Problem 62 - Algebra 1

Question

Translate the phrase "a number decreased by the quotient of three and four." (A)\(n-\frac{3}{4}\) (B) \(=\frac{3}{4}-n\) (C) \(\frac{n-3}{4}\) (D) \(\frac{3}{4-n}\)

Step-by-Step Solution

Answer

The correct translation of the phrase 'a number decreased by the quotient of three and four' is (A) \(n-\frac{3}{4}\).

Step 1: Understanding the Mathematical Terms

First, let's understand the mathematical terms. 'A number' can be represented by a variable, for example, \(n\). 'Decreased by' means subtraction in mathematics. 'The quotient of three and four' is a division operation and can be represented as \(\frac{3}{4}\).

Step 2: Translate 'Decreased by' Operation to Mathematical Symbolism

'Decreased by' is a subtraction operation. So 'a number decreased by the quotient of three and four' means we are subtracting \(\frac{3}{4}\) from our number \(n\).

Step 3: Formulating the Expression

Combining these understandings, 'a number decreased by the quotient of three and four' translates into the mathematical expression \(n - \frac{3}{4}\). This means we are first taking a number of decrease and then subtracting three-fourths from it.

Key Concepts

Translation of Phrases

Translating phrases into algebraic expressions is like turning words into numbers and operations. When given a phrase, we need to understand what each part represents. For instance, "a number" is usually represented by a variable, such as \( n \). Whenever you see "decreased by," it signals subtraction. This is a common phrase in algebra that tells you to subtract one value from another.

Understanding phrases also involves correctly interpreting math terms. In our case, "the quotient of three and four" signifies division. Such phrases are key to translating problems into solvable equations. Hence, knowing how to translate words accurately into math terms will help you form the correct algebraic expression.

Subtraction in Algebra

Subtraction is a fundamental aspect of algebra and mathematics in general. The term "decreased by" tells us to perform subtraction. When converting phrases into algebra, recognizing this terminology is crucial. In the expression from our exercise, "a number decreased by the quotient of three and four," we interpret this as subtracting \( \frac{3}{4} \) from \( n \).

Subtraction in algebra can sometimes be tricky, especially when dealing with negative numbers or variables. It's important to maintain the correct order of operations and understand that subtraction changes the size of the number you're dealing with. So, in our context, the expression \( n - \frac{3}{4} \) shows that \( n \) is being reduced by the value \( \frac{3}{4} \).

Division in Algebra

Division can be thought of as splitting a number into parts. In algebra, phrases like "the quotient of" often indicate a division. In the phrase "the quotient of three and four," the division is represented as \( \frac{3}{4} \). This quotient simply means dividing 3 by 4, resulting in a number less than 1.

When performing division in algebra, it's important to correctly understand what is being divided and what the divisor is. Mistaking the order can lead to incorrect expressions. So, in this exercise, by dividing 3 by 4, we get \( \frac{3}{4} \), which is then used in the subtraction operation with \( n \) in our expression \( n - \frac{3}{4} \). This careful attention to order ensures you arrive at the accurate algebraic interpretation.