Problem 26 - Elementary and Intermediate Algebra

Question

History. Delaware was the first state to enter the Union and Hawaii was the 50 th. If we order the positions of entry for the rest of the states, we find that Kentucky entered the Union right after Vermont, and the product of their order-of-entry numbers is \(210 .\) Use the given information to complete these statements: Kentucky was the th state to enter the Union. Vermont was the th state to enter the Union.

Step-by-Step Solution

Answer

Kentucky was the 15th state to enter the Union, and Vermont was the 14th.

Step 1: Understanding the Problem

We are given that Delaware was the 1st state, and Hawaii was the 50th. We need to find the order-of-entry numbers for Kentucky and Vermont given that the product of these numbers is 210.

Step 2: Defining Variables

Let's denote the order of Vermont entering the Union as \( V \) and Kentucky as \( K \). According to the problem, \( K = V + 1 \) and the product \( V \times K = 210 \).

Step 3: Setting up the Equation

We have two key equations: \( K = V + 1 \) and \( V \times K = 210 \). Substitute \( K = V + 1 \) into the product equation to get \( V \times (V + 1) = 210 \).

Step 4: Solving the Equation

The equation becomes \( V^2 + V = 210 \). Rearrange to form a quadratic: \( V^2 + V - 210 = 0 \). This can be factored into \( (V - 14)(V + 15) = 0 \).

Step 5: Finding Solutions

The solutions to the factored equation are \( V = 14 \) and \( V = -15 \). Since entry numbers must be positive, we choose \( V = 14 \). Thus, Vermont's order is 14.

Step 6: Calculating Kentucky's Order

Using \( K = V + 1 \) and \( V = 14 \), we find \( K = 14 + 1 = 15 \). Thus, Kentucky's order is 15.

Key Concepts

History in Mathematics

Understanding historical events can often be facilitated through mathematical methodologies. For example, historical order of statehood in the United States can be mathematically analyzed.
Mathematics not only allows us to calculate dates and positions but also helps us uncover relationships between past events.
For instance, using mathematical problem-solving techniques, we can deduce the chronological order of when different states entered the Union. This is often done by solving equations or using logical deductions based on given data points. Historical mathematics provides a structured approach to identify and confirm events, making it a pivotal tool in comprehending the timeline of historical events.

Quadratic Equations

Quadratic equations are expressions that take the form of \[ ax^2 + bx + c = 0 \].
They are called "quadratic" because the variable's highest exponent is two, meaning the variable is squared. These equations can model many scenarios, including the one we have for Vermont's entry to the Union.
To solve a quadratic equation, you can use various methods such as:

• Factoring
• Using the quadratic formula
• Completing the square

In our problem, you see how factoring was used to find Vermont's order of entry by rewriting the quadratic expression as \[ (V - 14)(V + 15) = 0 \]. This provided us with solution options for the values of \( V \). Quadratic equations are a powerful tool in algebra that can be used to find unknown values when given certain constraints.

Factors of a Product

When solving math problems, it's often useful to consider the factors of a product, especially when dealing with quadratic equations. For example, we see it in products like \( V \times K = 210 \).
Finding factors involves determining which pairs of numbers multiply to give the original product. For 210, factoring means identifying all combinations like \( 1 \times 210, 2 \times 105, \) and so on. These factors can become solutions or steps to solve larger equations.
In story problems, products and factors help form relationships between entities, such as states in a historical timeline problem. By knowing the relationship, we simplify complex scenarios into manageable calculations.

Order of Operations

The order of operations is a hierarchy that determines the sequence in which calculations should be performed. Essentially known as: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
This principle ensures that everyone interprets equations consistently. For instance, when solving equations in our historical problem, the sequence of operations ensures accurate rearrangement and calculation to reach the intended result.
Without following the correct order, you might end up with incorrect conclusions or solutions. Thus, understanding and applying the order correctly is crucial in all mathematical problem-solving processes. In practice, this might mean solving terms in parentheses first before handling multiplication or addition to maintain consistency across solutions.