Problem 69 - Intermediate Algebra

Question

Simplify each numerical expression. $$ \frac{2}{5}\left(-\frac{3}{4}\right)-\left(-\frac{1}{2}\right)\left(\frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Distribute the Fraction

First, we need to simplify the expression by distributing the fractions. We have two terms: \( \frac{2}{5} \times \left(-\frac{3}{4}\right) \) and \( -\left( -\frac{1}{2} \right) \times \frac{3}{5} \). Calculate the multiplication for each term separately.

Step 2: Calculate the First Term

Calculate \( \frac{2}{5} \times \left(-\frac{3}{4}\right) \). Multiply the numerators and the denominators: \[ \frac{2 \times (-3)}{5 \times 4} = \frac{-6}{20}. \] Simplify this by dividing both the numerator and denominator by 2: \( \frac{-6 \div 2}{20 \div 2} = \frac{-3}{10} \).

Step 3: Calculate the Second Term

Calculate \( -\left( -\frac{1}{2} \right) \times \frac{3}{5} \). The double negative becomes positive, so the expression becomes \( \frac{1}{2} \times \frac{3}{5} \). Multiply the numerators and denominators: \[ \frac{1 \times 3}{2 \times 5} = \frac{3}{10}. \]

Step 4: Combine the Results

Now let's combine the results from the two terms. We have \( \frac{-3}{10} \) from Step 2 and \( \frac{3}{10} \) from Step 3. Add these fractions: \( \frac{-3}{10} + \frac{3}{10} = 0 \).

Key Concepts

Numerical Expressions

Numerical expressions include numbers and operations, such as addition, subtraction, multiplication, and division. They are like math sentences without an equal sign. In the given exercise, we have a combination of fractions and arithmetic operations. Understanding how to simplify these expressions involves evaluating parts separately and then combining them once simplified. This often requires applying the rules of arithmetic operations orderly, especially multiplication and addition or subtraction of fractions.

Here, we look at the structure: within the expression, we have a subtraction of two products of fractions. Execution begins with resolving the multiplications first before dealing with addition or subtraction. It's important to remember:

• Always perform operations inside parentheses first.
• Apply the distributive property where it’s needed in expressions involving fractions.
• Any negative signs should be treated carefully during multiplication or addition and subtraction steps.

Fraction Multiplication

Fraction multiplication is straightforward: multiply the numerators together for a new numerator and the denominators together for a new denominator. For instance, when simplifying the given expression, you initially deal with two separate fraction multiplications.

To successfully multiply fractions:

• Identify the numerators and denominators of both fractions.
• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.
• Check if the result can be simplified by examining common factors.

For example, in the expression \( \frac{2}{5} \times \left(-\frac{3}{4}\right) \), the numerators \( (2 \times -3) \) give \(-6\) and the denominators \( (5 \times 4) \) give \(20\), resulting in \( \frac{-6}{20} \), which can then be simplified further.

Simplification Techniques

Simplifying fractions involves reducing them to their simplest form. This makes them easier to understand and work with. When you have a fraction, you check to see if both the numerator and the denominator share any common factors.

The simplification process is as follows:

• Look for the highest common factor (HCF) of the numerator and denominator.
• Divide both the numerator and the denominator by this factor to reduce the fraction.
• If no common factors other than 1 exist, the fraction is already in its simplest form.

In our exercise example, \( \frac{-6}{20} \) can be simplified by dividing both the numerator and denominator by their HCF, which is 2, resulting in \( \frac{-3}{10} \). Similarly, \( \frac{3}{10} \) is already in its simplest form since 3 and 10 have no common factors other than 1. Simplified fractions make combining and comparing results much easier.