Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Write a mathematical model for the number problem, and solve the problem. Find two consecutive natural numbers such that the difference of their reciprocals is one-fourth the reciprocal of the smaller number.

Step-by-Step Solution

Answer

The two consecutive natural numbers that fulfill the conditions given in the problem are 1 and 2.

Step 1: Identifying the variables

Let's identify two consecutive natural numbers as \( n \) and \( n + 1 \), where \( n \) is the smaller number.

Step 2: Establishing the mathematical model

As provided in the problem, the difference of their reciprocals is one-fourth the reciprocal of the smaller number. Translating words into math, we have: \[ \frac{1}{n+1} - \frac{1}{n} = \frac{1}{4n} \] This is our main equation and will be the foundation of our solution to the problem.

Step 3: Solve the mathematical model

Clear the fractions first by multiplying the equation by \( 4n(n+1) \), we get: \[ 4n - 4(n+1) = n+1 \] Simplifying the equation leaves us with: \[ 3n -3 = 0 \] which gives \( n = 1 \), yielding our two consecutive numbers as 1 and 2.

Key Concepts

Understanding Consecutive Natural Numbers

Consecutive natural numbers are essentially numbers that follow one another in order without any gaps. For instance, 1, 2, 3, and 4 are consecutive natural numbers because each number is one more than the one before it. When solving problems related to consecutive numbers, it's common to represent the first number with a variable, like \( n \), and the next consecutive number as \( n + 1 \).

This approach not only simplifies the formulation of algebraic equations but also makes calculations more straightforward. It's important to remember that natural numbers are positive integers, meaning they exclude zero and negative numbers, as well as fractions and decimals.

The Role of Reciprocals in Mathematics

A reciprocal of a number is defined as 1 divided by that number. For a natural number \( n \), the reciprocal is \( \frac{1}{n} \). Reciprocals are particularly interesting because they possess a unique property: the product of a number and its reciprocal is always 1. This property is often exploited in equations and algebraic manipulations.

When dealing with reciprocals in equations, it's crucial to pay attention to the signs and operations involved. As shown in the exercise, a subtle understanding of how to deal with the difference of reciprocals can be pivotal in finding the correct solution. Remember, with reciprocals, division turns into multiplication, which is a helpful transformation when solving complex equations.

Solving an Algebraic Equation

Algebraic equations are mathematical statements that show the equality of two expressions. When solving an algebraic equation involving reciprocals, as in the textbook exercise, the goal is to isolate the variable of interest. This often requires clearing fractions to simplify the equation, which can be done by finding a common denominator and multiplying through to eliminate the fraction.

Once the fractions are cleared, the resulting simplified equation can be solved through standard algebraic methods such as combining like terms and solving for the variable. Algebra requires practice, and working through steps methodically can lead to a better understanding of how to manipulate and solve equations.