Problem 47 - Intermediate Algebra

Question

Set up an equation and solve each problem. Suppose that the sum of two whole numbers is 9 , and the sum of their reciprocals is \(\frac{1}{2}\). Find the numbers.

Step-by-Step Solution

Answer

The numbers are 3 and 6.

Step 1: Define Variables

Let \( x \) and \( y \) represent the two numbers. These are the unknowns we need to solve.

Step 2: Set Up Equations

We know from the problem statement that the sum of the two numbers is 9. This gives us the first equation: \( x + y = 9 \). We also know that the sum of their reciprocals is \( \frac{1}{2} \), which gives us the second equation: \( \frac{1}{x} + \frac{1}{y} = \frac{1}{2} \).

Step 3: Solve the First Equation for y

Since \( x + y = 9 \), we can express \( y \) in terms of \( x \): \( y = 9 - x \).

Step 4: Substitute into Second Equation

Substitute \( y = 9 - x \) into the second equation: \( \frac{1}{x} + \frac{1}{9-x} = \frac{1}{2} \).

Step 5: Solve the Second Equation

To eliminate the fractions, find a common denominator, which is \( x(9-x) \). Multiply both sides of the equation by this common denominator: \( (9-x) + x = \frac{x(9-x)}{2} \). Simplify to get: \( 9 = \frac{x(9-x)}{2} \).

Step 6: Eliminate the Fraction

Multiply both sides by 2 to eliminate the fraction: \( 18 = x(9-x) \).

Step 7: Expand and Simplify

Rearrange and expand the equation: \( x^2 - 9x + 18 = 0 \).

Step 8: Solve the Quadratic Equation

Factor the quadratic equation: \( (x - 3)(x - 6) = 0 \). This gives two possible solutions: \( x = 3 \) or \( x = 6 \).

Step 9: Find Corresponding y values

Using \( y = 9 - x \), find \( y \) for each \( x \): - If \( x = 3 \), then \( y = 9 - 3 = 6 \).- If \( x = 6 \), then \( y = 9 - 6 = 3 \).

Step 10: Confirm Solution

Verify that both (3,6) and (6,3) satisfy the condition that their reciprocals add up to \( \frac{1}{2} \): \( \frac{1}{3} + \frac{1}{6} = \frac{2}{6} = \frac{1}{2} \). Both pairs satisfy the conditions.

Key Concepts

Whole Numbers

Whole numbers are the set of non-negative numbers that do not include any fractions or decimals. These numbers include 0, 1, 2, 3, and so on. They are often used in basic arithmetic right from counting objects to more advanced operations such as addition, subtraction, multiplication, and division.

• Whole numbers are easy to understand because they are numbers we use in everyday life.
• They do not include any negative numbers, just zero and the ones after it.

In the context of solving equations, whole numbers provide a straightforward way to identify solutions. In our exercise, since we are asked for whole numbers whose sum is 9 and whose reciprocals add up to 1/2, it becomes a matter of logical deduction within a limited set of possibilities.

Reciprocals

Reciprocals refer to a unique mathematical concept where one number is the inverse of another. When two numbers are multiplied together and the product is 1, they are considered reciprocals of each other. For example, 4 and 1/4 are reciprocals.

• The reciprocal of a number is computed by switching its numerator and denominator if it's a fraction, or dividing 1 by the number if it's an integer.
• Understanding reciprocals is crucial for solving equations that involve fractions.

In this problem, we know the reciprocals of the solution numbers have to add up to 1/2. Finding such a pair involves simple algebraic operations to rearrange and solve the equation formed.

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). This type of equation involves the variable squared (expressed as \( x^2 \)), and it often results in two possible solutions.

• Quadratic equations are fundamental in mathematics, frequently arising in various fields like physics, engineering, and finance.
• The solutions can be found using methods such as factoring, completing the square, or the quadratic formula.

In our example, the quadratic equation comes into play when we expand \( x(9-x) = 18 \) into \( x^2 - 9x + 18 = 0 \). This is a vital step to identify possible values for \( x \) that satisfy the conditions of the original problem.

Factoring

Factoring is a mathematical technique used to simplify quadratic equations, among other expressions. It involves breaking down the equation into a product of its simpler factors.

• The process can make it much easier to find solutions to equations because it reveals the roots directly.
• Factoring typically involves finding two numbers that add up to the coefficient of \( x \) and multiply to the constant term.

In the solution, we factor the quadratic equation \( x^2 - 9x + 18 = 0 \) into \( (x - 3)(x - 6) = 0 \). This reveals that the solutions for \( x \) are 3 and 6. Thus, factoring transforms a potentially complex quadratic equation into simple arithmetic operations.