Problem 42 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. four more than the quotient of 30 and a number

Step-by-Step Solution

Answer

The corresponding algebraic expression for the given English phrase is: \(\frac{30}{x} + 4\).

Step 1: Identify The Unknown

The problem statement mentions that \(x\) is the unknown number. So wherever we see 'a number' in the English phrase, we should substitute it with \(x\).

Step 2: Translate 'Quotient of 30 and a Number'

'Quotient of 30 and a number' means dividing 30 by the number. As \(x\) is the number, in algebraic expression, it will be \(\frac{30}{x}\).

Step 3: Translate 'Four More Than'

'Four more than' means adding 4 to the result. So we add 4 to the previously formed expression. This results in the final algebraic expression, \(\frac{30}{x} + 4\).

Key Concepts

Translating Phrases into Algebraic Expressions

When working with algebra, you will often need to translate English phrases into algebraic expressions. This process involves identifying specific words or phrases that point to mathematical operations. Splitting sentences into smaller parts can help. For example, words like 'more than' or 'increased by' typically suggest addition, while 'less than' or 'decreased by' indicate subtraction. In our example,

• 'Quotient of 30 and a number' suggests division because the word 'quotient' refers to the result of division.
• 'Four more than' signals an addition, implying that you should add 4 to the result of the division.

This method will help you accurately construct algebraic expressions from written phrases. You can think of each word as a clue to a mathematical operation.

Understanding the Unknown Variable

The unknown variable is a fundamental concept in algebra. It represents a number that you do not yet know, often denoted as variables like \(x\), \(y\), or \(z\). Think of it as a placeholder for a value. In our example, the sentence instructs us to use \(x\) as the unknown number.
Whenever an English phrase refers to a number without specifying it, you can replace that reference with the variable \(x\). For instance, 'a number' becomes \(x\). This approach allows us to generalize the situation and solve various problems using this versatile tool. Essentially, \(x\) lets us express many different numbers in the same mathematical language.

Applying Mathematical Operations

Once you've identified the operations from a phrase and substituted the unknown variable, it's time to apply those operations to form the complete algebraic expression.

• For division, take the number you are dividing (the dividend), and write it over the variable (the divisor). In our example, 30 is divided by \(x\), written as \(\frac{30}{x}\).
• For addition, simply add the number to the previous result. Here, 4 is added to the quotient, resulting in \(\frac{30}{x} + 4\).

By performing these simple operations step by step, you can transform any verbal statement into a clear mathematical form, making it manageable and solvable. This process is central to mastering algebra.