Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Write a mathematical model for the number problem, and solve the problem. One positive number is five times another positive number. The difference between the two numbers is 148 . Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 185 and 37.

Step 1: Define Variables

Let \( X \) be the first number and \( Y \) be the second positive number.

Step 2: Build Equations from the Problem

According to the problem, you can set up two equations as follows:\n\nEquation 1: One positive number is five times another positive number. This can be written as: \( X = 5Y \).\n\nEquation 2: The difference between the two numbers is 148. This can be written as: \( X - Y = 148 \).

Step 3: Substitution

Substitute \( X \) from Equation 1 into Equation 2: \( 5Y - Y = 148 \)

Step 4: Solve for Y

Combine like terms to find the value of Y: \( 4Y = 148 \). Then divide by 4: \( Y = 37 \).

Step 5: Solve for X

Substitute the value of \( Y \) into the Equation 1: \( X = 5*37 \). So, \( X = 185 \).

Key Concepts

Defining Variables in Algebra

To tackle any algebraic word problem effectively, one must first translate the given information into mathematical terms. That involves the crucial step of defining variables. Variables are symbols, usually letters like X or Y, that represent unknown values. It's akin to using placeholders for pieces of the puzzle not yet known.

In the context of our number problem, we started by naming the first positive number as X and the second as Y. By doing so, we created a reference point that allows us to translate the descriptive language of the problem ('one positive number is five times another positive number' and 'the difference between the two numbers is 148') into mathematical expressions.

Setting Up Equations

Once variables are established, the next foundational skill is setting up equations. An equation is a statement that asserts the equality of two expressions. Developing an equation from a word problem often involves identifying how these expressions relate to the real-world quantities in the problem.

For our number problem, two relationships were described—multiplication and subtraction. The first is expressed by the equation \( X = 5Y \) or equal to five times the second number. The second relationship reflects the difference between these numbers, \( X - Y = 148 \) translates the word 'difference' into an algebraic subtraction operation.

Substitution Method in Algebra

The substitution method is one of the essential techniques for solving systems of equations in algebra. It's particularly useful when you have one equation that can be solved explicitly for one variable in terms of the others.

In the exercise, we first expressed X in terms of Y (\( X = 5Y \) and then replaced X with \( 5Y \) in the other equation (\( X - Y = 148 \). This substitution gave us a single variable equation \( 5Y - Y = 148 \) that we could solve straightforwardly. By isolating Y, we eventually discovered that the second number is 37, leading us to find X (the first number) by substituting back into \( X = 5Y \) and finding it to be 185.