Question

The Narrowing Gender Gap Since the founding of the Equal Employment Opportunity Commission and the passage of equal-pay laws, the gap between men's and women's earnings has continued to close gradually. At the beginning of \(1990(t=0)\), women's wages were \(68 \%\) of men's wages; and by the beginning of \(2000(t=10)\), women's wages were projected to be \(80 \%\) of men's wages. If this gap between women's and men's wages continued to narrow linearly, what percent of men's wages were women's wages at the beginning of \(2004 ?\)

Short answer

At the beginning of 2004, women's wages were \(84.8\%\) of men's wages.

Step-by-step solution

Set up the coordinates

The problem tells us the percentage of women's wages to men's wages at t=0 (1990) and t=10 (2000). We can represent this as a coordinate (t, percentage) where t is the number of years since 1990. Coordinate 1: (0, 68) Coordinate 2: (10, 80)

Find the slope of the line

To find the linear relationship, first we'll find the slope of the line connecting these two points. We can use the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where m is the slope, \((x_1, y_1)\) and \((x_2, y_2)\) are the two coordinates. In this case, we get: \(m = \frac{80 - 68}{10 - 0}\)

Calculate the slope

Now calculate the slope: \(m = \frac{12}{10}\) \(m = 1.2\)

Find the y-intercept

Now that we have the slope, we can find the y-intercept (b) of the line equation (y = mx + b) using one of the given coordinates. We'll use the first coordinate (0, 68): \(68 = 1.2 \cdot 0 + b\)

Calculate the y-intercept

Now calculate the y-intercept: 68 = b

Write down the linear equation

Now that we have the slope and y-intercept, we can write down the linear equation describing the relationship between the years since 1990 and the percentage of women's wages to men's wages: \(y = 1.2t + 68\)

Find the value at t=14 (2004)

To find the percentage of women's wages to men's wages at the beginning of 2004, we need to find the value of y when t = 14 (14 years after 1990): \(y = 1.2 \cdot 14 + 68\)

Calculate the value at t=14

Now calculate the percentage: \(y = 16.8 + 68\) \(y = 84.8\) At the beginning of 2004, women's wages were 84.8% of men's wages.

Key concepts

Slope of a Line

The slope of a line is a measure of how steep a line is. It’s calculated by determining the vertical change between two points on the line (the rise) over the horizontal change between the same two points (the run). In mathematical terms, the slope (\(m\)) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where \(y_2\) and \(y_1\) are the y-coordinates and \(x_2\) and \(x_1\) are the x-coordinates of the two points. A positive slope means the line is going upward when moving from left to right, while a negative slope means it's going downward.

If the line is horizontal, the slope is \(0\), indicating no change in the y-value as x increases. Conversely, a vertical line would have an undefined slope as you would be dividing by \(0\), which is not allowed in mathematics. In our exercise, the slope was found to be \(1.2\), showing an upward, increasing pattern in women's wages as time progresses, which is a positive sign of improvement towards wage equality.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis of a graph. This is where the value of \(x=0\). It can give us initial or starting values in various real-life scenarios. For a linear equation in the form of \(y = mx + b\), the y-intercept is represented by the constant \(b\).

In contexts like our gender wage gap example, the y-intercept indicates the initial state before other influences or changes are accounted for. In this case, women’s wages were \(68\%\) of men’s wages in the base year 1990. Finding the y-intercept is straightforward when you have a point on the line where \(x=0\), but it can also be calculated using the line’s slope and any other point on the line. Here we confirmed the y-intercept as \(68\) by substituting the slope and the coordinates of an existing point into the line equation.

Linear Relationship

A linear relationship between two variables is represented by a straight line on a graph and can be defined by an equation in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This relationship implies that as one variable changes by a certain amount, the other variable changes by a fixed proportion.

For example, in the gender wage gap problem, the linear relationship indicates that for every year after 1990, the gap between men's and women's wages is projected to close by \(1.2\)% provided the trend continues unchanged. Such relationships are particularly useful for making predictions, as we did to estimate the percentage in the year 2004. Understanding linear relationships is vital since they are commonly assumed in various fields such as economics, science, and statistics.