Problem 30 - 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 \(27-42\) What is \(16 \%\) of \(90 ?\)

Step-by-Step Solution

Answer

16% of 90 is 14.4

Step 1: Identify the Numbers

Take the percentage as a decimal, which is \(P = 0.16\) (since 16% equals 0.16 when expressed as a decimal) and the base number, which is \(B = 90\).

Step 2: Insert the Numbers into the Formula

Substitute \(P = 0.16\) and \(B = 90\) into the percentage formula \(A = P \times B\). From this, the formula becomes \(A = 0.16 \times 90\).

Step 3: Calculate the Answer

Calculate \(0.16 \times 90\) to determine \(A\), yielding \(A = 14.4\). This is the result, which means that 16% of 90 equals 14.4.

Key Concepts

Percentage Calculation

Understanding percentage calculations is essential in math and daily life contexts, such as calculating discounts, determining grades, or analyzing statistical data. A percentage represents a fraction of 100, implying 'per hundred'. To work with percentages, one can convert them into decimals for ease of computation. For instance, a percentage like 16% is equivalent to the decimal 0.16, since a percentage is simply divided by 100 to convert it into a decimal.

Once you have the decimal form, you can calculate the percentage of any number by using the percent formula: \( A = P \times B \) where \textbf{A} represents the amount, \textbf{P} is the percentage in decimal form, and \textbf{B} is the base number the percentage is taken from. Applying this formula enables you to solve a wide array of practical problems involving percentages.

Algebraic Expressions

Manipulating the Percent Formula

At the heart of working with percentages is the manipulation of algebraic expressions. An algebraic expression is a mathematical phrase that includes numbers, variables, and operational symbols. In the context of the percent formula, \textbf{A}, \textbf{P}, and \textbf{B} are variables representing different quantities. To find the percentage of a number, the formula \( A = P \times B \) is adapted by plugging in the values for \textbf{P} and \textbf{B}.

Knowing how to work with algebraic expressions is crucial for solving problems in algebra. It involves understanding how to substitute values, simplify expressions, and follow the order of operations to find the correct value of the expression, in this case, the amount \textbf{A}. When students grasp these concepts, the seemingly complex world of algebra becomes much more approachable.

Decimal Conversion

Converting percentages to decimals, as seen in the percentage calculation process, is a fundamental skill in mathematics and practical situations. To perform decimal conversion, simply divide the percentage by 100. This step is crucial because arithmetic operations with decimals are more straightforward than with percentages.

For example, when given the task to find 16% of 90, first convert the 16% to a decimal, which results in 0.16. This conversion simplifies multiplication when using the percent formula. Decimal conversions also play a critical role in financial calculations, measurement conversions, and computational accuracy in various scientific and engineering fields, making it a versatile tool in a student's skill set.