Problem 24 - Elementary and Intermediate Algebra

Question

Consecutive Integer Problems Baseball. Catcher Thurman Munson and pitcher Whitey Ford are two of the sixteen New York Yankees who have had their uniform numbers retired. Their uniform numbers are consecutive integers whose product is \(240 .\) If Munson's was the smaller number, determine the uniform number of each player.

Step-by-Step Solution

Answer

Munson's number is 15 and Ford's number is 16.

Step 1: Understanding the Problem

We need to find two consecutive integers whose product equals 240. In this context, consecutive integers are numbers that follow each other in order, like 5 and 6. Let's assign the smaller number the variable \( x \), then the next consecutive number would be \( x + 1 \).

Step 2: Setting Up the Equation

Since Thurman Munson's number is the smaller number, we can say \( x \) is Munson's uniform number and \( x + 1 \) is Whitey Ford's. Therefore, the equation for their product should be: \( x(x + 1) = 240 \).

Step 3: Expanding the Product Equation

Expand the product \( x(x + 1) = 240 \) to form a quadratic equation: \( x^2 + x = 240 \). Subtract 240 from both sides to set the equation to zero: \( x^2 + x - 240 = 0 \).

Step 4: Solving the Quadratic Equation

We will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \) where \( a = 1 \), \( b = 1 \), and \( c = -240 \). First, calculate the discriminant: \( b^2 - 4ac = 1 - 4(1)(-240) = 961 \).

Step 5: Calculation of the Roots

Since the discriminant is positive, the quadratic equation has two real solutions. Substituting into the quadratic formula gives \( x = \frac{-1 \pm \sqrt{961}}{2} \). The square root of 961 is 31, hence: \[ x = \frac{-1 \pm 31}{2} \].

Step 6: Finding the Values of x

Calculate the two potential values for \( x \): \( x = \frac{30}{2} = 15 \) and \( x = \frac{-32}{2} = -16 \). Since uniform numbers cannot be negative, \( x = 15 \) is the only feasible solution. Thus, Thurman Munson's number is 15 and Whitey Ford's is \( 15 + 1 = 16 \).

Step 7: Verify the Solution

Verify by multiplying the consecutive numbers: \( 15 \times 16 = 240 \), which matches the problem statement. This confirms our solution is correct.

Key Concepts

Quadratic Equation

A quadratic equation is one of the most fundamental concepts in algebra and is defined as any equation in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic equations are essential in finding the roots or solutions that make the equation true.
In the context of our exercise, when determining the product of consecutive integers, we formed a quadratic equation: \( x^2 + x - 240 = 0 \). Identifying such patterns is crucial, as it simplifies the problem-solving process greatly.
Quadratic equations can typically have two solutions, which are calculated using various methods such as factoring, completing the square, or using the quadratic formula. In this exercise, the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) was employed.
Calculating the discriminant \( b^2 - 4ac \) is a critical first step in using the quadratic formula, as it determines the nature of the roots. In our exercise, a positive discriminant indicated two real and distinct solutions. Understanding how to manipulate and solve these equations provides a solid foundation for tackling a wide array of algebraic problems.

Product of Integers

Understanding the product of integers is key in many algebraic problems. A product is what you get when you multiply numbers together. In this problem, we were given that the product of two integers is 240.
The challenge arose from the condition that these integers had to be consecutive, meaning one number is directly next to the other, like 15 and 16 in our case. Setting the integers as \( x \) and \( x + 1 \) allowed us to write a concise equation. This formed the basis of our quadratic equation.
When working with integer products, especially in determining unknown consecutive numbers, forming an algebraic equation is a method to streamline tackling such problems.

• The two numbers are interconnected through their product.
• Expressing one in terms of the other can simplify the approach.

Recognizing standards that dictate such relationships is valuable, as it lets you encapsulate the problem in a solvable format without losing any necessary detail.

Problem-Solving in Algebra

Problem-solving in algebra requires a blend of conceptual understanding and systematic methodology. Analysing the problem carefully to extract known information is the first step. In this problem, recognizing that we are dealing with consecutive integers helped structure the approach.
We identified Thurman Munson's number as \( x \) and Whitey Ford's as \( x + 1 \). Next, formulating an equation based on the given product led us to a quadratic equation, unlocking the pathway to a solution. By transforming a word problem into algebraic expressions, hidden relationships come to light.
Effective problem-solving also involves verifying the solution to ensure alignment with the initial problem. In this example, calculating the product \( 15 \times 16 = 240 \) affirmed our solution's correctness. Practicing such strategies enhances one's ability to make logical connections and solve complex algebraic problems efficiently.

• First, understand the relationship between the elements involved.
• Next, translate this into an equation.
• Finally, solve confidently knowing each step builds logically on the last.

Such systematic methods ensure clarity and help students tackle even more intricate algebraic challenges in the future.