Problem 54 - Elementary and Intermediate Algebra

Question

Translate to an algebraic expression. \(38 \%\) of a number

Step-by-Step Solution

Answer

0.38x

Step 1: Understand the Problem

The problem asks to translate a given percentage of an unknown number into an algebraic expression.

Step 2: Convert Percentage to Decimal

Recall that a percentage can be converted to a decimal by dividing by 100. Therefore, \(38\%\) becomes \( \frac{38}{100} \) or \( 0.38 \).

Step 3: Represent the Unknown Number

Let's represent the unknown number by a variable, commonly \( x \).

Step 4: Formulate the Algebraic Expression

The phrase 'of a number' translates to multiplication in algebra. Therefore, \(38\%\) of a number \( x \) is written as \( 0.38 \times x \).

Key Concepts

Percentage Conversion

To solve problems about percentages, it's essential to understand how to convert percentages into decimals. A percentage is just a way to express a number as a part of 100. For example, when we say 38%, we are saying 38 out of 100. To convert a percentage to its decimal form, we divide the number by 100. This might seem like a small detail, but it's very important.

So, the percentage 38% becomes a decimal by dividing 38 by 100:

\(38 \%\underline{\phantom{xxx}}=\frac{38}{100}=0.38\).

Once you have the decimal, it becomes much easier to work with percentages in algebraic expressions. This step ensures you're ready for further steps in algebra.

Algebraic Representation

In algebra, we often deal with unknown numbers. These unknowns are typically represented by variables such as \(x\), \(y\), or \(z\). The task of translating a word problem into an algebraic expression means turning spoken or written statements into the language of algebra.

Let's break down the phrase '38% of a number.' Here, the unknown number can be represented by a variable, let's use \(x\).

This means we can rewrite 'a number' as \(x\). Substituting the percentage with its decimal form we converted earlier, we get \(0.38\).

The entire phrase '38% of a number' now begins to take shape as \(0.38\) in terms of our unknown variable \(x\).

Multiplication in Algebra

In algebra, the term 'of' usually means multiplication. So, when we encounter a phrase like '38% of a number,' we're actually talking about multiplying 38% (or its decimal form) by a variable.

Let's combine the pieces we have now: We know that 38% is \(0.38\) and 'a number' is \(x\). Therefore, '38% of a number' can be written as:

\(0.38 \times x\).

In algebra, it's more common to write multiplication without the multiplication sign. So, \(0.38 \times x\) is usually simplified to \(0.38x\). This algebraic expression tells us that we're looking for 0.38 times some number \(x\); essentially, 38% of that number.

Mastering this idea helps in a lot of math problems, making this a very useful concept to understand.