Problem 39 - 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 \(8,\) the increase is what percent of the original number?

Step-by-Step Solution

Answer

The increase is 60% of the original number.

Step 1: Calculating the Amount of Increase

The amount of increase can be found by subtracting the original number from the increased number: \(\Delta = 8 - 5 = 3\). So, the amount of increase is 3.

Step 2: Calculating the Percentage Increase

The percentage increase can be computed by using the formula mentioned in the analysis section, which is \(P = \frac{A}{B} \times 100\). By substituting \(\text{A} = 3\) and \(\text{B} = 5\) into the formula, the percentage increase can be calculated as \(P = \frac{3}{5} \times 100 = 60%\).

Key Concepts

Percent Calculation

When we talk about percent calculation, it's all about finding a part of a whole in terms of hundredths. Percentages make it easier to understand proportions: a percentage always tells you how many parts of 100 something represents. For example, 50% means 50 parts out of 100. When calculating how much a number increases in percentage terms, we start by finding the difference or change between the original number and the new number. This difference is then compared to the original number by using the percent formula: \[ P = \frac{A}{B} \times 100 \] In this formula, \(A\) is the change, \(B\) is the original number, and \(P\) is the percentage calculated. For a simple example, if you increase a number from 5 to 8, the increase is 3. The next step is to figure out what fraction that increase (3) is of the original number (5) and convert that into a percentage.

Algebraic Formulas

Algebraic formulas are powerful tools that help us translate real-world problems into solvable math equations. For percentage increase problems, we use a specific algebraic formula that links the change in value to the original amount. The formula used in our percent increase calculation relies on the basic algebraic principle: \[ P = \frac{A}{B} \times 100 \] Where \( A \) is the increase (or decrease) amount and \( B \) is the original value. This formula can be rearranged for different uses, such as finding out the new value when the percentage increase and the original value are known. Understanding algebraic formulas allows flexibility in problem-solving, enabling you to tackle a series of related problems by just adjusting the knowns and unknowns in the formula. Mastery of such formulas is foundational for mathematics and beyond, in finance, business, and everyday life.

Problem Solving

Problem solving in mathematics is all about step-by-step thinking and applying known strategies to find a solution. When faced with a problem asking for a percent increase, it's crucial to approach it methodically:

• First, determine the increase by subtracting the original value from the new value.
• Next, apply the percent increase formula to translate that increase into a percentage.
• Finally, interpret the result to ensure it answers the question correctly.

Having a structured approach reduces errors and increases the chances of arriving at the correct solution efficiently. Additionally, practicing problem solving builds confidence and proficiency in handling similar tasks, making it a critical skill in academics and real-world situations alike. Remembering these steps can simplify complex-sounding problems, transforming them into manageable and solvable tasks.