Problem 37 - Intermediate Algebra

Question

Perform the operations. See Example 3 . $$ \left(-\frac{3}{5}\right)\left(\frac{10}{7}\right) $$

Step-by-Step Solution

Answer

The result of \((-\frac{3}{5})(\frac{10}{7})\) is \(-\frac{6}{7}\).

Step 1: Identify the Numbers

We have two fractions, \(-\frac{3}{5}\) and \(\frac{10}{7}\), that we need to multiply together.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \(-3\) (from \(-\frac{3}{5}\)) and \(10\) (from \(\frac{10}{7}\)).\[(-3) \times 10 = -30\]

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \(5\) (from \(-\frac{3}{5}\)) and \(7\) (from \(\frac{10}{7}\)).\[5 \times 7 = 35\]

Step 4: Form the New Fraction

Combine the results from Steps 2 and 3 to form the new fraction: \[\frac{-30}{35}\]

Step 5: Simplify the Fraction

Identify the greatest common divisor (GCD) of \(-30\) and \(35\), which is \(5\). Divide both the numerator and the denominator by \(5\).\[\frac{-30 \div 5}{35 \div 5} = \frac{-6}{7}\]The simplified result is \(-\frac{6}{7}\).

Key Concepts

Understanding the Numerator

In the world of fractions, the term "numerator" refers to the top number. In a fraction like \(\frac{a}{b}\), \(a\) is the numerator. The role of the numerator is crucial because it tells you how many parts of the whole you are considering. For example, in the fraction \(-\frac{3}{5}\), the numerator is \(-3\). This indicates that we are looking at \(-3\) parts of something divided into 5 equal parts. The negative sign in the numerator means that the value of the fraction is negative when applied to real-world quantities, such as owing 3 out of 5 dollars.When multiplying fractions, you multiply the numerators together. In this exercise, when you multiply the numerators of \(-\frac{3}{5}\) and \(\frac{10}{7}\), you calculate \((-3) \times 10 = -30\). This new numerator tells us how the quantity relates to the product of the fractions. The larger the absolute value of the numerator, the greater the number of parts you're dealing with, as seen in our result from the multiplication.

Decoding the Denominator

The denominator, found beneath the line in a fraction, is a key element that shows us how many total equal parts there are. In the fraction \(\frac{a}{b}\), \(b\) is the denominator. For instance, in \(\frac{10}{7}\), the denominator is \(7\), indicating the whole is divided into 7 equal parts. Understanding the denominator helps us grasp the scale or size of each part of the whole.When multiplying fractions, it’s straightforward: you multiply the denominators of both fractions. For our example, the denominators are \(5\) and \(7\). Multiplying these gives us \(35\), resulting in the denominator for the product of the fractions: \(\frac{-30}{35}\). This means in our final fraction, we are still dealing with parts of a whole divided into 35 parts, but now related to the newly calculated numerator. It's an essential step that ensures the fraction remains in proportion as we manipulate it with multiplication.

Greatest Common Divisor Unveiled

Simplifying fractions involves one crucial step: finding the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest integer that can completely divide both numbers. Simplifying a fraction makes it easier to understand and use, especially when you want to compare or add fractions.In the original fraction \(\frac{-30}{35}\), one needs to identify the GCD to simplify it. Here, the GCD of \(-30\) and \(35\) is \(5\). Both \(-30\) and \(35\) can be divided by \(5\), leading to a simplified fraction. Divide \(-30\) by \(5\) to get \(-6\), and \(35\) by \(5\) to get \(7\), resulting in the simplified fraction \(-\frac{6}{7}\).By simplifying the fraction, using the GCD, you create a version of the fraction that is more intuitive to interpret and easier to work with in future calculations. This step concludes the multiplication and ensures clarity and precision in fractional mathematics.