Problem 30 - Algebra 2

Question

Evaluate each expression if \(a=\frac{2}{5}, b=-3, c=0.5,\) and \(d=6\). \(\frac{1}{c}+\frac{1}{d}\)

Step-by-Step Solution

Answer

The expression evaluates to approximately 2.1667.

Step 1: Understand the Expression

The expression requires you to compute the sum of the reciprocals of the given variables. Specifically, you're asked to evaluate \( \frac{1}{c} + \frac{1}{d} \) using the values for \(c\) and \(d\) provided.

Step 2: Identify the Known Values

From the problem, you know that \( c = 0.5 \) and \( d = 6 \). You need these values to substitute into the expression for evaluation.

Step 3: Substitute the Known Values into the Expression

Replace \( c \) with 0.5 and \( d \) with 6 in the expression: \( \frac{1}{0.5} + \frac{1}{6} \).

Step 4: Calculate the Reciprocal of Each Variable

Find \( \frac{1}{0.5} \) which equals 2, and \( \frac{1}{6} \) which equals approximately 0.1667.

Step 5: Add the Reciprocals

Calculate the total by adding the two results: \( 2 + 0.1667 = 2.1667\).

Step 6: Final Result

Therefore, the evaluated result for the expression \( \frac{1}{c} + \frac{1}{d} \) is approximately 2.1667.

Key Concepts

Reciprocal

The concept of a reciprocal is quite straightforward but powerful in mathematics. When you hear "reciprocal," think of flipping a number. To find the reciprocal of any number, you simply take one divided by that number.

For example, if you have a number like 4, its reciprocal is found by calculating \( \frac{1}{4} \). Similarly, for a fraction like \( \frac{2}{5} \), the reciprocal would be \( \frac{5}{2} \).

When dealing with whole numbers like 6, they can be rewritten as fractions like \( \frac{6}{1} \), so their reciprocal would be \( \frac{1}{6} \).

Some interesting points about reciprocals include:

• The reciprocal of 1 is 1, because \( \frac{1}{1} = 1 \).
• Zero does not have a reciprocal because division by zero is undefined.
• Reciprocals are especially handy when solving equations that involve division, as they can help simplify the expressions.

Substitution Method

The substitution method is a powerful technique often used to solve algebraic expressions and equations. It involves replacing variables with known values to simplify the problem.

Here's how you can put it into practice:

1. Identify the known values: Take note of which variables have been assigned numbers. In our example, the values for \(c\) and \(d\) were provided: \(c = 0.5\) and \(d = 6\).
2. Replace the variables: In the expression \(\frac{1}{c} + \frac{1}{d}\), substitute \(c\) and \(d\) with their respective values. This changes the expression to \(\frac{1}{0.5} + \frac{1}{6}\).

This method helps in simplifying complex expressions by breaking them into more manageable calculations. When numbers replace variables, it's easier to perform arithmetic operations and achieve the final result.

Arithmetic Operations

Arithmetic operations include four basic mathematical operations: addition, subtraction, multiplication, and division. In evaluating algebraic expressions, you often perform these operations to find solutions.

In the given exercise, the following operations are carried out:

• Division: This is seen when finding the reciprocal, such as \(\frac{1}{0.5}\), which is a division operation.
• Addition: After calculating the reciprocals \(2\) and \(0.1667\), you add them (\(2 + 0.1667\)) to find the total.

It's important to remember:

• Addition combines values, increasing the total.
• Division is the process of determining how many times one number is contained within another, which is crucial for finding reciprocals.
• Every operation is carried out following the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Knowing these basics can help solve more complex expressions systematically and accurately.