Problem 46 - Elementary Algebra

Question

Translate each sentence to a mathematical statement and then simplify. A nurse has 30 milliliters of saline solution but needs 75 milliliters of the solution. How much more does she need?

Step-by-Step Solution

Answer

The nurse needs 45 more milliliters.

Step 1: Identify Given Quantities

The nurse currently has 30 milliliters of saline solution. She needs a total of 75 milliliters of the solution.

Step 2: Set Up the Equation

We want to find out how many more milliliters, let's symbolize this quantity as \( x \), the nurse needs. The equation we set up will be based on the total amount needed.\[ 30 + x = 75 \]

Step 3: Solve for the Unknown

Rearrange the equation to isolate \( x \) on one side.\[ x = 75 - 30 \]

Step 4: Calculate the Result

Subtract 30 from 75 to find \( x \).\[ x = 45 \]

Key Concepts

Equation Solving

When faced with real-world problems, translating them into equations is a powerful tool. An equation is a mathematical statement that shows the relationship between different quantities. In elementary algebra, we often use them to find unknown values. Here, we started by identifying what we knew and what was missing - the nurse has 30 milliliters of solution and needs 75 milliliters in total. The unknown quantity, represented as \( x \), is the additional amount needed. Setting up an equation involves forming a mathematical sentence that mirrors the problem at hand. We expressed it as \( 30 + x = 75 \). The process of solving involves rearranging the equation to find \( x \). We isolated \( x \) by subtracting 30 milliliters from both sides, resulting in the solution \( x = 75 - 30 \). This logical step-by-step approach shows the power of equations in problem-solving.

Simplifying Expressions

Simplifying expressions involves making a complex mathematical expression easier to understand and work with by reducing it to its simplest form. In our scenario, after setting up the equation, we had: \( 30 + x = 75 \). The process of simplifying in this context involved arranging terms to solve for \( x \). By subtracting 30 from 75, we directly calculated the value of \( x \), which simplifies the equation to a straightforward numerical answer, \( x = 45 \). Simplification is crucial as it transforms questions into clear and concise answers, making it easier to interpret and solve the problem.

Mathematical Statements

Mathematical statements help us express problems using numbers and symbols, providing a universal language for solving problems. In this exercise, the statement "A nurse has 30 milliliters of solution but needs 75 milliliters" was translated into \( 30 + x = 75 \). Creating mathematical statements involves identifying known values and variables for unknown quantities within a verbal problem. These statements are essential as they set the groundwork for systematically approaching algebraic problems. They allow us to move step-by-step, translating complex real-world scenarios into manageable mathematical equations that can be analyzed and solved.