Problem 32 - Intermediate Algebra

Question

The difference of two numbers is 75 . The larger number is three less than four times the smaller number. Find the numbers.

Step-by-Step Solution

Answer

The smaller number is 26, and the larger number is 101.

Step 1: Understand the Problem

We are given that the difference between two numbers is 75, and the larger number is three less than four times the smaller number.

Step 2: Define the Variables

Let the smaller number be \( x \). Then, the larger number is \( 4x - 3 \).

Step 3: Set Up the Equation

According to the problem, the larger number minus the smaller number equals 75. This gives us the equation: \((4x - 3) - x = 75\).

Step 4: Simplify the Equation

Combine like terms: \( 4x - x - 3 = 75 \), which simplifies to \( 3x - 3 = 75 \).

Step 5: Solve for the Smaller Number

Add 3 to both sides to isolate the term with \( x \): \( 3x = 78 \). Then divide by 3 to solve for \( x \): \( x = 26 \).

Step 6: Find the Larger Number

Substitute \( x = 26 \) back into the expression for the larger number: \( 4x - 3 = 4(26) - 3 = 104 - 3 = 101 \).

Key Concepts

Linear Equations

Linear equations are at the heart of solving many algebraic word problems. Generally, these equations are expressions that involve variables raised to the power of one. They are fundamental in describing relationships between numbers or quantities in a straightforward manner.

In the context of the given problem, the relationship between the larger and smaller numbers can be expressed with a linear equation. Specifically, we look for an equation that directly translates the words of the problem into a mathematical form. For example, the problem states that the larger number, minus the smaller number, equals 75. This translates to the equation: \[ (4x - 3) - x = 75 \]
This equation models the difference between the two numbers using a straightforward algebraic approach. Linear equations like these are key tools because they transform word problems into solvable mathematical formats, allowing us to find specific values for unknowns.

Variable Expressions

Variable expressions use letters to represent numbers or unknowns in mathematical problems. They are incredibly useful because they allow us to formulate general solutions applicable to a wide variety of specific instances.

When we introduce a variable, such as \( x \), to represent an unknown quantity, we can create expressions that define relationships between different parts of a problem. In the given exercise, we let the smaller number be represented by \( x \). The larger number was described as being three less than four times the smaller number, allowing us to write this as the expression \( 4x - 3 \).

• Smaller Number: \( x \)
• Larger Number: \( 4x - 3 \)

This concept of using variable expressions helps in forming equations that solve problems. By representing the numbers with variables, algebra becomes a powerful tool that can efficiently solve for unknowns.

Problem Solving Steps

Solving algebraic word problems systematically requires a series of well-thought-out steps. Each step leads you closer to finding the solution by breaking the problem down into manageable parts.

In our problem, the first step is understanding the given information. This involves clearly writing down what is known and what we are trying to find. Next, we define our variables. For instance, the smaller number is \( x \), and based on the problem, the larger one is \( 4x - 3 \).

We then set up an equation based on the problem statement. This is where we translate our understanding and variable definitions into a form that can be mathematically manipulated. For example, we use the equation \((4x - 3) - x = 75\) to express the condition given in the problem.

The following steps involve solving the equation: combining like terms, solving for the variable, and substituting back to find other unknowns. By taking it step by step, one can methodically work towards finding solutions efficiently and correctly. Following a structured approach ensures that no part of the problem is overlooked or wrongly interpreted.