Problem 5 - Intermediate Algebra

Question

Find the value of each algebraic expression at the given replacement values. See Examples 1 and 2 $$ a b \text { when } a=\frac{1}{2} \text { and } b=\frac{3}{4} $$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{3}{8} \).

Step 1: Understand the expression

The algebraic expression we have is \( ab \), which means \( a \) multiplied by \( b \). We are given the values \( a = \frac{1}{2} \) and \( b = \frac{3}{4} \). Our task is to substitute these values into the expression and then calculate the product.

Step 2: Substitute the values into the expression

Replace \( a \) with \( \frac{1}{2} \) and \( b \) with \( \frac{3}{4} \) in the expression. This gives us:\[ab = \frac{1}{2} \times \frac{3}{4}\].

Step 3: Calculate the product

Multiply the fractions by multiplying the numerators together and the denominators together:\[ab = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}\].

Key Concepts

Substituting Values

In algebra, substituting values means replacing variables in an expression with given numbers. This is a crucial skill for solving problems where specific values are provided for each variable. Let's take an algebraic expression like \( ab \). Here, \( a \) and \( b \) are variables, and we might be asked to find the value of the expression for specific numbers assigned to \( a \) and \( b \).

• To start, identify which values correspond to each variable. In our example, \( a = \frac{1}{2} \) and \( b = \frac{3}{4} \).
• Next, substitute these values into the expression. This means wherever you see \( a \), you write \( \frac{1}{2} \), and where there's \( b \), you write \( \frac{3}{4} \).

Substituting accurately is important to ensure that the calculations that follow are correct. It helps to keep track of the variables and ensure each one is replaced with the right value.

Multiplying Fractions

Multiplying fractions is a straightforward process that students often encounter in math problems involving algebra. To multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Let's take the example of multiplying \( \frac{1}{2} \) by \( \frac{3}{4} \):

• Multiply the numerators: \( 1 \times 3 = 3 \).
• Multiply the denominators: \( 2 \times 4 = 8 \).

This results in the fraction \( \frac{3}{8} \). It's always a good practice to check if the product can be simplified, though in this case, \( \frac{3}{8} \) is already in its simplest form. Multiplying fractions doesn't require finding a common denominator, which simplifies the process compared to adding or subtracting fractions.

Step-by-Step Solution

Following a step-by-step solution is like following a recipe. It ensures that you approach the problem systematically. Let's break down what we did to solve our expression \( ab \) with \( a = \frac{1}{2} \) and \( b = \frac{3}{4} \):
First, we understood the task: to multiply two variables. Then, we identified the values for each variable, substituting \( a \) and \( b \) with \( \frac{1}{2} \) and \( \frac{3}{4} \) respectively.

Next, we proceeded to multiplication. We tackled the multiplication of fractions by taking the numerators and the denominators separately. In our example, \( ab = \frac{1}{2} \times \frac{3}{4} \) led to \( \frac{3}{8} \). Each step builds on the previous, so it's crucial to follow them in order to reach the correct result.