Problem 69 - Elementary Algebra

Question

For exercises \(67-82\), use the five steps and a proportion. Find the number of 725,000 women in their mid \(-40 \mathrm{~s}\) with a history of normal pregnancy who would be expected to have a heart attack or stroke some 10 years later. Of 100 women in their mid-40's with a history of normal pregnancy, about 4 would be expected to have a heart attack or stroke some 10 years later. (Source: www.nytimes.com, March 17, 2009)

Step-by-Step Solution

Answer

29,000 women are expected to have a heart attack or stroke.

Step 1: Understand the Problem

The problem states that out of 100 women with a history of normal pregnancy, 4 are expected to have a heart attack or stroke within 10 years. Given 725,000 women in the same category, the task is to find how many are expected to have a heart attack or stroke.

Step 2: Set Up the Proportion

Use the given data to set up a proportion. We know that 4 women out of 100 will have a heart attack or stroke, which can be written as \ \ \ \[ \frac{4}{100} = \frac{x}{725000} \] \ \ \ Here, \(x\) represents the number of women out of 725,000.

Step 3: Solve for x

To solve for \(x\), cross-multiply the proportion: \ \ \ \[ 4 \times 725000 = 100 \times x \] \ \ \ Calculate the left-hand side: \[ 2900000 = 100x \]

Step 4: Isolate x

Divide both sides by 100 to isolate \(x\): \ \ \ \[ x = \frac{2900000}{100} \] \ \ \ Simplify the division to find \(x\): \[ x = 29000 \]

Step 5: Interpret the Result

The solution shows that 29,000 women out of 725,000 are expected to have a heart attack or stroke over the next 10 years.

Key Concepts

Proportions

Proportions are an essential concept in algebra that allow us to relate two ratios or fractions. A proportion is an equation stating that two ratios are equal. For example, if we know that 4 out of every 100 women are expected to have a heart attack within 10 years, we can use proportions to predict outcomes for different groups.

When setting up a proportion, the key is to ensure that the ratios are equivalent, meaning you can compare 'part to whole' relationships accurately. Explicitly, the proportion from the example is: ewline \[ \frac{4}{100} = \frac{x}{725000}\] ewline
In this case, 4 women out of 100 is equivalent to 'x' women out of 725,000. Understanding how to set up these proportional relationships is the first step to finding a solution.

Cross-Multiplication

Once a proportion is set up, cross-multiplication is a technique used to solve it. This method involves multiplying diagonally across the proportion equation. For the equation ewline \[ \frac{4}{100} = \frac{x}{725000}\] ewline
Cross-multiplying it involves these steps:

• Multiply the numerator of the left fraction (4) by the denominator of the right fraction (725,000).
• Multiply the denominator of the left fraction (100) by the numerator of the right fraction (x).

This gives us a new equation: ewline \[ 4 \times 725000 = 100 \times x\] ewline
By performing cross-multiplication, the equation now becomes a linear equation that we can easily solve for 'x'.

Solving Equations

After cross-multiplying the proportions, the next step is to solve the equation for the unknown variable 'x'. Here, the equation ewline \[ 4 \times 725000 = 100 \times x\] ewline
First, perform the multiplication on the left side: ewline \[ 2900000 = 100x\] ewline
To isolate 'x', divide both sides of the equation by 100: ewline \[ x = \frac{2900000}{100} = 29000\] ewline
This step completes the algebraic manipulation needed to solve for the unknown variable, yielding 29,000 women. It is crucial to understand each step in isolation to avoid confusion. Breaking the problem into smaller parts often makes it clearer.

Word Problems

Word problems like the one in the exercise are common in algebra. They involve translating a real-world scenario into a mathematical problem. The steps to solving such problems typically include:

• Understanding the problem: Read carefully to identify the given data and what you need to find out.
• Setting up the equation: Use proportions or other algebraic methods to represent the problem mathematically.
• Solving the equation: Use techniques like cross-multiplication to find the unknown variable.
• Interpreting the result: Translate the mathematical solution back into the context of the problem to ensure it makes sense.

In the given example, we first understood the rates of heart attacks before setting up our proportion. Solving the proportion gave us a final answer related to the real-world context of the problem. Mastering these steps allows students to tackle similar problems with confidence.