Problem 77 - Elementary Algebra Within Reach

Question

Give two interpretations of "the quotient of 5 and a number times 3 ." Explain why \(\frac{3 n}{5}\) is not a possible interpretation.

Step-by-Step Solution

Answer

The two possible interpretations of the phrase are \(3*\frac{5}{n}\) and \(\frac{5}{3n}\) . The interpretation \(\frac{3n}{5}\) is not possible because it implies the multiplication is done before the division, which contradicts the original phrase.

Step 1: Interpretation 1: Quotient First, Then Multiply

In this interpretation, take the quotient of 5 and a number, call it n, first. The mathematical form is \(\frac{5}{n}\). Then that quotient is multiplied by 3, leading to the expression \(3*\frac{5}{n}\) or \(\frac{15}{n}\).

Step 2: Interpretation 2: Multiply First, Then Quotient

In this interpretation, multiply the number by 3 first, resulting in \(3*n\) or simply \(3n\). Then this result is divided by 5, leading to the expression \(\frac{3n}{5}\).

Step 3: Why \(\frac{3n}{5}\) is not a possible interpretation

The statement doesn't follow the order of the original phrase. The phrase 'the quotient of 5 and a number times 3' implies the quotient is taken first, and then the result is multiplied by 3, not the other way around.

Key Concepts

Order of Operations

Understanding the order of operations is essential when solving mathematical problems. It tells you which operations to perform first in an expression to get the correct answer. The common acronym for remembering this sequence is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the context of interpreting the phrase 'the quotient of 5 and a number times 3', we must first identify the operations involved: division and multiplication. Using the order of operations, we focus on division before multiplication unless grouping symbols dictate otherwise. Therefore, we interpret the quotient (division) before multiplying by 3, leading to the expression \(\frac{5}{n} * 3\) or equivalently \(\frac{15}{n}\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \( n \)), and operation symbols. Understanding how to construct and interpret these expressions is a foundational skill in algebra.

An accurate interpretation of 'the quotient of 5 and a number times 3' leads to the algebraic expression \(\frac{5}{n} * 3\), which incorporates a variable \( n \) to represent the unknown number. Misinterpreting the wording can lead to erroneous expressions, such as \(\frac{3n}{5}\), which does not match the intended sequence of operations described by the phrase.

Multiplication and Division

Multiplication and division are inverse operations; one undoes the effect of the other. Understanding how to correctly apply these operations within algebraic expressions is crucial.

When given a complex phrase like 'the quotient of 5 and a number times 3', it is vital to determine the correct order in which to perform multiplication and division. The correct interpretation of this phrase leads to multiplying the quotient \(\frac{5}{n}\) by 3. Conversely, \(\frac{3n}{5}\) suggests division after multiplication, which doesn't align with the provided phrase. This understanding prevents common mistakes when translating worded problems into numerical operations.

Algebraic Interpretation

Algebraic interpretation involves translating words into mathematical symbols and expressions. This skill translates a sentence like 'the quotient of 5 and a number times 3' into an algebraic expression that can be evaluated or simplified.

The proper algebraic interpretation of the given phrase is \(3 * \frac{5}{n}\), while \(\frac{3n}{5}\) is an incorrect interpretation because it reverses the order of operations. The structure of the sentence indicates that the division should be completed before multiplication. Mastery of algebraic interpretation is vital to solve algebra problems accurately.