Problem 67 - Elementary Algebra

Question

For exercises \(67-82\), use the five steps and a proportion. About five of 100 pregnant women have pre-eclampsia, a condition that results in high blood pressure. About 300,000 pregnant women per year in the United States have pre-eclampsia. Find the number of pregnant women in the United States used to create this ratio. (Source: www.nytimes.com, March 17, 2009)

Step-by-Step Solution

Answer

The number of pregnant women in the US used to create this ratio is 6,000,000.

Step 1: Understand the Given Ratio

The problem states that 5 out of 100 pregnant women have pre-eclampsia. This can be written as a ratio: \ \( \frac{5}{100} \).

Step 2: Identify Known Values

The problem provides the information that 300,000 pregnant women per year in the United States have pre-eclampsia. This is our known value.

Step 3: Set up the Proportion

Set up a proportion to find the total number of pregnant women (\( x \)) using the given ratio and the known value: \ \( \frac{5}{100} = \frac{300,000}{x} \).

Step 4: Solve for the Unknown

Cross-multiply to solve for \( x \): \ \( 5x = 100 \times 300,000 \).

Step 5: Calculate the Result

Divide both sides by 5 to isolate \( x \): \ \( x = \frac{100 \times 300,000}{5} \). \ Compute the value to find \( x \): \ \( x = 6,000,000 \).

Key Concepts

Ratio

A ratio is a way to compare two quantities by showing how many times one value contains or is contained within the other. For example, in the given problem, the ratio of women who have pre-eclampsia to those who don’t is 5 to 100. This can be expressed as \( \frac{5}{100} \), which means for every 100 pregnant women, 5 have pre-eclampsia.
Ratios can be simplified just like fractions. For instance, \( \frac{5}{100} \) can also be written as \( \frac{1}{20} \), showing that 1 out of every 20 pregnant women has pre-eclampsia. Ratios are critical in comparing quantities and solving problems involving proportions. Understanding ratios is the first step in tackling proportion problems.

Proportion

A proportion is an equation that states that two ratios are equal. In the given problem, we formed a proportion to find the total number of pregnant women in the U.S. using the ratio of women with pre-eclampsia. The proportion is set up as follows: \( \frac{5}{100} = \frac{300,000}{x} \).
Here, the ratio of 5 women out of 100 having pre-eclampsia is set equal to 300,000 women having pre-eclampsia out of the total number of pregnant women, denoted by \( x \). Proportions are useful in so many areas of math and everyday problem-solving, helping us to find unknown quantities by setting two ratios equal to each other.

Cross-Multiplication

Cross-multiplication is a key technique used to solve proportions. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), you cross-multiply to find \( a \cdot d = b \cdot c \). This method helps to eliminate the fractions, making it easier to solve for the unknown variable.
In our problem, we used cross-multiplication to solve for \( x \): \( \frac{5}{100} = \frac{300,000}{x} \). By cross-multiplying, we get \( 5 \cdot x = 100 \cdot 300,000 \), which simplifies to \( 5x = 30,000,000 \). Cross-multiplication simplifies solving proportions, making it a fundamental skill in algebra.

Problem-Solving Steps

Solving proportion problems often involves a structured approach. Here are the key steps demonstrated in the given exercise:

• **Step 1:** Understand the given ratio. In our problem, this was \( \frac{5}{100} \), showing the rate of pre-eclampsia.
• **Step 2:** Identify the known values. We know that 300,000 women have pre-eclampsia annually in the U.S.
• **Step 3:** Set up the proportion. We formed \( \frac{5}{100} = \frac{300,000}{x} \).
• **Step 4:** Solve for the unknown using cross-multiplication: \( 5x = 30,000,000 \).
• **Step 5:** Calculate the result: \( x = \frac{30,000,000}{5} = 6,000,000 \).

Following these steps ensures a clear path to the solution, helping you understand not just how to solve the problem, but why each step is necessary.