Problem 6 - Algebra for College Students

Question

A student translated the phrase "the quotient of a number \(n\) and \(12^{\prime \prime}\) incorrectly as \(\frac{12}{n}\).Give the correct mathematical expression.

Step-by-Step Solution

Answer

\( \frac{n}{12} \)

Step 1: Understand the Phrase

The phrase is 'the quotient of a number n and 12.' The quotient means division.

Step 2: Identify the Numerator

The 'number n' is the numerator, so it will be on the top part of the fraction.

Step 3: Identify the Denominator

The number '12' is the denominator, so it will be on the bottom part of the fraction.

Step 4: Write the Expression

Combine the numerator and denominator in the correct order to write the expression: \( \frac{n}{12} \)

Key Concepts

Quotients

In algebra, understanding quotients is very important. A quotient is the result of dividing one quantity by another.
When you see phrases like 'the quotient of' in word problems, this indicates a division operation.
For example: In the phrase 'the quotient of a number n and 12,' 'quotient' signals that the number n is divided by 12.
Learning to identify these key words can help you translate word problems into mathematical expressions correctly.

Numerators and Denominators

Fractions consist of two parts - the numerator and the denominator. The numerator is the top part and the denominator is the bottom part.
In mathematical expressions involving division, the number being divided (the 'part' or 'portion') is the numerator, and the number by which it is divided (the 'whole' or 'total') is the denominator.
For instance, with the phrase 'the quotient of a number n and 12,' the number 'n' is the numerator and the number '12' is the denominator.
So we write the mathematical expression as: \(\frac{n}{12}\). Understanding this structure is crucial because reversing the order of the numerator and denominator will result in a completely different meaning.
Always make sure to identify the numerator and the denominator correctly when creating fractions from word problems.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operators. Learning to translate word phrases into algebraic expressions is a fundamental skill in algebra.
Variables like 'n' represent unknown values and are used frequently in algebraic expressions. When given a phrase like 'the quotient of a number n and 12,' it challenges you to form this into a mathematical expression using a variable.
First, identify the operation: 'quotient' indicates division. Then recognize which numbers are involved: 'a number n' and '12.'
Next, assign 'n' to the numerator and '12' to the denominator, resulting in the expression: \(\frac{n}{12}\).
This is an important step in solving algebraic problems as it bridges word problems to numerical and variable expressions.~Learning this translation can make complex word problems more manageable and solvable.