Problem 15 - Introductory Algebra for College Students

Question

perform the indicated multiplication. $$-\frac{3}{5} \cdot\left(-\frac{4}{7}\right)$$

Step-by-Step Solution

Answer

The result of the multiplication is \( \frac{12}{35} \).

Step 1: Understand the multiplication of fractions

When multiplying fractions, one simply multiplies the numerators (the top numbers) together to get the new numerator, and does the same with the denominators (the bottom numbers) for the new denominator.

Step 2: Perform the operation

Multiply the numerators: \( -3 \times -4 = 12 \). Next, multiply the denominators: \( 5 \times 7 = 35 \). The fraction is thus \( \frac{12}{35} \).

Step 3: Consider the sign rule

Since a negative number times a negative number yields a positive number, the answer is positive. Hence, the answer remains \( \frac{12}{35} \).

Key Concepts

Numerators and Denominators

In the world of fractions, numerators and denominators play the leading roles. Simply put, the numerator is the number above the fraction bar, indicating how many parts we have. The denominator, sitting below the fraction bar, tells us into how many equal parts the whole is divided.

When multiplying fractions, the key is to understand that you multiply the numerators together to find the numerator of the product. Then, you do the same with the denominators to find the denominator of the product. For instance, with \(-\frac{3}{5} \cdot \left(-\frac{4}{7}\right)\), you'd multiply -3 and -4 to get the new numerator, and 5 and 7 for the new denominator.

To visualize it:

• Numerator (top number): -3 \(\times\) -4 = 12.
• Denominator (bottom number): 5 \(\times\) 7 = 35.

Thus the fraction formed by the multiplication process would be \(\frac{12}{35}\).

Multiplication of Negative Numbers

Negative numbers might seem tricky at first, but there's one golden rule that makes multiplication with them a breeze: a negative times a negative is a positive. This might sound counterintuitive, but it's a fundamental principle in mathematics that ensures consistency across various mathematical operations.

When you come across a multiplication like \(-3 \times -4\), both numbers are negative. According to the rule, their product will be positive. So, \(-3 \times -4 = 12\), not -12. Remembering this rule is crucial, especially when dealing with fractions that have negative numerators, denominators, or both.

Applying this to fraction multiplication, like in our example \(-\frac{3}{5} \cdot \left(-\frac{4}{7}\right)\), you'll notice that both fractions have a negative sign. Hence, the resulting fraction after multiplication will always have a positive numerator and denominator due to the multiplication of the negative numbers.

Simplifying Fractions

Once you've multiplied the numerators and denominators, the next step is often simplifying the fraction. This step is about making the fraction as simple as possible by removing any common factors that the numerator and denominator share.

To simplify a fraction, you look for the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both without leaving a remainder. You then divide both the numerator and the denominator by the GCD to get the simplified fraction.

Let's look at our fraction \(\frac{12}{35}\). The numbers 12 and 35 don't have a common factor other than 1, so this fraction is already in its simplest form. If our numerator and denominator had a common factor (like if we had \(\frac{14}{28}\)), we'd divide both by 14 to simplify the fraction to \(\frac{1}{2}\).