Problem 36 - Introductory Algebra for College Students

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve. 18 is what percent of \(90 ?\)

Step-by-Step Solution

Answer

18 is 20% of 90.

Step 1: Identifying the Known Variables

In this problem, we are given that \(A=18\) and \(B=90\). We need to find the percent 'P', which is unknown.

Step 2: Rearranging the Percent Formula

We need to rearrange the percent formula to solve for \(P\). Let's do that: \(P = \frac{A}{B}\). After this, we substitute the given values of \(A\) and \(B\) into formula.

Step 3: Substitute and calculate

Substitute \(A = 18\) and \(B = 90\) into the formula. That gives us \(P = \frac{18}{90} = 0.2\). But remember! We need to express \(P\) as a percentage, not as a decimal.

Step 4: Convert the Decimal to a Percentage

To convert it to a percentage, we multiply \(P\) by \(100\), yielding the result \(P = 0.2 * 100 = 20\%\).

Key Concepts

Percent Formula

Percent problems often involve the percent formula, which is a useful tool to determine how one number relates to another in terms of percentage. The formula can be mathematically expressed as \( A = P \times B \), where \( A \) represents "the part" or whole amount that results, \( B \) is "the base" or initial quantity, and \( P \) is the percentage or the rate expressed as a decimal. If any two values are known, the third can be found by rearranging the equation appropriately. This foundational formula highlights the relationship between the part, the whole, and the percentage.

Practically, this formula helps in scenarios like determining discounts, interest rates, or test scores, where understanding parts to wholes in percentage form is crucial. The key is recognizing which variable represents which part of the equation.

Solving for Percentages

Solving for percentages usually requires rearranging the percent formula to find the unknown percentage. In situations where you know the part and the base, you can solve for the percentage \( P \) by reconfiguring the formula as \( P = \frac{A}{B} \). This method effectively isolates the percentage variable on one side of the equation.

In the example provided, we are told that \( A = 18 \) and \( B = 90 \). By substituting these values into the formula \( P = \frac{18}{90} \), we find that the decimal equivalent is 0.2. This result needs to be converted to a more familiar percentage form to more effectively communicate the answer.

It's important to follow the steps carefully and ensure calculations are accurate to avoid errors.

Converting Decimals to Percentages

Converting a decimal to a percentage is a straightforward, yet essential part of working with percent problems. After solving the percentage formula, you may end up with a result expressed in decimal form, which is not as intuitive as a percentage. To convert this, simply multiply the decimal by 100.

This conversion makes understanding results easier. For example, multiplying a fraction like 0.2 by 100 converts it to 20%. Therefore, the result reads as "18 is 20% of 90," offering a clearer context.

• Multiply the decimal result by 100.
• Add the percentage sign (%) to the result.

This method reinforces your results by transforming a mathematical answer into a more comprehensible statement.

Algebraic Expressions

Algebraic expressions provide a convenient way to manipulate and solve for unknown variables through rearrangement and substitution. In percent problems, we often need to rework the original formula, \( A = P \times B \), into forms like \( P = \frac{A}{B} \) to isolate the desired variable.

Substitution involves replacing the known variables with their respective values to solve for the unknown. This systematic approach allows consistency and accuracy in solving problems.

Algebraic expressions allow for flexibility in mathematical operations, giving you the ability to handle a variety of problems efficiently. By mastering this technique, you gain the tools needed to tackle more complex mathematical scenarios.