Problem 88 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{1}{3}\left(-\frac{1}{5}\right)\left(-\frac{1}{7}\right) $$

Step-by-Step Solution

Answer

\( \frac{1}{105} \)

Step 1: Start by Handling the Signs

The expression given is \( \frac{1}{3} \left(-\frac{1}{5}\right) \left(-\frac{1}{7}\right) \). First, consider the signs of the fractions. A negative multiplied by a negative results in a positive, so \( \left(-\frac{1}{5}\right) \times \left(-\frac{1}{7}\right) \) will be positive.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions. We have: \( 1 \times 1 \times 1 = 1 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \( 3 \times 5 \times 7 = 105 \).

Step 4: Write the Final Fraction

The result of multiplying the given fractions is \( \frac{1}{105} \). No further simplification is needed as 1 is the numerator.

Key Concepts

Understanding Numerators

In every fraction, the numerator is the top number, which represents the number of equal parts being considered. For example, in the fraction \( \frac{1}{3} \), 1 is the numerator.
It's important to note that the numerator tells us about the parts of the whole that we have or are discussing.
When multiplying fractions, as seen in the exercise \( \frac{1}{3} \cdot \left(-\frac{1}{5}\right) \cdot \left(-\frac{1}{7}\right) \), focus on multiplying the numerators together. Since our numerators were all 1, the multiplication step is easy:

• First numerator = 1
• Second numerator = 1
• Third numerator = 1
• Total = 1 \( \times \) 1 \( \times \) 1 = 1

The main takeaway is that multiplying numerators is straightforward, especially when the numbers are small, as they are here.

Grasping the Role of Denominators

The denominator is the bottom number of a fraction and tells us into how many equal parts the whole is divided. In the fraction \( \frac{1}{3} \), the denominator is 3.
When multiplying fractions, the denominators need to be multiplied together just like the numerators.
In our exercise, this involved calculating:

• First denominator = 3
• Second denominator = 5
• Third denominator = 7
• Total = 3 \( \times \) 5 \( \times \) 7 = 105

This process means that the resulting fraction \( \frac{1}{105} \) has a denominator of 105, reflecting how the whole is divided into many tiny parts.

Navigating Negative Signs

In multiplication, negative signs can sometimes be tricky but follow simple rules. When you multiply a negative number by another negative number, the product is positive.
In the exercise \( \frac{1}{3} \cdot \left(-\frac{1}{5}\right) \cdot \left(-\frac{1}{7}\right) \):

• \(-\frac{1}{5}\) is a negative fraction.
• \(-\frac{1}{7}\) is also negative.

Multiplying these two gives a positive fraction, \( \frac{1}{35} \), before any further operations. When dealing with negatives, always remember:

• Negative \( \times \) Negative equals Positive
• Negative \( \times \) Positive equals Negative
• Positive \( \times \) Positive equals Positive

This simple rule helps keep track of signs during multiplication.

Simplification In Fractions

Simplification means reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This involves dividing them by their greatest common divisor (GCD).
In our question, the product was \( \frac{1}{105} \). Since 1 is the numerator, this fraction is already in its simplest form because no other factor besides 1 divides evenly into 1 or 105.
Simplification is essential for efficient computation and for making comparisons between fractions easier.
However, always check if the numerator can be reduced further with the denominator to ensure simplicity.