Problem 37 - Elementary and Intermediate Algebra

Question

Multiply. $$ -\frac{5}{9} \cdot \frac{3}{4} $$

Step-by-Step Solution

Answer

\(\frac{-5}{12}\)

Step 1: Identify the fractions

The two fractions to be multiplied are \(-\frac{5}{9} \) and \(\frac{3}{4}\).

Step 2: Multiply the numerators

Multiply the numerators of the fractions: \(-5 \times 3 = -15\)

Step 3: Multiply the denominators

Multiply the denominators of the fractions: \(9 \times 4 = 36\)

Step 4: Combine the results

Write the product of the numerators over the product of the denominators: \(\frac{-15}{36}\)

Step 5: Simplify the fraction

Simplify \(\frac{-15}{36} \) by finding the greatest common divisor (GCD) of 15 and 36, which is 3. Divide both the numerator and the denominator by 3: \(\frac{-15 \div 3}{36 \div 3} = \frac{-5}{12}\)

Key Concepts

Numerator

In a fraction, the numerator is the top number. It tells us how many parts of a whole we are considering.
For example, in the fraction \(\frac{5}{9}\), 5 is the numerator.
When multiplying fractions, we start by multiplying the numerators together.
For instance, in our exercise, we have the fractions \(-\frac{5}{9}\) and \(\frac{3}{4}\). Multiplying the numerators, we get \(-5 \times 3 = -15\).

Denominator

The denominator is the bottom number in a fraction.
It shows the total number of equal parts the whole is divided into.
In the fraction \(\frac{3}{4}\), 4 is the denominator.
To multiply fractions, we multiply the denominators together.
For our example, with the fractions \(-\frac{5}{9}\) and \(\frac{3}{4}\), we multiply the denominators: \9 \times 4 = 36\.

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest form.
This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For instance, the fraction \(\frac{-15}{36}\) can be simplified.
We find the GCD of 15 and 36, which is 3. Dividing both the numerator and the denominator by 3, we get \(\frac{-5}{12}\).
Simplified fractions are easier to work with and understand.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder.
To simplify a fraction, finding the GCD is essential.
For the numbers 15 and 36, the GCD is 3.
We use the GCD to divide the numerator and denominator to get the simplest form of the fraction.
In our example, \(\frac{-15}{36}\) simplified becomes \(\frac{-5}{12}\) by dividing both by their GCD, 3.