Problem 4 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\left(\frac{5}{6}\right)\left(-\frac{4}{15}\right)$$

Step-by-Step Solution

Answer

\(-\frac{2}{9}\)

Step 1: Multiply the Numerators

To multiply the fractions, first multiply the numerators together. The numerators are 5 and -4.\(5 \times (-4) = -20\).

Step 2: Multiply the Denominators

Next, multiply the denominators together. The denominators are 6 and 15.\(6 \times 15 = 90\).

Step 3: Form the Fraction

Combine the results from the previous steps to form a fraction. The fraction is:\(-\frac{20}{90}\).

Step 4: Simplify the Fraction

To simplify the fraction, find the greatest common divisor (GCD) of 20 and 90, which is 10. Divide both the numerator and the denominator by 10:\(-\frac{20 \div 10}{90 \div 10} = -\frac{2}{9}\).

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing them to the smallest possible equivalent fraction. This involves finding an equivalent fraction where the numerator and the denominator are as small as possible. This process makes fractions easier to understand and compare.

• To simplify a fraction, you want to divide the numerator and the denominator by their greatest common divisor (GCD).
• A simplified fraction has no common factors, other than 1, between the numerator and the denominator.

Let's illustrate it with an example: Suppose you have the fraction \(-\frac{20}{90}\). You will need first to find the GCD of 20 and 90, which is 10, and then divide both parts of the fraction by 10:\[-\frac{20 \div 10}{90 \div 10} = -\frac{2}{9}\].This is how this fraction becomes simplified. The result is a fraction where both the numerator and the denominator are smaller and have no divisors in common other than 1.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. It's essential for simplifying fractions.

• The GCD helps to find the smallest form of a fraction by showing which numbers you can divide both parts of the fraction by.
• Finding the GCD can be done by listing out the divisors of both numbers and finding the largest one common to each.

For the fraction \(-\frac{20}{90}\), the divisors of 20 are 1, 2, 4, 5, 10, 20, and those of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. The greatest common divisor here is 10. Finding the GCD allows you to divide both the numerator and the denominator by this number, thus simplifying your fraction efficiently.

Numerator and Denominator Manipulation

In the process of multiplying fractions, both the numerator and the denominator come into play. Understanding how to manipulate them is crucial for successful fraction multiplication and simplification.

• When you multiply fractions, you multiply the numerators with each other and the denominators with each other.
• It is important to pay attention to negative signs, as in with multiplying any numbers, to determine the sign of the resulting fraction.

For example, if you multiply \(\frac{5}{6}\) by \(-\frac{4}{15}\), multiply the numerators together: \(5 \times (-4) = -20\), and then the denominators: \(6 \times 15 = 90\). This results in \(-\frac{20}{90}\). After finding the GCD and simplifying, remember to divide both parts of the fraction to reduce it to \(-\frac{2}{9}\). Numerator and denominator manipulation is at the heart of fraction multiplication and simplification.