Problem 40 - Introductory Algebra for College Students

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve. 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 represents an 80% increase.

Step 1: Identify the increase

To begin with, let's determine the amount of increase between the original number and the new number. This can be calculated by subtracting the original number from the new number, which is \(9-5 = 4\)

Step 2: Apply the percent formula

Now that we have the amount of increase, we can apply the percent formula \(A=P B\), where \(A\) is the amount, \(P\) is the percent we want to find and \(B\) is the base quantity. In this case, our base quantity is the original number, 5, and the amount is the increase, 4. Therefore, the formula will be \(4 = P * 5\)

Step 3: Solve for P

We can solve for \(P\) by dividing both sides of the equation by 5. This gives us \(P = \frac{4}{5}\)

Step 4: Convert to percentage

Finally, to express the solution in percentage terms, we multiply by 100. This gives us \(P = \frac{4}{5} * 100 = 80\% \)

Key Concepts

Percent Calculation

Understanding percent calculation is essential in solving problems involving percentages. The percent formula can be represented as \(A = P \times B\), where:

• \(A\) is the amount of part of interest.
• \(P\) is the percentage expressed as a decimal.
• \(B\) is the base or the original amount.

In our exercise, we calculated the percentage increase from 5 to 9. First, compute the difference between the new number and the original number, which is 4. Then use the percent formula to find the percentage this increase represents in relation to the original number. Rewriting the formula in terms of \(P\), we express this as \(P = \frac{A}{B}\). To convert \(P\) into a percentage, multiply it by 100.

Increase Percentage

The increase percentage is a useful measure to understand how much a quantity has grown relative to its original value. In this context:

• The formula for finding the increase is \(\text{increase} = \text{new value} - \text{original value}\).
• Then, the percentage increase is found using the formula \(\text{percentage increase} = \left(\frac{\text{increase}}{\text{original value}}\right) \times 100\).

In our example, we start by calculating the increase, which is \(9 - 5 = 4\). Next, we calculate the percentage this increase represents of the original number, 5. Thus, we find the percentage increase as \(\frac{4}{5} \times 100 = 80\%\). This result means there is an 80% increase from the original number.

Algebra Problem Solving

Algebra problem solving often involves manipulating equations to find unknown values. In our exercise, we see this by expressing the increase as a percentage. The steps involve:

• Identifying known and unknown variables.
• Setting up the equation based on the problem statement using a known formula (in this case, the percent formula).
• Solving the equation by isolating the variable of interest (\(P\)) through division or other algebraic operations.

In this task, we use algebra to determine the percentage representation of an increase from the original value, illustrating how algebraic techniques can address real-world problems like percentage calculations. By breaking the problem into smaller steps, complex algebraic expressions become much easier to manage and solve.