Problem 32 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

-\frac{10}{21}

Step 1: Identify the fractions

The given fractions are \(\frac{5}{7}\) and \(-\frac{2}{3}\).

Step 2: Multiply the numerators

Multiply the numerators of both fractions: \(5 \times (-2) = -10\).

Step 3: Multiply the denominators

Multiply the denominators of both fractions: \(7 \times 3 = 21\).

Step 4: Write the result as a single fraction

Place the product of the numerators over the product of the denominators: \(-\frac{10}{21}\).

Step 5: Simplify the fraction if possible

Check if the fraction \(-\frac{10}{21}\) can be simplified. Since 10 and 21 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Exploring Numerators

In fraction multiplication, the numerator represents the top part of each fraction.
For the exercise, our numerators were 5 and -2, found in the fractions \(\frac{5}{7}\) and \(-\frac{2}{3}\).
To multiply fractions, start by multiplying these numerators together.
Here's what we did: 5 \(\times\) -2 = -10.

Remember, any numerical value at the top (above the line) in a fraction is the numerator.
Understanding this helps in keeping track while performing calculations like multiplying or simplifying fractions.

Understanding Denominators

The denominator is the bottom part of a fraction.
In our exercise, the denominators were 7 and 3 in the fractions \(\frac{5}{7}\) and \(-\frac{2}{3}\).
When multiplying fractions, it's crucial to multiply these denominators.

Here's what we did: 7 \(\times\) 3 = 21.

It's important to note that when you're multiplying fractions, you multiply the numerators together and the denominators together.
This keeps the process organized and ensures that the result is a new fraction where the numerator is the product of the original numerators and the denominator is the product of the original denominators.

Simplifying Fractions

After multiplying fractions, the next step is to simplify the resulting fraction if possible.
Simplifying means making a fraction as simple as possible—essentially, reducing it.

In our example, we ended up with the fraction \(-\frac{-10}{21}\).
To check if we can simplify, we need to see if there are common factors between the numerator and the denominator.
If they share a common factor other than 1, we divide by that common factor to reduce the fraction.
Since 10 and 21 don't have common factors other than 1, our fraction \(-\frac{10}{21}\) is already in its simplest form.