Problem 42 - Elementary and Intermediate Algebra

Question

Multiply. $$ -\frac{1}{2} \cdot \frac{3}{5} \cdot\left(-\frac{2}{7}\right) $$

Step-by-Step Solution

Answer

\(\frac{3}{35}\)

Step 1 - Multiply the Numerators

To begin, multiply the numerators of the fractions: \[-1 \times 3 \times -2 = 6\]

Step 2 - Multiply the Denominators

Next, multiply the denominators of the fractions: \[2 \times 5 \times 7 = 70\]

Step 3 - Combine the Results

Now combine the results from Steps 1 and 2 to form a new fraction: \[\frac{6}{70}\]

Step 4 - Simplify the Fraction

Finally, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2: \[\frac{6 \div 2}{70 \div 2} = \frac{3}{35}\]

Key Concepts

Numerators

When multiplying fractions, the numerators play a crucial role. The numerator is the top number of a fraction, like the '1' in \(-\frac{1}{2}\) or the '3' in \(\frac{3}{5}\). These numbers tell us how many parts of a whole are being considered. When you multiply fractions, you multiply the numerators together. For the exercise, the numerators are \(-1\), \(3\), and \(-2\). So, multiplying them gives: \[-1 \times 3 \times -2 = 6\]. It's important to keep track of the signs (positive or negative) when doing this multiplication. Remember, multiplying two negative numbers results in a positive number.

Denominators

The denominator is the bottom part of a fraction and indicates the total number of equal parts. For instance, in \(-\frac{1}{2}\), the denominator is '2'. When multiplying fractions, you also multiply their denominators together. In our exercise, the denominators are 2, 5, and 7. So, multiplying them gives: \[2 \times 5 \times 7 = 70\]. This step is usually straightforward, but make sure to perform the multiplication carefully to avoid mistakes. The resulting denominator helps form the initial product in fraction form.

Simplifying Fractions

Once you have your new fraction, it's time to simplify. Simplifying makes the fraction easier to understand by reducing it to its smallest form. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD). In our exercise, the fraction after multiplication is \[\frac{6}{70}\]. The GCD of 6 and 70 is 2. So, we divide both the numerator and the denominator by 2: \[\frac{6 \div 2}{70 \div 2} = \frac{3}{35}\]. This gives us a simpler, equivalent fraction. Simplifying helps make fractions more manageable and easier to work with in further calculations.