Problem 83 - Algebra 2

Question

Evaluate \(\frac{a}{1-b}\) for the given values of \(a\) and \(b\) $$ a=\frac{1}{2}, b=\frac{1}{4} $$

Step-by-Step Solution

Answer

The evaluated expression is \(\frac{2}{3}\).

Step 1: Substitute the Values

Start by substituting the given values of \(a\) and \(b\) into the expression \(\frac{a}{1-b}\). This gives us \(\frac{\frac{1}{2}}{1-\frac{1}{4}}\).

Step 2: Simplify the Denominator

Compute the value of the denominator \(1 - b\). Substitute \(b = \frac{1}{4}\): \(1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}\).

Step 3: Divide the Fractions

Now, divide the fraction \(\frac{1}{2}\) by \(\frac{3}{4}\). Dividing by a fraction is equal to multiplying by its reciprocal, so we get \(\frac{1}{2} \times \frac{4}{3}\).

Step 4: Multiply the Fractions

Perform the multiplication of the fractions: \(\frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6}\).

Step 5: Simplify the Result

Simplify \(\frac{4}{6}\) by dividing both the numerator and the denominator by 2, resulting in \(\frac{2}{3}\).

Key Concepts

Evaluating Expressions

When working with mathematical expressions, evaluating them means finding their value by replacing variables with given numbers. In the exercise, we are asked to evaluate the expression \( \frac{a}{1-b} \) using specific values for \( a \) and \( b \). This requires us to manually substitute the values into the expression and compute it step by step.
Here’s a quick guide to evaluating an expression:

• Identify the variables and their given values.
• Substitute these values into the expression, replacing each occurrence of the variables.
• Simplify the expression if necessary to reach a final numerical result.

This process forms the bedrock of solving mathematical problems efficiently and accurately.

Substitution

Substitution is a fundamental technique in mathematics where specific values are used in place of variables to make an expression or equation simpler to solve. In our exercise, we substitute \( a = \frac{1}{2} \) and \( b = \frac{1}{4} \) into the expression \( \frac{a}{1-b} \).
Here's how to do it step by step:

• Identify the parts of the expression that contain variables.
• Replace each variable with its corresponding given value.
• Ensure that the substitution is correct by checking that all instances of the variables are replaced.

By substituting these values correctly, we transform the expression into a numerical form that is easier to handle in subsequent steps.

Simplifying Expressions

Simplifying an expression involves reducing it to its simplest form, making it easier to understand and solve. After substitution in the exercise \( \frac{\frac{1}{2}}{1-\frac{1}{4}} \), the next step is simplifying the denominator.
To simplify the expression:

• Start with simplifying inside parentheses or grouping symbols first, such as \( 1 - \frac{1}{4} \).
• Convert whole numbers to fractions if necessary to perform operations like subtraction.
• Carry out basic arithmetic operations like subtraction or addition to simplify fractions.

After simplifying the denominator to \( \frac{3}{4} \), the expression is ready for the next operation. Simplification is crucial as it helps in reducing complicacy, allowing for easier computation or further manipulation.

Dividing Fractions

Dividing fractions might seem tricky at first, but it becomes simple with the right approach. The exercise involves dividing \( \frac{1}{2} \) by \( \frac{3}{4} \), which is accomplished by multiplying by the reciprocal.
Here’s the step-by-step process for dividing fractions:

• Identify the two fractions involved.
• Replace the division operation with multiplication, and take the reciprocal (invert the numerator and denominator) of the second fraction.
• Multiply the fractions as you would when performing multiplication of fractions: multiply the numerators together and the denominators together.
• Simplify the resulting fraction if needed by reducing it to its simplest form.

Following these steps, the division in our exercise results in \( \frac{2}{3} \), which is the simplified answer. Remember, dividing by a fraction is the same as multiplying by its reciprocal, a handy rule that can make fraction operations clearer!