Problem 92 - Introductory Algebra for College Students

Question

Writing about mathematics will help you to learn mathematics. For all writing exercises in this book, use complete sentences to respond to the questions. Some writing exercises can be answered in a sentence; others require a paragraph or two. You can decide how much you need to write as long as your writing clearly and directly answers the question in the exercise. Standard references such as a dictionary and a thesaurus should be helpful. If \(x\) represents the number, explain the difference between translating the following phrases: a number decreased by 5 a number subtracted from 5 .

Step-by-Step Solution

Answer

The main difference between 'a number decreased by 5' and 'a number subtracted from 5' is in the subtraction order. The first signifies \(x - 5\), while the latter implies \(5 - x\).

Step 1: Interpretation of the first phrase

The phrase 'a number decreased by 5' means you have some number, denoted as \(x\), and you subtract 5 from it. In mathematical language, this translates to \(x - 5\).

Step 2: Interpretation of the second phrase

The phrase 'a number subtracted from 5' signifies that you have the number 5 and you subtract some number (denoted as \(x\)) from it. Formulated easily into an equation this becomes \(5 - x\).

Step 3: Delineating the Difference

The primary difference between these two phrases lies in the order of subtraction. In 'a number decreased by 5', the original number \(x\) has 5 subtracted from it. In contrast, in 'a number subtracted from 5', the number \(x\) is what's subtracted from 5, reversing the order of operation.

Key Concepts

Mathematical Language

Mathematical language is a precise way to express ideas and operations using symbols and formulas. In math, we use specific terminology to convey exact meanings, and this often involves translating everyday language into mathematical expressions. When we talk about a number, we typically represent it with a variable such as \(x\).

Understanding the precise meaning of phrases in mathematical context is crucial. It helps to avoid confusion in calculations and ensure correct interpretation of problems. Math terms are like a universal language, and learning them lets you turn complex sentences into straightforward equations.

When reading a mathematical phrase, always identify key terms and their roles within the context. This allows you to translate ordinary language into mathematical notation accurately.

Order of Subtraction

The order of operations matters a lot in mathematics, and subtraction is no exception. When we subtract, the sequence or order in which we perform the subtraction affects the result. Recognizing this helps prevent mistakes in solving problems.

Consider the two phrases:

• 'a number decreased by 5' which translates to \(x - 5\)
• 'a number subtracted from 5' which translates to \(5 - x\)

While they seem similar, the difference in wording completely changes the calculation.

In the first scenario, you start with \(x\), then subtract 5. Yet, in the second, you start with 5 and then subtract \(x\). These produce different outcomes, clearly showing how critical the order of subtraction can be in changing the equation's meaning and result.

Translating Phrases in Mathematics

Translating phrases in mathematics involves interpreting everyday language and converting it into mathematical symbols and operations. This skill is fundamental in solving math problems accurately, as it establishes a connection between verbal descriptions and mathematical expressions.

For example, phrases such as 'a number decreased by 5' or 'a number subtracted from 5' illustrate this translation process. Hence, knowing how to break down phrases helps define and solve equations correctly. Here’s a quick guide:

• Identify the key operations: words like 'decreased by' or 'subtracted from' should signal subtraction.
• Note the order: understand which number is subtracted from which.
• Use symbols: translate these phrases into mathematical expressions, like \(x - 5\) and \(5 - x\).

Effective translation ensures clarity in mathematical communication and solution processes, aiding in achieving accurate results.