Problem 38 - Prealgebra

Question

Write an expression in simplest form that represents the total amount in situation. Alicia earned \(d\) dollars baby-sitting. Her friend earned twice as much. You earned \(\$ 2\) less than Alicia's friend earned.

Step-by-Step Solution

Answer

The expression in simplest form is \(2d - 2\).

Step 1: Identify Earnings

First, identify how much Alicia's friend earns compared to Alicia. Alicia earned \(d\) dollars, and her friend earned twice as much as Alicia. Therefore, Alicia's friend earned \(2d\) dollars.

Step 2: Calculate Your Earnings

Next, calculate your earnings. You earned \$2 less than what Alicia's friend earned. Since Alicia's friend earned \(2d\) dollars, your earnings would be \(2d - 2\) dollars.

Step 3: Write the Expression

Now, we write the expression for the total amount earned. In this problem, your earnings alone are what we are looking for after considering the conditions given. Thus, the simplest form of the expression representing the total amount you earned is \(2d - 2\).

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. Unlike equations, algebraic expressions do not have an equality sign. They are used to represent a value that can change depending on the value of the variables within the expression.

In the exercise involving Alicia and her friend, the expression represents the total amount of money earned in terms of the variable \(d\). Here, \(d\) is the unknown quantity or variable, representing the dollars Alicia earned. By using algebraic expressions, we can describe complex situations in a simple mathematical form.

This representation allows us to manipulate the expression later on either to find specific values or to simplify it further, just as we simplified your earnings to \(2d - 2\). Understanding how to build these expressions is key in solving real-life math problems.

Variable Manipulation Simplified

Variable manipulation is the process of rewriting expressions or equations to make solving problems easier. It involves using arithmetic operations with variables.

In our situation, we started with expressions involving \(d\), then used variable manipulation to find a new expression for your own earnings. Here's a breakdown on how we did it:

• Realize Alicia's friend earns twice as much as Alicia, which is \(2d\).
• You earned \$2 less than Alicia's friend, so subtract \(2\) from \(2d\).

Combining these steps gives us the expression \(2d - 2\). Manipulating an expression essentially allows us to focus on the desired quantity of interest—in this case, your total earnings.

Mastering Arithmetic Operations in Algebra

Arithmetic operations such as addition, subtraction, multiplication, and division are foundational concepts not only in basic math but also when dealing with algebraic expressions. To simplify expressions or manipulate variables, being comfortable with these operations is crucial.

In the exercise, we see how multiplication and subtraction play a role:

• Multiplication was used to calculate Alicia's friend's earnings, \(2 \times d = 2d\).
• Subtraction helped determine your earnings relative to Alicia's friend, hence \(2d - 2\).

These basic operations allow us to transform complex word problems into manageable mathematical expressions. As we practice more expressions like these, applying arithmetic operations becomes second nature and aids in tackling more intricate problems later on.