Problem 5 - Prealgebra

Question

Evaluate each expression if \(a=5, b=12,\) and \(c=4\) $$\frac{2 b}{8}$$

Step-by-Step Solution

Answer

The expression evaluates to 3.

Step 1: Plug in Variable Values

First, substitute the value of \(b\) into the expression. Since \(b = 12\), the expression becomes \(\frac{2 \times 12}{8}\).

Step 2: Simplify the Numerator

Calculate the multiplication in the numerator: \(2 \times 12 = 24\). So the expression now is \(\frac{24}{8}\).

Step 3: Simplify the Fraction

Divide the numerator by the denominator: \(\frac{24}{8} = 3\). So the expression simplifies to \(3\).

Key Concepts

Substitution

Substitution is like filling in the blanks. It’s a way to replace variables with known values to simplify equations or expressions. For the given exercise, we have the expression \( \frac{2b}{8} \) and are told that \( b = 12 \). To substitute, simply replace \( b \) with 12. The expression now becomes \( \frac{2 \times 12}{8} \). This step transforms the problem from an abstract expression to one with concrete numbers, making it easier to analyze and solve. The key is to ensure you don't accidentally replace other symbols or miss out any part of the expression. Always double-check that each variable is replaced with its corresponding number.

Simplifying Fractions

Simplifying fractions is about finding an equivalent fraction where the numerator and denominator have no common factors other than 1. Consider \( \frac{24}{8} \). To simplify:

• First, identify the greatest common divisor (GCD) of the numerator (24) and the denominator (8). Here, 8 is the largest number that evenly divides both 24 and 8.
• Divide both the numerator and the denominator by their GCD. So, \( \frac{24}{8} = \frac{24 \div 8}{8 \div 8} = \frac{3}{1} \).

The resulting expression is simplified to 3. This process reduces complexities in problems and helps in spotting simpler patterns within numbers. It’s always helpful to double-check your simplification by multiplying back to ensure consistency.

Multiplication in Numerators

In arithmetic expressions, multiplication within numerators is a crucial step before simplifying any fractions. In our example expression \( \frac{2 \times 12}{8} \), the focus is on the numerator, \( 2 \times 12 \). Here's how to approach it:

• Perform the multiplication first: \( 2 \times 12 = 24 \).
• Work methodically and ensure the operation is correctly executed. Multiplication is sequential and should be processed before any addition, subtraction, or division in numerators.

This gives you a new, often simpler expression like \( \frac{24}{8} \). Handling multiplication efficiently can help prevent mistakes and make remaining steps, like simplification, more straightforward.