Problem 13 - Prealgebra

Question

"Midway" is the name of 252 towns in the United States. "Pleasant Hill" occurs \(\frac{5}{9}\) as many times. How many towns named "Pleasant Hill" are there in the United States?

Step-by-Step Solution

Answer

There are 140 towns named "Pleasant Hill" in the United States.

Step 1: Understand the Problem

We are given two names of towns, "Midway" and "Pleasant Hill". There are 252 towns named "Midway" in the United States. We need to find how many towns are named "Pleasant Hill" given that they occur \( \frac{5}{9} \) as many times as "Midway".

Step 2: Set Up the Equation

Let the number of towns named "Pleasant Hill" be represented by the variable \( x \). According to the problem, \( x = \frac{5}{9} \times 252 \). This equation represents the relationship between the number of "Midway" and "Pleasant Hill" towns.

Step 3: Calculate the Expression

To find \( x \), we calculate the expression \( \frac{5}{9} \times 252 \). Start by multiplying 252 by 5, which gives 1260. Then divide 1260 by 9.

Step 4: Perform the Division

Carry out the division \( \frac{1260}{9} \). When 1260 is divided by 9, it equals 140. Therefore, \( x = 140 \).

Step 5: Interpret the Result

The calculation shows that there are 140 towns named "Pleasant Hill" in the United States. This conclusion is based on the fraction of the number of "Midway" towns.

Key Concepts

Fraction Multiplication

Fraction multiplication is an essential concept in mathematics that allows us to solve various problems, including the one involving the number of towns named "Pleasant Hill." In this problem, we need to find a fraction of a whole number, which requires multiplying the fraction by that number.
In our case, we have the fraction \( \frac{5}{9} \) and the whole number 252. To perform the multiplication, follow these steps:

• Multiply the numerator of the fraction (5) by the whole number (252).
• This multiplication gives us 1260, which is an intermediate step in our calculation.
• Next, we'll divide the result by the denominator of the fraction (9) to find the final value.

Thus, fraction multiplication helps us easily calculate how many towns are named "Pleasant Hill" by scaling down the original number proportionally.

Basic Arithmetic

Basic arithmetic encompasses simple mathematical operations like addition, subtraction, multiplication, and division. Understanding these operations is crucial for solving the town name problem effectively.
Here, multiplication and division play key roles:

• First, we multiply 252 by 5, making use of multiplication to find a part of the whole number. The multiplication result is 1260.
• After that, we perform division to find the fractional part represented by \( \frac{5}{9} \). Dividing 1260 by 9 gives us the final answer.

This exercise highlights how multiplication can expand a number while division can reduce it to fit the required ratio, demonstrating the power of basic arithmetic in solving real-world problems.

Problem Solving Steps

Problem-solving involves a systematic approach to tackling a mathematical question. By following these steps, we can easily solve the problem at hand:

• Understand the problem: Identify what is given and what needs to be found—in this case, the number of "Pleasant Hill" towns relative to "Midway" towns.
• Set up the equation: Use given information to form an equation, here represented by \( x = \frac{5}{9} \times 252 \).
• Calculate the expression: Implement fraction multiplication, multiplying 252 by 5.
• Perform division: Divide the result by 9 to find the solution.
• Interpret the result: Conclude that there are 140 towns named "Pleasant Hill," understanding the relationship between the towns' names.

Following these steps ensures clarity and accuracy in achieving the final answer, paving the way for successful problem resolution.