Problem 40 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \left(\frac{1}{4}\right)\left(-\frac{8}{9}\right) $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{-2}{9}\).

Step 1: Multiply the Numerators

Identify the numerators of the fractions: 1 and -8. Multiply these numerators together: \(1 \times -8 = -8\).

Step 2: Multiply the Denominators

Identify the denominators of the fractions: 4 and 9. Multiply these denominators together: \(4 \times 9 = 36\).

Step 3: Simplify the Fraction

Put the result of the numerators over the result of the denominators to form a new fraction: \(\frac{-8}{36}\). Simplify this fraction by finding the greatest common divisor (GCD) of 8 and 36, which is 4, and dividing both the numerator and denominator by it: \(\frac{-8 \div 4}{36 \div 4} = \frac{-2}{9}\).

Key Concepts

Numerators

The numerator is the top part of a fraction. It tells you how many parts of a whole you have. In the fraction \(\frac{1}{4}\), the numerator is 1. When multiplying fractions, you first multiply the numerators together. In our example, you multiply 1 by -8. This gives you \(-8\) because multiplying a positive number by a negative number results in a negative product. Understanding numerators is key to working with fractions.

Denominators

The denominator is the bottom part of a fraction. It shows the total number of equal parts the whole is divided into. For instance, in the fraction \(\frac{8}{9}\), the denominator is 9. When multiplying fractions, you also multiply the denominators together. In our exercise, we multiply 4 by 9, which gives us 36. Denominators help us understand the size of the parts we are dealing with in fractions.

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest form where the numerator and the denominator are as small as possible. In the fraction \(\frac{-8}{36}\), we can simplify by finding a common factor. Both -8 and 36 can be divided by 4. When we divide the numerator and the denominator by 4, we get the simplified fraction \(\frac{-2}{9}\). Simplifying makes fractions easier to understand and work with.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that divides both the numerator and the denominator without leaving a remainder. To simplify \(\frac{-8}{36}\), we find the GCD of 8 and 36. The GCD is 4, as it's the largest number that fits into both 8 and 36 evenly. By dividing both parts of the fraction by the GCD, you effectively simplify the fraction. This makes the fraction equivalent, but easier to read or use in further calculations.