Problem 22 - Intermediate Algebra

Question

Translate each statement into mathematical symbols. Do not solve. What is \(83.5 \%\) of \(245 ?\)

Step-by-Step Solution

Answer

Translate to \(0.835 \times 245\).

Step 1: Convert 83.5% to a Decimal

To convert a percentage to a decimal, divide it by 100. So, we convert 83.5% to a decimal by computing \(83.5 \div 100 = 0.835\).

Step 2: Write the Expression for 'of' as Multiplication

In mathematics, the word 'of' typically indicates multiplication. So, we write the expression as \(0.835 \times 245\).

Key Concepts

Understanding Mathematical Expressions

Mathematical expressions are a way to represent numbers and operations on those numbers in a concise and efficient manner. In the problem, we are asked to find what 83.5% of 245 is. This translates into a mathematical expression using the main concept of operations.

• The percentage symbol (%) is often used in real-world problems and it signifies a part per hundred.
• Expressions use symbols like numbers and operation signs (such as +, -, ×) to convey a calculation.
• Mathematical expressions are built to analyze and solve problems in a structured and systematic way.

The problem involves understanding how to transform a word problem into a mathematical expression by understanding the meaning of phrases like 'of', which signifies multiplication. This step is crucial because it organizes how we approach solving the problem mathematically.

Decimal Conversion Process

Decimal conversion is an essential step when working with percentages. It allows for simpler calculations by transforming a percentage into a decimal form. When converting a percentage to a decimal, you divide the percentage value by 100. This process transforms the percentage, a value representing a fraction of 100, into a decimal, which is easier to work with in various calculations.

• For instance: Converting 83.5% involves calculating \(83.5 \div 100 = 0.835\).
• This conversion is crucial for arithmetic operations, particularly multiplication and division.
• Decimal forms simplify processes and help align numbers for accurate computation.

The conversion from percentage to decimal form is a foundational skill in mathematics that aids in solving problems where percentages are involved. Understanding and mastering this process forms the basis for more advanced calculations.

Multiplication in Mathematics

In mathematics, multiplication is a fundamental operation that represents combining equal groups. It is used extensively in many types of calculations. When you see the word 'of' in mathematical problems, it usually indicates multiplication. This is particularly true for problems involving percentages, where finding a part of a whole uses multiplication.

• In the provided exercise, the task is to calculate 0.835 multiplied by 245.
• This operation will yield the part of 245 that 83.5% represents.
• Multiplication allows for scaling numbers up or down according to various factors.

Knowing when and how to use multiplication, especially when dealing with percentage questions, is critical. It is a key arithmetic skill that helps to break down complex word problems into more manageable tasks.