Problem 120 - Prealgebra

Question

Perform the indicated operation. $$-\frac{4}{3}\left(\frac{6}{5}\right)$$

Step-by-Step Solution

Answer

The result is \(-\frac{8}{5}\).

Step 1: Understand the Operation

The exercise requires us to multiply two fractions: \(-\frac{4}{3}\) and \(\frac{6}{5}\). Multiply the numerators together and the denominators together.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \(-4\) from \(-\frac{4}{3}\) and \(6\) from \(\frac{6}{5}\). \[-4 \times 6 = -24\]

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \(3\) from \(-\frac{4}{3}\) and \(5\) from \(\frac{6}{5}\). \[3 \times 5 = 15\]

Step 4: Form the New Fraction

Combine the results from Steps 2 and 3 to form a new fraction: \[-\frac{24}{15}\]

Step 5: Simplify the Fraction

Simplify the fraction \(-\frac{24}{15}\) by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 24 and 15 is 3. Divide both the numerator and the denominator by 3: \[-\frac{24 \div 3}{15 \div 3} = -\frac{8}{5}\]

Key Concepts

Fraction Simplification

Simplifying fractions means reducing them to their simplest form possible. This involves dividing both the numerator and the denominator by a common factor.
When simplifying a fraction, it's essential to find the greatest common factor that both the numerator and the denominator share.

• Take the fraction \(-\frac{24}{15}\).
• Identify common factors of 24 and 15.
• The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.
• The factors of 15 are: 1, 3, 5, and 15.

The greatest common factor is 3, which divides both numbers evenly. By dividing both the numerator and the denominator by their GCD, we obtain a fraction in its simplest form.
After simplification, \(-\frac{24}{15}\) reduces to \(-\frac{8}{5}\). This fraction is now simpler but still holds the same value as before.

Negative Fractions

When we encounter negative fractions like \(-\frac{4}{3}\) in multiplication or any arithmetic operation, it's crucial to handle the sign properly. A negative fraction implies that either
the numerator or the denominator is negative, but not both at the same time.

• If the negative sign is in front of the fraction, it means the whole fraction is negative.
• For example, \(-\frac{4}{3}\) is equivalent to \(\frac{-4}{3}\) as well as \(\frac{4}{-3}\).

Handling negative fractions in multiplication is straightforward: * If you multiply a negative fraction with a positive one, you will end up with a negative result.
* In contrast, multiplying two negative fractions results in a positive outcome.The original exercise of multiplying \(-\frac{4}{3}\) by \(\frac{6}{5}\) results in the negative fraction \(-\frac{24}{15}\) because one of the fractions was negative.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a vital concept in simplifying fractions. It refers to the largest number that can divide two integers without leaving a remainder.
To determine the GCD of any two numbers, like the 24 and 15 in our fraction, follow these steps:

• List the factors of the first number (24). These are: 1, 2, 3, 4, 6, 8, 12, 24.
• List the factors of the second number (15). These are: 1, 3, 5, 15.
• Identify the greatest factor they have in common.

For 24 and 15, the greatest common factor is 3. Once determined, it allows us to divide both the numerator and denominator by 3, reducing the fraction to its simplest form.
Knowing how to find the GCD is essential as it streamlines many mathematical processes that involve fractions, ensuring they are easy to interpret and work with.