Problem 64 - Prealgebra

Question

Find quotient. Write in simplest form. \(\frac{1}{6} \cdot\left(-\frac{8}{9}\right)\)

Step-by-Step Solution

Answer

The quotient in simplest form is \(-\frac{4}{27}\).

Step 1: Multiply the Numerators

Multiply the numerators of the two fractions: \(1\) and \(-8\). This gives us \(1 \times (-8) = -8\).

Step 2: Multiply the Denominators

Multiply the denominators of the two fractions: \(6\) and \(9\). This gives us \(6 \times 9 = 54\).

Step 3: Form the Resulting Fraction

Combine the results from the previous two steps to form the new fraction: \(\frac{-8}{54}\).

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{-8}{54}\) by finding the greatest common divisor (GCD) of \(8\) and \(54\), which is \(2\). Divide both the numerator and the denominator by \(2\) to get \(\frac{-4}{27}\).

Step 5: Write in Simplest Form

The simplified fraction is \(\frac{-4}{27}\), which is the quotient in its simplest form.

Key Concepts

Simplifying Fractions

Fraction simplification involves reducing a fraction to its simplest form. This process requires finding the greatest common divisor (GCD) of the numerator and the denominator. By dividing both the numerator and denominator by their GCD, you can simplify the fraction without changing its value.

• For instance, in the fraction \(\frac{-8}{54}\), we look for a number that divides both \(8\) (ignoring the sign for simplification purposes) and \(54\) without leaving a remainder.
• Here, the GCD is \(2\). Thus, \(\frac{-8}{54}\) can be simplified to \(\frac{-4}{27}\) by dividing both top and bottom by \(2\).

Simplifying fractions is useful for making them easier to read and compare. A fraction in its simplest form often reveals critical mathematical relationships and insights.

Negative Fractions

Dealing with negative fractions is not as daunting as it might seem. Negative fractions are simply fractions where either the numerator or the denominator is negative. This signifies a negative quantity or value.

• If there is a negative sign, it usually goes in front of the whole fraction, such as \(\frac{-8}{54}\).
• It's the same as \(-\frac{8}{54}\) or \(\frac{8}{-54}\), but traditionally we prefer writing it as \(-\frac{8}{54}\).

Remembering this can help maintain clarity in mathematical operations and ensure you're interpreting the values correctly, particularly when it comes to calculations and graphing.

Numerator and Denominator

In a fraction, the numerator is the top number, and the denominator is the bottom number. These components represent parts of a whole.

• The numerator signifies how many parts you have, while the denominator indicates the total number of equal parts the whole is divided into.
• In our example, \(\frac{-8}{54}\), \(-8\) is the numerator, showing the number of parts we are considering, some of which might be negative.
• The \(54\) is the denominator, revealing that the whole consists of \(54\) equal sections.

Understanding these roles is crucial for performing operations correctly, like finding common denominators or simplifying fractions to more easily understand or compare their values.

Greatest Common Divisor

The greatest common divisor (GCD), often referred to as the greatest common factor (GCF), is the largest number that divides two or more numbers without leaving a remainder. It's a fundamental concept for simplifying fractions.

• To simplify a fraction, like \(\frac{-8}{54}\), you find the GCD of \(8\) and \(54\), which is \(2\).
• You then divide both the numerator and the denominator by this GCD. So, dividing \(8\) and \(54\) by \(2\) gives \(\frac{-4}{27}\).
• This results in the simplest form of the fraction.

The GCD helps reduce fractions efficiently, allowing you to simplify complex calculations and understand equivalences between different fractions easily.