Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve Exercises \(27-42\) If 4 is decreased to \(1,\) the decrease is what percent of the original number?

Step-by-Step Solution

Answer

75%

Step 1: Identify A, P, and B

In this problem, we're told the original number decreases from 4 to 1, so our \(A\) (the amount of the decrease) is \(4 - 1 = 3\). \(B\) is the original number, which is \(4\). \(P\) is what we're trying to find - the percent decrease.

Step 2: Plug Into the Formula

Now we'll plug \(A\) and \(B\) into our formula: \(3 = P \cdot 4\). Then we need to solve for \(P\).

Step 3: Solve for P

To isolate \(P\), we'll divide both sides of the equation by 4: \(P = 3 / 4\).

Step 4: Convert to Percentage

To express \(P\) as a percentage, we multiply it by 100: \(P = (3 / 4) \cdot 100\).

Key Concepts

Percent Formula

The percent formula is a helpful tool in many mathematical scenarios. It is particularly useful in problems involving percentage increases or decreases. The formula is represented as \(A = P \cdot B\), where \(A\) is the part of the whole, \(P\) is the percentage (in decimal form), and \(B\) is the whole or initial value. Understanding this formula helps in figuring out what percentage a certain value represents of an original amount.

In the given exercise, the task is to find out what percent decrease happens when a number reduces from 4 to 1. Here, the original number 4 is \(B\), and the decrease \(A\) is calculated as \(4 - 1 = 3\). The objective is then to determine \(P\), the percentage that \(A\) is of \(B\). By using this formula, we can see how the components come together to find out the percent decrease.

Arithmetic Operations

Arithmetic operations, such as subtraction, multiplication, and division, are the foundation of solving percent problems. In our exercise, the first step required subtracting the smaller number from the original larger one: \(4 - 1 = 3\). This operation helps us determine the actual decrease that occurred.

Once the decrease, or \(A\), is known, multiplication and division help isolate the unknown \(P\). First, the percent formula \(3 = P \cdot 4\) requires us to divide to find \(P\). By dividing both sides by 4, we get \(P = \frac{3}{4}\). This operation reduces complex problems into simpler, solvable equations. Finally, to convert this decimal to a percent, a multiplication by 100 is necessary, illustrating the versatility and necessity of arithmetic operations in daily math problems.

Problem Solving

Effective problem solving in percentage calculations involves carefully analyzing the problem step-by-step. The procedure begins by correctly identifying all known and unknown quantities from the problem statement. In this case, identifying the original value \(B\), the amount of decrease \(A\), and what needs to be calculated (the percent \(P\)).

After identifying the elements, incorporate them into the appropriate formula. Always ensure that calculations are performed in the correct order, and take care to convert fractions to percentages where necessary. This enhances accuracy and helps avoid common errors.

Problem solving is also about verifying each step. As demonstrated, each part breaks down a complex task into manageable parts, ensuring clarity throughout the solving process. This method not only provides the right answer but also deepens understanding of how percentages work in practical, real-world situations.