Problem 41 - Prealgebra

Question

The city of Heath makes up \(\frac{1}{10}\) of the population in Rockwall County. Use the table to find the fraction of Rockwall County's population that lives in other cities. Write each fraction in simplest form. $$\begin{array}{|l|c|}\hline \text { City } & \begin{array}{c}\text { Decimal Part of } \\\\\text { Rockwall County's }\text { Population }\end{array} \\\\\hline \text { Fate } & 0.018 \\\\\hline \begin{array}{l}\text { Malendon- } \\\\\text { Chisholm }\end{array} & 0.02 \\\\\hline \text { Rockwall } & 0.42 \\\\\hline \text { Royse City } & 0.07 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The fraction of Rockwall County's population living in other cities is \(\frac{93}{250}\).

Step 1: Identify the Decimal Parts

From the table, we see the decimal parts for each city in Rockwall County: Fate is 0.018, Malendon-Chisholm is 0.02, Rockwall is 0.42, and Royse City is 0.07.

Step 2: Sum of Decimal Parts

Add the decimal parts from the table for all cities except Heath. The sum is: \[ 0.018 + 0.02 + 0.42 + 0.07 = 0.528 \]

Step 3: Fraction for "Other Cities"

Since Heath makes up \(\frac{1}{10}\) or 0.1 of the population, subtract the decimal part of Heath from 1 to find the fraction that represents all cities not listed:\[ 1 - 0.1 = 0.9 \]

Step 4: Calculate the Remaining Cities' Fraction

Subtract the sum of the other cities' decimal parts from the remaining fraction:\[ 0.9 - 0.528 = 0.372 \]

Step 5: Convert to Fraction and Simplify

Convert 0.372 to a fraction. 0.372 can be written as \(\frac{372}{1000}\). Simplifying this fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 4:\[ \frac{372}{1000} = \frac{93}{250} \]

Key Concepts

Turning Decimals into Fractions

Understanding how to convert decimals into fractions makes complex calculations easier. It helps to express numbers in a clear and precise way.
Here's how to convert a decimal to a fraction:

• Write down the decimal divided by 1. For example, 0.372 becomes \( \frac{0.372}{1} \).
• Multiply both the numerator and the denominator by 10 for every digit after the decimal point. In the case of 0.372, you will multiply by 1000 to remove the three decimal places: \( \frac{0.372 \times 1000}{1 \times 1000} \), resulting in \( \frac{372}{1000} \).
• Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD of 372 and 1000 is 4, so \( \frac{372}{1000} \) simplifies to \( \frac{93}{250} \).

This process not only helps in mathematical calculations but also strengthens your ability to switch between forms of numbers, aiding in various math and real-world applications.

Understanding Population Fractions

Population fractions articulate the share of the population in a specific area, helping analyze demographic information.
Suppose a county's population is shared among different cities, with each portion expressed as fractions or decimals. To find out each city's share:

• Look at the given fraction or decimal for a specific city. For Heath, it is \( \frac{1}{10} \) or 0.1.
• For other cities, add their decimal parts to determine what percentage they comprise of the total. Here, the other cities sum up to 0.528.
• Subtract this sum from the whole (1) to compute the fraction corresponding to cities not explicitly listed. For instance, you subtract the provided city's share from 1 to calculate the rest.

By effectively analyzing these calculations, you gain a better grasp over the distribution of populations, providing insights into demographic composition.

The Art of Adding Decimals

Adding decimals is a skill that’s handy in solving real-life problems, like calculating total costs or determining combined measurements.
When adding decimals, it’s crucial to ensure that all the decimal points line up vertically.

• Consider 0.018, 0.02, 0.42, and 0.07. Align the decimals and add them like you would whole numbers: \( 0.018 + 0.02 + 0.42 + 0.07 = 0.528 \).
• Start from the rightmost decimal place and move left, carrying over any numbers greater than 10 to the next column.
• After summing, ensure all digits remain in proper decimal places for accuracy.

Mastering this helps with efficiency and precision in mathematical operations, enhancing numerical understanding in various fields.