Problem 103 - Elementary Algebra

Question

\(\frac{1}{2}(12-7)-\frac{1}{4}(17-14)\)

Step-by-Step Solution

Answer

1.75

Step 1: Simplify within the parentheses

Calculate the values inside the parentheses. For \(12 - 7\): \(12 - 7 = 5\). For \(17 - 14\): \(17 - 14 = 3\).

Step 2: Apply the multiplication

Multiply the simplified values by their respective fractions. For \( \frac{1}{2} (5) \): \( \frac{1}{2} \times 5 = 2.5 \). For \( \frac{1}{4} (3) \): \( \frac{1}{4} \times 3 = 0.75 \).

Step 3: Subtract the results

Subtract the result of the second multiplication from the result of the first. \( 2.5 - 0.75 = 1.75 \).

Key Concepts

Order of Operations

Understanding the order of operations is crucial in solving algebraic expressions correctly.
The order of operations ensures we perform mathematical operations in the correct sequence.
The common acronym PEMDAS is used to remember this order:

• P: Parentheses
• E: Exponents
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

For the provided exercise, follow these rules strictly. Always start by simplifying expressions within the parentheses first.

Fractions

Fractions represent parts of a whole and are written as one number over another.
They are useful in breaking down complex problems into manageable parts.
When solving algebraic expressions with fractions, remember:

• Multiplying fractions involves multiplying the numerators and denominators separately.
• Understanding reciprocal values helps; for example, the reciprocal of \(\frac{1}{2}\) is 2.

In our example, we used \(\frac{1}{2}\) and \(\frac{1}{4}\) to multiply the simplified values, crucially converting them to decimals afterward.

Basic Arithmetic

Basic arithmetic operations include addition, subtraction, multiplication, and division.
Mastering these operations is essential as they form the foundation of more advanced mathematical concepts. For our exercise:

• Subtract within the parentheses first.
• Multiply the results by the given fractions.
• Subtract the final results.

Practicing these steps repeatedly helps in solidifying your understanding of arithmetic operations.

Step-by-Step Solution

Breaking down problems into step-by-step solutions simplifies complex expressions. Let's revisit our exercise: \(\frac{1}{2}(12-7)-\frac{1}{4}(17-14)\). Follow these steps precisely:

• Step 1: Simplify within the parentheses. For \(12 - 7\): the result is 5. For \(17 - 14\): the result is 3.
• Step 2: Apply the fractions. \( \frac{1}{2} \times 5 = 2.5 \) and \( \frac{1}{4} \times 3 = 0.75 \).
• Step 3: Subtract the results: \( 2.5 - 0.75 = 1.75 \).

Understanding and practicing each step ensures you can handle similar algebraic expressions with confidence.