Question

Four years ago, Jenny's annual salary was \(\$ 22,625\). This year, her salary was \(\$ 32,433\). What will her new salary be in 5 years if it continues to rise at the same linear rate? A) \(\$ 34,885\) B) \(\$ 42,241\) C) \(\$ 44,693\) D) \(\$ 55,058\)

Short answer

The new salary in 5 years will be \(\$44,693\).

Step-by-step solution

Calculate the increase in salary

First, let's find out how much the salary has increased in 4 years. Increase in salary = This year's salary - Annual salary 4 years ago Increase in salary = \(32,433 - 22,625 = 9,808\)

Calculate the linear rate of increase in salary

Now, to calculate the rate of increase in salary per year, we divide the increase in salary by the number of years (4 years). Linear rate of increase in salary = Increase in salary / Number of years Linear rate of increase in salary = \(9,808 / 4 = 2,452\)

Calculate the future salary in 5 years

To find out the salary after 5 years, we multiply the linear rate of increase in salary by the number of years in the future and add it to the current salary. New salary in 5 years = This year's salary + (Linear rate of increase in salary × Number of years in the future) New salary in 5 years = \(32,433 + (2,452 × 5) = 32,433 + 12,260 = 44,693\) The correct option is (C) \(\$44,693\).

Key concepts

Understanding Linear Equations

When it comes to GED math practice, grasping the concept of linear equations is crucial. A linear equation represents a straight line when plotted on a graph. It corresponds to the formula of the line, which is typically written in the form \( y = mx + b \) where \( m \) is the slope of the line indicating the steepness, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

In the context of predicting salaries, as in Jenny's case, we can imagine each year as a point on the x-axis (the independent variable) and the salary for that year as a point on the y-axis (the dependent variable). The change in salary from one year to the next can be plotted to form a straight line, which is our linear equation. Understanding linear equations helps us to model and solve real-world problems involving continuous growth or decline.

Rate of Change

The term rate of change is a pivotal concept in various mathematical contexts, including GED math. It describes how one quantity changes in relation to another. In the scenario provided, the rate of change is the amount Jenny's salary increases each year.

To calculate the linear rate of change, we simply take the difference in values over time. In this case, you have found Jenny's total increase over four years (\(32,433 - 22,625 = 9,808\)) and then divided it by the time period to get the annual rate of change (\(9,808 ÷ 4 = 2,452\)).

This calculation assumes a constant rate of increase, meaning her salary grows by the same amount every year. This is often a key assumption when using linear models for prediction, although you should be aware that in real life, such consistency is rare. The rate of change is a powerful tool for comparing different scenarios and understanding trends in everyday situations.

Predicting Future Values

The ability to predict future values is an essential application of math that extends to numerous fields such as economics, finance, and science. In mathematics, particularly when dealing with linear models, we can use the rate of change to forecast future scenarios based on current trends.

As with Jenny's salary, once we have the linear rate of change, we can predict future values by extending the line. The equation \( \text{future value} = \text{present value} + (\text{rate of change} \times \text{number of periods}) \) allows us to calculate what a particular value will be after a certain amount of time has passed, assuming the rate of change remains constant.

This predictive power is not only useful for solving textbook problems but also in life decisions such as planning for retirement, investments, and even climate change models. However, while these predictions are based on the assumption that current trends will continue, one has to consider potential variations that could affect the outcome.