Problem 14 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. \(12 \%\) of a number

Step-by-Step Solution

Answer

The expression is \( 0.12x \).

Step 1: Understanding Percentages

First, recall that a percentage is a way of expressing a number as a fraction of 100. Thus, 12% means 12 out of 100 or \( \frac{12}{100} \).

Step 2: Setting up the Expression

Choose a variable to represent the unknown number, commonly used ones are \( x \) or \( n \). For this problem, let's use \( x \). The phrase '12% of a number' refers to finding 12% of \( x \).

Step 3: Converting the Percent to a Decimal

Convert the percentage to a decimal by dividing by 100: \( \frac{12}{100} = 0.12 \).

Step 4: Formulating the Algebraic Expression

To find 12% of the number \( x \), multiply the decimal equivalent of 12% by \( x \). The algebraic expression is \( 0.12 \times x \) or simply \( 0.12x \).

Key Concepts

Understanding Percentages

Percentages are a way to express numbers as a part of a whole, specifically out of 100. When we say '12%', we're really saying 12 parts out of 100. It's like slicing a pizza into 100 equal pieces and taking 12 of those slices. That's what a percentage is all about: showing how many parts you have out of 100.

• Percent means "per hundred." So, 50% means 50 out of 100.
• A percentage can very often be found on sales discounts or statistics in a newspaper.
• Understanding percentages is crucial for reading statistics and dealing with real-life data.

By getting the hang of percentages, you'll better understand how to compare and contrast different quantities directly.

Variables in Algebra

Variables are symbols used to represent unknown numbers or values. In algebra, we often use letters like \( x \), \( y \), or \( n \) for this purpose. They act like placeholders in expressions and equations, which help us solve problems where we don't know all the numbers yet.

• Think of variables as boxes that can hold any number. You can change it anytime to see how it affects the outcome.
• Variables are crucial because they let us write general formulas.For instance, if \( x \) is the number of apples you have, and each apple costs $0.50, then the cost of apples could be expressed as \( 0.50x \).
• The use of variables helps us generalize mathematical solutions and apply them to various situations.

Using variables is like solving a mystery; with the right clues (information), you find out what those boxes (variables) contain.

Converting Percentages to Decimals

Converting percentages to decimals is one of the first steps in solving many mathematical problems, especially those involving algebraic expressions. It's an essential skill because decimals are often easier to work with in equations than percentages. To convert a percentage to a decimal, you simply divide by 100.For example:

• Convert 15% to a decimal: \( \frac{15}{100} = 0.15 \)
• Convert 75% to a decimal: \( \frac{75}{100} = 0.75 \)
• To convert a percentage to a decimal, think of moving the decimal point two places to the left. So, 12% becomes 0.12.

The ability to convert percentages to decimals quickly is super helpful in tasks ranging from calculating discounts while shopping, to interpreting data in a science experiment. Once you've mastered this, you'll find tackling real-world math problems a lot easier.