Problem 71 - Elementary Algebra

Question

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(426.0\) cases of chlamydia per 100,000 Americans with a total of \(1,307,893\) cases of chlamydia. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc .gov, 2011)

Step-by-Step Solution

Answer

The population of Americans used to create the ratio is 307,078,400.

Step 1: Identify the Given Ratio

There are 426.0 cases of chlamydia per 100,000 Americans. This can be written as the ratio \(\frac{426.0}{100,000}\).

Step 2: Write Down the Total Cases

The problem states that there were a total of 1,307,893 cases of chlamydia in 2010.

Step 3: Set Up a Proportion

Write the proportion using the given ratio and the total number of cases. Let \(P\) represent the total population of Americans:\[\frac{426.0}{100,000} = \frac{1,307,893}{P}\]

Step 4: Solve for Population (P)

Cross-multiply to solve for \(P\):\[426.0 \times P = 100,000 \times 1,307,893\]\[426.0P = 130,789,300,000\]Divide both sides by 426.0 to isolate \(P\):\[P = \frac{130,789,300,000}{426.0} = 307,078,398.12\]Round to the nearest hundred: 307,078,400.

Key Concepts

Ratios

A ratio is a relationship between two numbers showing how many times one value contains or is contained within the other. In this exercise, the ratio given is \(\frac{426.0}{100,000}\), which means there were 426.0 cases of chlamydia per 100,000 Americans. Ratios are essential in comparing different quantities. They are often used in various aspects of real life, from comparing ingredients in recipes to analyzing financial data. When working with ratios, it's important to clearly understand what each part of the ratio represents. For example, in our given ratio, the numerator (426.0) represents the number of chlamydia cases, while the denominator (100,000) represents the population size.

Cross-Multiplication

Cross-multiplication is a technique used to solve proportions. A proportion is an equation that states two ratios are equal. In this exercise, we set up the proportion as follows: \(\frac{426.0}{100,000} = \frac{1,307,893}{P}\). To solve for the unknown variable P (total population), we use cross-multiplication.

This involves multiplying the numerator of one ratio by the denominator of the other ratio. So we get: \(426.0 \times P = 100,000 \times 1,307,893\). After solving, we find \(P\) to be approximately 307,078,400 when rounded to the nearest hundred. Cross-multiplication simplifies solving proportions and helps find accurate results for real-life problems such as population estimation.

Population Estimation

Population estimation involves using mathematical techniques to determine the number of people in a group. In our example, we know the number of chlamydia cases and their incidence rate (cases per 100,000 people). By setting up a proportion, we calculated the total estimated population. This approach is often used in public health to understand disease spread, plan resources, and implement preventive measures. For instance, knowing the population helps officials allocate vaccines, plan health campaigns, and predict future trends of disease spread.

By using ratios and proportions effectively, we can transform specific data points into meaningful estimates that guide public policy and community health initiatives.