SAT Linear Growth/Decay Model Calculator
This calculator helps you solve common SAT math problems involving linear growth or decay. Many SAT questions present scenarios where a quantity changes at a constant rate over time or a series of steps. Use this tool to practice identifying the initial value, the rate of change, and the number of units/steps to find the final value.
The starting amount or quantity.
The constant amount added or subtracted per unit of time or step. Use a negative number for decay.
The total number of time periods or steps over which the change occurs.
Result:
Enter values and click 'Calculate'.
Understanding Linear Growth and Decay on the SAT
The SAT often tests your ability to interpret and apply linear relationships. A linear model describes a situation where a quantity increases or decreases by a constant amount over equal intervals. This is in contrast to exponential models, where a quantity changes by a constant percentage.
Key Components of a Linear Model:
- Initial Value (y-intercept): This is the starting point or the quantity at "time zero" (or zero units/steps). On a graph, it's where the line crosses the y-axis.
- Rate of Change (Slope): This is the constant amount by which the quantity increases or decreases for each unit of time or step. If the quantity is increasing, the rate of change is positive (growth). If it's decreasing, the rate of change is negative (decay). On a graph, this is the slope of the line.
- Number of Units/Steps: This represents how many times the rate of change has been applied. It could be years, hours, miles, or any other unit of measurement for the independent variable.
The Formula:
The general formula for a linear model is often expressed as:
Final Value = Initial Value + (Rate of Change × Number of Units/Steps)
Or, in function notation:
f(x) = mx + b
Where b is the initial value, m is the rate of change, and x is the number of units/steps.
SAT Example Scenario:
Imagine a plant that is initially 15 centimeters tall and grows at a constant rate of 2.5 centimeters per week. How tall will the plant be after 8 weeks?
- Initial Value: 15 cm
- Rate of Change: +2.5 cm/week
- Number of Units/Steps: 8 weeks
- Calculation: 15 + (2.5 × 8) = 15 + 20 = 35 cm
Using the calculator above, you would input 15 for Initial Value, 2.5 for Rate of Change, and 8 for Number of Units/Steps, and it would give you 35.
Another Example (Decay):
A car's fuel tank starts with 18 gallons of gas. It consumes gas at a rate of 0.05 gallons per mile. How much gas is left after driving 150 miles?
- Initial Value: 18 gallons
- Rate of Change: -0.05 gallons/mile (negative because it's consumption/decay)
- Number of Units/Steps: 150 miles
- Calculation: 18 + (-0.05 × 150) = 18 – 7.5 = 10.5 gallons
For this, you'd input 18, -0.05, and 150 into the calculator.
Mastering these types of problems is crucial for success on the SAT math section, as they appear in various forms, from word problems to graph interpretation.