Problem 66 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{7}{12}\).

Step 1: Multiply the First Pair of Fractions

Start by simplifying the expression \(-\frac{2}{3} \times \frac{1}{4}\). Multiply the numerators and the denominators: Numerator: \(-2 \times 1 = -2\)Denominator: \(3 \times 4 = 12\). Thus, the result is \(-\frac{2}{12}\), which simplifies to \(-\frac{1}{6}\) by dividing the numerator and the denominator by 2.

Step 2: Multiply the Second Pair of Fractions

Next, simplify the expression \(-\frac{1}{3} \times \frac{5}{4}\).Multiply the numerators and the denominators: Numerator: \(-1 \times 5 = -5\)Denominator: \(3 \times 4 = 12\). Thus, the result is \(-\frac{5}{12}\).

Step 3: Add the Results of the Products

Combine the results from Steps 1 and 2 by adding them together:\(-\frac{1}{6} + -\frac{5}{12}\).To add these fractions, find a common denominator. The least common denominator of 6 and 12 is 12.Convert \(-\frac{1}{6}\) to \(-\frac{2}{12}\), so:\(-\frac{2}{12} + -\frac{5}{12} = -\frac{7}{12}\).

Step 4: Simplify the Final Result

The expression \(-\frac{7}{12}\) is already in its simplest form since 7 is a prime number and cannot divide 12 without a remainder.

Key Concepts

Numerical Expressions

Numerical expressions use numbers and operations to represent a specific quantity or value. In our exercise, the given expression is \(-\frac{2}{3}\left(\frac{1}{4}\right)+\left(-\frac{1}{3}\right)\left(\frac{5}{4}\right)\). This expression involves fractions and multiple operations like multiplication and addition. To simplify numerical expressions, the order of operations must be followed. This order is usually represented by PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). For our problem, we start with simplifying the fractions through multiplication before moving to addition. Understanding numerical expressions is key as they are foundational for algebra and many other areas of mathematics.

Least Common Denominator

When adding or subtracting fractions, it is essential to have a common denominator for the fractions involved. The Least Common Denominator (LCD) is the smallest number that can be a common denominator for a given set of fractions. Finding the LCD allows us to add or subtract fractions more easily.For example, in the expression \(-\frac{1}{6} + -\frac{5}{12}\), we first identify that the denominators are 6 and 12.

• The multiples of 6 are 6, 12, 18, etc.
• The multiples of 12 are 12, 24, 36, etc.

The smallest common value is 12, hence it is the LCD. This allows us to convert fractions into a form that makes addition straightforward. The LCD simplifies complex fraction operations, making calculations easier and the results more accurate.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together for the new numerator and the denominators together for the new denominator. It is straightforward compared to other fraction operations.In our problem, we first address \(-\frac{2}{3} \times \frac{1}{4}\):

• Multiply the numerators: \(-2 \times 1 = -2\)
• Multiply the denominators: \(3 \times 4 = 12\)
• This results in \(-\frac{2}{12}\)

To simplify: divide both the numerator and denominator by their greatest common divisor, which is 2, resulting in \(-\frac{1}{6}\).The same process applies to \(-\frac{1}{3} \times \frac{5}{4}\), giving us \(-\frac{5}{12}\). This method keeps calculations simple and systematic.

Adding Fractions

Adding fractions requires a common denominator. Once all fractions are expressed with this common denominator, the numerators can simply be added together. For the expression \(-\frac{1}{6} + -\frac{5}{12}\), we previously found the LCD, which is 12. Convert \(-\frac{1}{6}\) into \(-\frac{2}{12}\) so both fractions have the denominator of 12. Now add the numerators:

• \(-2 + -5 = -7\)
• The result is \(-\frac{7}{12}\)

The final result didn't need further simplification since 7 is a prime number and doesn't divide 12 evenly. This method ensures accuracy in the addition of fractions, maintained through careful alignment of denominators.