Problem 40 - 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 5 is increased to \(9,\) the increase is what percent of the original number?

Step-by-Step Solution

Answer

The increase from 5 to 9 is 80% of the original number 5.

Step 1: Determine the Initial and Final Numbers

The initial number here is 5 (since 5 was the original amount). The final number is 9 (since 5 was increased to 9).

Step 2: Plug the Numbers into the Percent Change Formula and Solve

Now, insert these values into the formula: \((\text{Final Number - Initial Number}) / \text{Initial Number} \times 100 \%\). Calculating, this results in: \((9 - 5) / 5 \times 100 \% = 80 \% \)

Step 3: Interpret the Result

The result of 80% means that the increase from 5 to 9 is 80% of the original number 5. This is the percent increase.

Key Concepts

Understanding the Percent Formula

When tackling percent change problems, it is important to employ the percent formula. This formula helps in determining the percentage that one number is of another. The percent formula can be represented as \( A = P \times B \), where \( A \) is the part, \( P \) is the percentage (expressed as a decimal), and \( B \) is the whole. This formula allows you to solve various problems involving percentages by rearranging it, depending on what you need to find. By understanding how each element of this formula relates, you can easily determine either the part, the percentage, or the whole. Remembering that percentages are simply a way to express numbers as parts of 100 can make percentages an easy tool for comparison and calculation.

Identifying Initial and Final Values

In any problem involving percent change, accurately identifying the initial and final values is crucial. The initial value is typically the starting value or amount before any change has occurred. In the given exercise, the initial value is 5, because that is the number you start with.

The final value, on the other hand, represents the resulting value after the change. The final value here is 9, as the number increased from 5 to 9. These values form the basis of the calculation and are necessary to observe the magnitude of the change. By pinpointing these initial and final metrics, you lay the groundwork for successfully determining the percentage change.

Calculating Percent Increase

The next step once you've identified your initial and final values, is to calculate the percent increase. To find this, you can use the percent change formula: \( (\text{Final Number} - \text{Initial Number}) / \text{Initial Number} \times 100 \% \). This formula calculates the proportion of change relative to the initial number, and expresses this as a percentage.

In this specific problem, you subtract the initial value (5) from the final value (9), which gives you 4. Then, divide that increase (4) by the initial value (5) to find how many times the increase fits into the original number. Multiply by 100 to express it as a percentage, resulting in an 80% increase. Hence, the number 5 was increased by 80%.

Practical Mathematical Problem-Solving

Solving mathematical problems, like those involving percent change, involves a structured approach to applying mathematical concepts. Initially, you dissect the problem to understand what is being asked. Identifying the initial and final values provides a clear picture.

Next, you apply the appropriate formula, which in this case is the percent increase formula. By incrementally breaking down the task into smaller, manageable steps, you avoid confusion and enhance accuracy. Finally, interpreting the results correctly is essential to grasp the real-world implications of the mathematics.

• Always verify your figures to rule out common calculation errors.
• Practice regularly to strengthen familiarity with percent calculations.
• Use real-world examples to see the practical application of percent changes.

By nurturing a methodical approach to mathematical problem-solving, complexities dissolve and clarity shines through.