Problem 33 - College Algebra

Question

According to the National Center for Health Statistics, in \(1990,28 \%\) of babies in the United States were born to parents who were not married. Throughout the 1990 s, this increased by approximately \(0.6 \%\) per year. Use this information to solve Exercises \(33-34\) If this trend continues, in which year will \(37 \%\) of babies be born out of wedlock?

Step-by-Step Solution

Answer

The year in which 37% of babies will be born out of wedlock, if the given trend continues, would be 2005.

Step 1: Formulate a linear equation using the given information

First, you need to translate the given information into a linear equation. Let's denote \(P\) as the percentage of babies born out of wedlock, and \(y\) as the year.Given that the initial percentage in 1990 is 28% and it increases at 0.6% per year, we can formulate the following linear equation: \[ P = 28 + 0.6(y-1990) \]

Step 2: Set up the equation to solve for the year when percentage is 37%

To find the year in which 37% of babies will be born out of wedlock, we need to set \(P\) equal to 37 and solve for \(y\):\[ 37 = 28 + 0.6(y-1990) \]

Step 3: Solve the equation

To solve for \(y\), first subtract 28 from both sides of the equation, which will give: \[ 9 = 0.6(y - 1990) \]Then, divide both sides by 0.6:\[15 = y-1990 \]Finally, add 1990 to both sides of this equation to solve for \(y\):\[y = 2005 \]

Key Concepts

Slope-intercept form

The slope-intercept form of a linear equation is a foundational concept in algebra. It's expressed as \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( m \) is the slope, or rate of change.
• \( x \) is the independent variable.
• \( b \) is the y-intercept, or the starting value when \( x = 0 \).

In the context of the exercise, the slope-intercept form helps us model the situation in which the percentage of babies born out of wedlock increases over time. Here, the initial percentage in 1990 is 28%, which corresponds to the y-intercept, \( b = 28 \). The annual increase of 0.6% is the slope, \( m = 0.6 \).
By using this model, we can predict future percentages by plugging in different values for \( y \) (the year) in relation to \( x \) (the percentage of babies born out of wedlock). The equation \( P = 28 + 0.6(y-1990) \) transitions perfectly from the slope-intercept form, with adjustments for the specific year point of 1990.

Solve for a variable

Solving for a variable means isolating it on one side of the equation, allowing us to find its value. In this exercise, we are tasked with finding the year when 37% of babies will be born out of wedlock. Here's how we do it step by step:1. We know our equation from Step 1 is \( P = 28 + 0.6(y-1990) \).2. To solve for \( y \) when \( P = 37 \), set up the equation \( 37 = 28 + 0.6(y-1990) \).
Subtraction and division are essential tools for solving such equations:3. Subtract 28 from both sides: \( 37 - 28 = 0.6(y - 1990) \).
This simplifies to the equation: \( 9 = 0.6(y - 1990) \).4. Next, divide both sides by 0.6 to isolate \( y - 1990 \): \( \frac{9}{0.6} = y - 1990 \).5. Solve the division: \( 15 = y - 1990 \).6. Finally, add 1990 to both sides to get \( y = 2005 \).Solving for \( y \) involved a strategic clearing of operations that surrounded it, until the variable stood alone. This process shows the strength of algebra as a tool for prediction and problem solving.

Percentage increase

Understanding percentage increase is crucial for interpreting situations that involve growth over time. A percentage increase is calculated by multiplying the original amount by a multiplication factor. This factor comes from the percentage rate given.
In our exercise, the situation presents a steady percentage gain year by year. The initial 28% represents the proportion of babies born unmarried in 1990, and this percentage grows by 0.6% per year. Let's break it down:

• Think of the initial value as our baseline: 28% in 1990.
• Each year, add 0.6% to this base percentage: this is the annual increase.
• To calculate the percentage for any year, multiply the number of years since 1990 by 0.6% and then add this to 28%.

For example, in the year 2005 (15 years after 1990, as identified in our solution), we calculate the increase:

• Years since 1990: 15
• Percentage increase: \(15 \times 0.6 = 9\)
• New percentage: \(28 + 9 = 37\)%

Understanding this mechanical, step-wise increase is essential for solving similar growth-related problems.