Problem 80 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The result is 2.

Step 1: Identify the Expression

We need to evaluate the expression \( \frac{a}{1-b} \) with the given values for \(a\) and \(b\).

Step 2: Substitute the Given Values

Substitute \(a = 1\) and \(b = \frac{1}{2}\) into the expression. This gives us \( \frac{1}{1 - \frac{1}{2}} \).

Step 3: Simplify the Denominator

Calculate the denominator \(1 - \frac{1}{2}\). This simplifies to \(\frac{1}{2}\) because \(1\) can be written as \(\frac{2}{2}\), and \(\frac{2}{2} - \frac{1}{2} = \frac{1}{2}\).

Step 4: Divide the Numerator by the Denominator

Now, divide the numerator, 1, by the simplified denominator value, \(\frac{1}{2}\). So, \(\frac{1}{\frac{1}{2}} = 1 \times \frac{2}{1} = 2\).

Step 5: Final Result

The final evaluated result of the expression \(\frac{a}{1-b}\) with \(a=1\) and \(b=\frac{1}{2}\) is \(2\).

Key Concepts

Substitution

Substitution is a fundamental process in algebra where you replace the variables in an expression with their given values. This technique allows us to transform an algebraic expression into a numerical one, which can then be easily evaluated.

Let's look at the expression \( \frac{a}{1-b} \). We need to evaluate it using specific values for the variables: \( a = 1 \) and \( b = \frac{1}{2} \). Begin by substituting these values into the expression:

• Replace \( a \) with 1.
• Replace \( b \) with \( \frac{1}{2} \).

This changes our expression to \( \frac{1}{1 - \frac{1}{2}} \). Now, the expression is entirely numerical, and we can move on to solving it.

Simplifying Fractions

Simplifying fractions involves reducing them to their most basic form, making them easier to work with. In this case, simplifying happens in the denominator of our expression.

The denominator in our expression \( \frac{1}{1 - \frac{1}{2}} \) is \( 1 - \frac{1}{2} \). To simplify, you need to perform the subtraction:

• Express 1 as a fraction to match the denominator of \( \frac{1}{2} \). So, \( 1 \) becomes \( \frac{2}{2} \).
• Subtract \( \frac{1}{2} \) from \( \frac{2}{2} \): \( \frac{2}{2} - \frac{1}{2} = \frac{1}{2} \).

After simplification, the denominator is \( \frac{1}{2} \), allowing us to proceed to the final operation of division.

Division of Fractions

Division of fractions can sometimes seem tricky, but it's mainly about multiplying by the reciprocal. The reciprocal of a fraction is simply the numbers in the numerator and denominator swapped.

Once we simplify the denominator in our expression to \( \frac{1}{2} \), we need to perform the division:

• The expression becomes \( \frac{1}{\frac{1}{2}} \).
• To divide by \( \frac{1}{2} \), multiply by its reciprocal, which is \( 2 \) (since the reciprocal of \( \frac{1}{2} \) is \( \frac{2}{1} \)).
• Therefore, \( \frac{1}{\frac{1}{2}} = 1 \times 2 = 2 \).

This straightforward multiplication gives us the final result of \( 2 \). Division of fractions this way is just a shortcut via multiplication!