Question

The average home attendance at New York Yankees' baseball games was a little more than \(150 \%\) of the average home attendance at Anaheim Angels' baseball games. The average home attendance at New York Yankees' baseball games was about \(33,000 .\) Write and solve an equation to estimate the average home attendance at Anaheim Angels" baseball games.

Short answer

The average home attendance at the Anaheim Angels' baseball games is approximately 22,000.

Step-by-step solution

Understand the Problem

The problem is presenting a comparison of the average home attendance of two baseball teams. The comparison is given in terms of percentages, where the average home attendance of the New York Yankees is 150% of that of the Anaheim Angels. Moreover, the exact number for the Yankees' attendance is also given as 33,000.

Form an equation to represent the problem

If we let \(x\) represent the average home attendance at the Anaheim Angels' games, then the problem tells us that 150% of \(x\), or \(1.5x\), is equal to the average home attendance at the New York Yankees' games, which is 33,000. So, we can form the equation \(1.5x=33000\).

Solve the equation

To solve for \(x\), we need to isolate \(x\) on one side of the equation. We can do this by dividing both sides of the equation by 1.5. That gives us \(x = \frac{33000}{1.5}\). Solving this equation gives us \(x = 22000\).

Interpret the result

The solution to the equation gives us the average home attendance at the Anaheim Angels' games. So, the average home attendance at the Anaheim Angels' games is approximately 22,000.

Key concepts

Writing Algebraic Equations

Writing algebraic equations is a fundamental skill in algebra that allows us to translate word problems into mathematical statements. This process involves identifying variables, which represent unknown values, and then constructing an equation based on the relationships described.

For example, in the context of the provided exercise, we're dealing with a comparison between two quantities: the average home attendance at New York Yankees' games and that at Anaheim Angels' games. The problem states that the Yankees' attendance is 'a little more than 150%' of the Angels' attendance. Translated into algebra, this means we need to use a variable (let's call it \(x\)) to represent the Angels' attendance, and then express the Yankees' attendance as 150% of this variable, or \(1.5x\). The equation \(1.5x = 33000\) emerges directly from this translation. This equation forms a bridge between a verbal statement and its mathematical representation, and is the foundation upon which we can solve the problem.

To enhance understanding, it's important to always define your variables clearly, and to start by breaking down the sentence structure of the problem into pieces that can easily be converted into mathematical expressions.

Percentages in Algebra

Percentages in algebra can often cause confusion, but they are just a different way to express fractions or ratios. A percentage represents a number out of 100. To use percentages in algebra, we convert them into decimal form, which can then easily be used in equations.

In the exercise, the phrase '150%' is converted to a decimal by dividing 150 by 100, giving us 1.5. This number is then used as a multiplier to represent '150% of' something—in this case, \(x\), the average attendance at Angels' games. Thus, translating percentages into algebra involves turning the percent into a decimal and then applying it to the appropriate variable or value as a multiplier. Remember, when a value is '100%', that's equivalent to the entire thing, or '1' in decimal form, so '150%' means 1.5 times the quantity.

It's helpful to practice converting percentages into decimals to become comfortable with using them in algebraic expressions. Always double-check the conversion, ensuring that the percentage has indeed been divided by 100, which is a frequent oversight.

Solving Equations

Solving equations, put simply, is the process of finding the value (or values) of the variable(s) that make the equation true. This usually involves isolating the variable on one side of the equation while keeping the equality balanced. To do this, we apply inverse operations such as addition, subtraction, multiplication, or division.

Considering our exercise, we have the equation \(1.5x = 33000\). To isolate \(x\), we divide both sides by 1.5 because division is the inverse operation of multiplication. As such, \(x = \frac{33000}{1.5}\) which simplifies to \(x = 22000\), giving us the average home attendance for the Angels' games.

When solving equations, it is crucial to perform the same operation on both sides to maintain the balance of the equation. This principle is rooted in the property of equality, which states that if you do something to one side of an equation, you must do the same to the other side. Practice and reinforcement of these steps are valuable for mastering the art of solving equations.