Problem 78 - Elementary Algebra Within Reach

Question

Give two interpretations of "the difference of 6 and a number divided by \(3 . "\) Explain why \(\frac{n-6}{3}\) is not a possible interpretation.

Step-by-Step Solution

Answer

The two possible interpretations of 'the difference of 6 and a number divided by 3' can be mathematically represented as \(\frac{n - 6}{3}\) and \(6 - \frac{n}{3}\). The expression \(\frac{n-6}{3}\) is not a valid interpretation because the phrase 'the difference of 6 and a number' signifies '6 - n' and not 'n - 6'.

Step 1: Interpretation 1

The first interpretation can be 'subtract 6 from a number, then divide the result by 3'. This can be mathematically represented as \(\frac{n - 6}{3}\). Here the operation of 'difference' is calculated first, followed by the division.

Step 2: Interpretation 2

Another interpretation can be 'the difference of 6 and a third of a number'. This can be mathematically represented as \(6 - \frac{n}{3}\). Here the operation of 'division' is performed first on 'n', followed by the operation of 'difference'.

Step 3: Reasoning why \(\frac{n-6}{3}\) is not a possible interpretation

The statement 'the difference of 6 and a number' denotes '6 - n' and not 'n - 6'. The expression \(\frac{n-6}{3}\) represents 'subtract 6 from a number and then divide by 3' and not 'the difference of 6 and a number divided by 3' which corresponds to \(6 - \frac{n}{3}\) as per the rules of mathematical operations and the order of operations (BODMAS).

Key Concepts

Order of Operations

The order of operations is a fundamental principle in algebra that dictates the sequence in which mathematical procedures must be performed to accurately calculate an expression. This set of rules ensures that every mathematician or student will arrive at the same result when calculating an expression. The standard order to follow in mathematics is often abbreviated as PEMDAS or BODMAS:

• Parentheses/Brackets: Perform all operations inside parentheses or brackets first.
• Exponents/Orders: Next, solve exponents including powers and roots.
• Multiplication and Division: These operations are performed from left to right, whichever comes first.
• Addition and Subtraction: Like multiplication and division, these are also performed from left to right.

Understanding the order of operations is crucial when interpreting algebraic expressions. For example, in the exercise provided, one interpretation of 'the difference of 6 and a number divided by 3' leads to the expression \(6 - \frac{n}{3}\), which respects the order of operations by first dividing the number by 3 and then subtracting from 6.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations that represent mathematical relationships. They are fundamental in algebra and help to describe patterns, formulate equations, and solve problems. Variables, represented by letters, are placeholders for values that can change.

• Terms: An expression is made up of terms, which can be numbers, variables, or the product of numbers and variables.
• Coefficients: Numbers multiplying the variables in an expression are known as coefficients.
• Constants: Numbers on their own in an expression are called constants.

In the context of our exercise, the different interpretations of 'the difference of 6 and a number divided by 3' yield distinct algebraic expressions. The correct interpretation hinges on the phrase's semantics as well as the algebraic rules for operations. Expression \(\frac{n-6}{3}\) is not a valid interpretation of the given statement, as it implies that 6 is being subtracted from the variable 'n', rather than describing the 'difference of 6 and a number'.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to solve mathematical problems. It involves analyzing problems, drawing on existing knowledge, deducing new information, and applying the appropriate methods to find solutions. This form of reasoning is essential for interpreting expressions and solving equations accurately.

When encountering a phrase like 'the difference of 6 and a number divided by 3', careful reasoning is required to determine the accurate algebraic expression. Misinterpretation can lead to an incorrect representation of the problem. As demonstrated in the exercise, the phrase suggests we consider the number as being divided by 3 before taking the difference with 6, which is correctly expressed as \(6 - \frac{n}{3}\). The alternative interpretation, \(\frac{n-6}{3}\), does not convey the intended meaning and demonstrates why mathematical reasoning, coupled with an understanding of language, is critical in solving algebraic problems.