Problem 49 - Algebra A Combined Function

Question

During the \(1982-1983\) term, the Supreme Court made 151 decisions while during the \(2007-2008\) term, they only made 72 . Find the percent decrease in number of decisions. Round to the nearest tenth of a percent.

Step-by-Step Solution

Answer

The percent decrease in decisions is 52.3%.

Step 1: Identify the Initial and Final Values

The initial value is the number of decisions during the 1982-1983 term, which is 151. The final value is the number of decisions during the 2007-2008 term, which is 72.

Step 2: Find the Decrease in the Number of Decisions

Subtract the final value from the initial value to find the decrease: \[\text{Decrease} = 151 - 72 = 79\]

Step 3: Calculate the Percent Decrease

Use the formula for percent decrease:\[\text{Percent Decrease} = \left(\frac{\text{Decrease}}{\text{Initial Value}}\right) \times 100\]Substitute the values you found:\[\text{Percent Decrease} = \left(\frac{79}{151}\right) \times 100 \approx 52.3\]

Step 4: Round to the Nearest Tenth of a Percent

The calculated percent decrease is approximately 52.315. Round to the nearest tenth, which results in a percent decrease of 52.3%.

Key Concepts

Initial Value

When calculating a percent decrease, it’s important to identify what we start with; this is known as the initial value. In our problem, the initial value is the number of decisions made by the Supreme Court during the 1982-1983 term. In that term, they made 151 decisions. It's crucial to clearly define this starting point because our calculations for percent changes are based on how the final number compares to this initial figure. Knowing the initial value provides the baseline for measuring any increase or decrease, ensuring clarity and consistency in your calculations.

Final Value

The term 'final value' refers to the quantity at the end of the period we are examining. For this exercise, the final value is the number of Supreme Court decisions made during the 2007-2008 term. This is 72 decisions. This final count is what you compare to the initial value to determine any difference or change. By understanding what the situation looked like at the end, you can effectively measure how much it has changed from the beginning point. This comparison is vital for calculating the actual difference needed for understanding how substantial the change is.

Rounding to the Nearest Tenth

Rounding numbers makes them easier to work with, especially in the context of percentages. To round a number to the nearest tenth, consider the digit in the hundredth place (second digit after the decimal point). If this digit is 5 or higher, the digit in the tenth place (first digit after the decimal point) is increased by 1. If it is less than 5, the digit in the tenth place remains the same. For instance, with a calculated result of 52.315, you look at the "1" after the decimal. Since 1 is less than 5, you retain the tenths digit "3," resulting in a rounded percent decrease of 52.3%.

Percent Formula

The percent formula is a tool to express the size of a change relative to its starting value, which can be either an increase or decrease. For percent decrease, the formula is:

• Calculate the decrease: Subtract the final value from the initial value.
• Divide this difference by the initial value.
• Multiply the result by 100 to convert it to a percentage.

In the problem at hand, this translates to finding how 79 decisions (the decrease from 151 to 72) relate to 151 (the initial value). You then express this relationship as a percentage, which makes it easier to grasp the magnitude of the change. Ultimately, understanding this basic formula allows you to measure and describe changes in any similar scenario effectively.