Step 1: Convert Mixed to Improper Fraction

First, convert the mixed number -1 \frac{1}{8} to an improper fraction. Multiply the whole number by the denominator and add the numerator: \(1 \times 8 + 1 = 9\). Since it's a negative mixed number, it becomes \(-\frac{9}{8}\).

Step 2: Multiply the Fractions

Now multiply the improper fraction \(-\frac{9}{8}\) by the other fraction \(-\frac{3}{8}\). When multiplying fractions, multiply the numerators and the denominators: \[ (-9) \times (-3) = 27 \] \[ 8 \times 8 = 64 \] So the product is \(\frac{27}{64}\).

Step 3: Simplify the Fraction (if necessary)

Check if the fraction \(\frac{27}{64}\) can be simplified. Both 27 and 64 share no common factors other than 1, so \(\frac{27}{64}\) is already in its simplest form.

Key Concepts

Improper Fractions

In mathematics, improper fractions are fractions where the numerator is greater than or equal to the denominator. For example, in the fraction \(-\frac{9}{8}\), 9 is the numerator and 8 is the denominator, making it an improper fraction because 9 is greater than 8. Converting a mixed number like \(-1 \frac{1}{8}\) into an improper fraction involves a few simple steps:

• Multiply the whole number by the denominator: in this case, \(1 \times 8 = 8\).
• Add the numerator: \(8 + 1 = 9\).
• Place the result over the original denominator, resulting in \(-\frac{9}{8}\).

The importance of improper fractions lies in their ability to simplify complex calculations, especially when multiplying or dividing fractions. They provide a uniform way to handle fractions without dealing with whole numbers, streamlining calculations and reducing mistakes.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This process makes fractions easier to work with and understand. For instance, consider the fraction \(\frac{27}{64}\). To check if it can be simplified, determine if there are any common factors between 27 and 64:

• List the factors of 27 (1, 3, 9, 27) and 64 (1, 2, 4, 8, 16, 32, 64).
• The only common factor is 1.

Since 27 and 64 have no common factors apart from 1, the fraction is already simplified. It’s crucial to simplify fractions in your final answer to convey the simplest form of the result and to keep mathematical expressions clear and concise.

Mixed Numbers

Mixed numbers consist of a whole number and a fractional part, like \(-1 \frac{1}{8}\). They are often used in everyday situations where quantities might not be whole, such as recipe measurements or dividing tasks. Mixed numbers can be a bit cumbersome in mathematical operations, which is why they are often converted to improper fractions for calculations. Here's how to understand and work with mixed numbers effectively:

• The whole number provides a clear quantity, making it intuitive for representing measures.
• The fractional part adds the precision needed for exact measurements.
• When converting to improper fractions, the goal is to apply a consistent method for simplifying calculations across various mathematical operations.

Understanding mixed numbers is key to navigating both simple and complex problems requiring fraction multiplication, ensuring accurate results and maintaining clarity throughout your calculations.