Problem 85 - Algebra 2

Question

Evaluate \(\frac{a}{1-b}\) for the given values of \(a\) and \(b\) $$ a=0.9, b=-0.5 $$

Step-by-Step Solution

Answer

The evaluated expression is 0.6.

Step 1: Substitute the Given Values

First, substitute the given values of \(a\) and \(b\) into the expression \(\frac{a}{1-b}\). The expression becomes \(\frac{0.9}{1 - (-0.5)}\).

Step 2: Simplify the Denominator

In the denominator, you have the term \(1 - (-0.5)\). This simplifies to \(1 + 0.5 = 1.5\).

Step 3: Perform the Division

Now, divide the numerator by the simplified denominator: \(\frac{0.9}{1.5}\).

Step 4: Simplify the Fraction

Convert \(\frac{0.9}{1.5}\) to a simpler fraction or decimal. By converting them both into fractions and finding a common term, \(\frac{0.9}{1.5} = \frac{3}{5} = 0.6\).

Key Concepts

Evaluating Expressions

Evaluating expressions involves calculating the value of an expression by using specific values for the variables. In this exercise, our expression is \(\frac{a}{1-b}\), and we are given values \(a=0.9\) and \(b=-0.5\). To evaluate means to systematically substitute these values into the expression and simplify.
To begin, insert the given values into the expression so it becomes \(\frac{0.9}{1 - (-0.5)}\). The process doesn't end with substitution; you must continue by simplifying any arithmetic operations. This step ensures you arrive at a precise numerical value for the expression.

Substitution

Substitution is a key concept when evaluating expressions. It involves replacing variables with specific numbers. Here, the given values for \(a\) and \(b\) are substituted into the expression \(\frac{a}{1-b}\).
When substituting, the target expression should reflect these replacement values accurately. For our problem:

• The variable \(a\) is replaced with \(0.9\).
• The variable \(b\) is replaced with \(-0.5\), resulting in the modified expression \(\frac{0.9}{1 - (-0.5)}\).

This step is crucial because it allows us to transition from an algebraic form to a numeric form, simplifying our path to the solution.

Fractions

Understanding fractions is essential in solving expressions like \(\frac{0.9}{1-b}\). A fraction consists of a numerator and a denominator. In this context, \(0.9\) is the numerator and \(1 - b\) is the denominator.
Simplifying the denominator is often necessary before you perform the division. Here, simplifying \(1 - (-0.5)\) to \(1.5\) is a critical step. Once simplified, the expression becomes \(\frac{0.9}{1.5}\). Below, we'll convert this fraction into simpler terms through basic arithmetic or converting it to decimals.

Decimal to Fraction Conversion

Decimal to fraction conversion can simplify expressions further, making understanding easier. In our example, we expressed \(\frac{0.9}{1.5}\) as a fraction.
Both numbers, 0.9 and 1.5, can be rewritten in terms of fractions:

• 0.9 becomes \(\frac{9}{10}\)
• 1.5 becomes \(\frac{15}{10}\)

When dividing fractions, multiply the first fraction by the inverse of the second:\[\frac{9}{10} \div \frac{15}{10} = \frac{9}{10} \times \frac{10}{15} = \frac{9 \times 1}{1 \times 15} = \frac{3}{5}\]Thus, \(\frac{0.9}{1.5}\) simplifies to \(\frac{3}{5}\), which can also be represented as the decimal 0.6. Understanding these conversions aids in interpreting the results accurately across different forms.