Problem 21 - Intermediate Algebra

Question

Translate each statement into mathematical symbols. Do not solve. 32.5 is \(74 \%\) of what number?

Step-by-Step Solution

Answer

Equation: \( 32.5 = 0.74 \times x \).

Step 1: Understand the Problem Statement

The problem states that 32.5 is 74% of an unknown number. We need to translate this sentence into an equation using mathematical symbols.

Step 2: Define the Variables

Let's denote the unknown number by the variable \( x \). So, we have to express the fact that 32.5 is 74% of \( x \).

Step 3: Convert Percentage to Decimal

To handle the percentage mathematically, convert 74% to a decimal. 74% can be written as 0.74 in decimal form.

Step 4: Write the Mathematical Equation

Now, translate the statement into an equation: "32.5 is 74% of \( x \)" can be written as \( 32.5 = 0.74 \times x \).

Key Concepts

Mathematical Symbols

Translating word problems into mathematical symbols is like changing a sentence into a special code that mathematicians understand. In math, symbols act as a bridge to represent real-world situations in a simplified form. When you see words like "is," "of," or numbers, they each have specific symbols or operations they relate to in math.

• "is" often means "equals," represented by the symbol \( = \).
• "of" in percentage problems usually implies multiplication, represented by the symbol \( \times \).
• Numerical values like 32.5 stay as is when translated into symbols.

For example, in the statement "32.5 is 74% of what number?", "32.5" remains a number, "is" becomes \( = \), "74% of" turns into \( 0.74 \times \), and the question "what number?" is the unknown value usually represented by a variable like \( x \). So, the problem translates to \( 32.5 = 0.74 \times x \). Understanding these connections helps solve math problems accurately.

Percentages to Decimals

Converting percentages to decimals is a crucial step in working with percentages in math equations. Percentages indicate parts per hundred, so to express them in a simpler numerical form, we convert them to decimals.

• To convert a percentage to a decimal, you divide by 100.
• This is because a percentage represents a fraction out of 100.

For instance, to convert 74% to a decimal:

• Remove the percent sign.
• Translate it directly by dividing by 100, giving you \( 0.74 \).

This transformation from percentage to decimal can then be used in equations, making calculations easier. For the problem "32.5 is 74% of what number?", converting 74% to 0.74 lets you easily express this comparison in a clear mathematical equation \( 32.5 = 0.74 \times x \). This conversion is vital in keeping math expressions consistent and manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical symbols that represent a value. When we encounter a statement involving unknown values, such as "of what number?", we use variables to indicate that we're working with something that is currently unknown.

• A variable is often symbolized by letters like \( x, y, \) or \( z \).
• They hold the place of a number we aim to find.

In translating the statement "32.5 is 74% of what number?", we used \( x \) to represent the unknown number. Therefore, the algebraic expression becomes \( 32.5 = 0.74 \times x \).Another important point about algebraic expressions:

• They often make it easier to solve equations by simplifying complex word problems into straightforward equations.

This method of using algebraic expressions makes dealing with real-life situations simpler and is a core part of solving math problems effectively.