Problem 119 - Prealgebra

Question

Perform the indicated operation. $$-\frac{5}{4}\left(\frac{8}{15}\right)$$

Step-by-Step Solution

Answer

The result is \(-\frac{2}{3}\).

Step 1: Identify the Operation

We need to multiply a negative fraction, \(-\frac{5}{4}\), by another fraction, \(\frac{8}{15}\). The operation is multiplication between two fractions.

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions together. For \(-\frac{5}{4}\) and \(\frac{8}{15}\), this means multiplying \(-5\) by \(8\), which equals \(-40\).

Step 3: Multiply the Denominators

Multiply the denominators of the two fractions. For \(-\frac{5}{4}\) and \(\frac{8}{15}\), this means multiplying \(4\) by \(15\), which equals \(60\).

Step 4: Combine the Results

Combine the products from Steps 2 and 3 into a single fraction: \(\frac{-40}{60}\).

Step 5: Simplify the Fraction

Simplify the fraction \(\frac{-40}{60}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 20. This gives \(\frac{-2}{3}\).

Key Concepts

Negative Fractions: Understanding the Basics

Negative fractions can seem a bit tricky at first, but they're just like regular fractions with a little twist. A negative fraction simply has a negative sign in front of it, indicating a value less than zero.
This means that \(-\frac{5}{4}\) is read as negative five-fourths, which is the same as \(-1.25\) in decimal form.

• Negative fractions can be the result of dividing a negative number by a positive one, a positive number by a negative one, or a negative by another negative.
• When multiplying fractions, it’s important to carry over the negative sign. So, \(-\frac{5}{4} \times \frac{8}{15}\) becomes \(\frac{-40}{60}\) after multiplying the numerators and denominators.
• If both the numerator and the denominator have a negative sign, the fraction becomes positive. But in this exercise, only the numerator was negative, resulting in a negative fraction.

Negativity in fractions functions just like negativity in whole numbers. If you multiply or divide by another negative, the negative signs would cancel out. This is a crucial point to remember as you work through fraction problems.

Simplifying Fractions: Making Math Simple

Simplifying fractions means reducing them to their simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor. This reduction makes fractions easier to understand at a glance.
In our exercise, we simplified \(-\frac{40}{60}\) to \(-\frac{2}{3}\).

• To simplify a fraction, find a common factor that divides both the numerator and the denominator equally.
• Continue dividing until you can no longer use a common factor other than 1.
• A fraction is in its simplest form when the only common factor between its numerator and denominator is 1.

Finding the simplest form helps make mathematical operations cleaner and often reveals equivalent relations with other fractions or numbers.

Greatest Common Divisor: The Key to Simplification

The greatest common divisor (GCD) is a very useful tool for simplifying fractions. It refers to the largest number that can equally divide both the numerator and the denominator. Using the GCD, you can quickly and efficiently reduce a fraction to its simplest form.
Let's explore how to find the GCD:

• List all divisors of the numerator and the denominator.
• Find the largest number present in both lists. That number is the GCD.
• Using the GCD, divide both the numerator and the denominator to get the simplified fraction.

For \(-\frac{40}{60}\), the GCD is 20, and when both the numerator and the denominator are divided by 20, the resulting fraction is \(-\frac{2}{3}\). Remembering to find the GCD is a key step in fraction problems, helping you work efficiently and achieve clearer results.