Problem 69 - Algebra 1

Question

Write the verbal sentence as an equation. Eleven is two fifths of the quantity \(n\) decreased by thirteen.

Step-by-Step Solution

Answer

The equation is \(11 = \frac{2}{5}(n-13)\).

Step 1: Identification of Mathematical Equivalents

Identify the corresponding mathematical equivalents. 'Eleven' becomes 11, 'two fifths' is 2/5, 'the quantity \(n\) decreased by thirteen' translates to \(n - 13\), and 'is' signifies the equal sign (=).

Step 2: Formulate the equation

Now that all the mathematical equivalents are known, the sentence 'Eleven is two fifths of the quantity \(n\) decreased by thirteen' can be written as the equation \(11 = \frac{2}{5}(n-13)\).

Key Concepts

Verbal Expressions

Verbal expressions are phrases or sentences that describe mathematical relationships using words. When solving algebraic problems, translating these verbal expressions into mathematical equations is crucial. By understanding the language of math, students can interpret word problems or verbal instructions effectively. In the provided exercise, the verbal sentence was: "Eleven is two fifths of the quantity \(n\) decreased by thirteen." This sentence contains specific clues which indicate mathematical relationships:

• 'Eleven' refers to the number 11.
• 'Two fifths' translates to the fraction \(\frac{2}{5}\).
• 'The quantity \(n\) decreased by thirteen' represents the expression \(n - 13\).
• 'Is' means 'equals' and signifies the equation's balance.

By breaking down each segment of the sentence, students can methodically convert verbal expressions into precise mathematical equations, reinforcing comprehension of algebraic language.

Mathematical Translation

Mathematical translation is the process of converting verbal expressions into corresponding algebraic equations. It involves identifying keywords and phrases in a problem and using mathematical symbols to represent them. This skill is fundamental for tackling word problems, allowing students to visualize the problem within mathematical parameters.
In the original problem, we started by identifying words like 'is', 'two fifths', and 'decreased by', which pointed to specific mathematical equivalents:

• 'Is' translated to '='.
• 'Two fifths' became \(\frac{2}{5}\).
• 'Decreased by' suggested subtraction, leading to \(n - 13\).

Thus, the sentence formed the equation \(11 = \frac{2}{5}(n-13)\). Mastering this translation skill enables students to break down complex problems into manageable equations, bridging verbal language with mathematical notation.

Fractional Coefficients

Fractional coefficients are numbers or terms in equations that contain fractions and are multiplied by variables. In algebra, they modify the magnitude or direction of a relationship by scaling it according to the fraction. Understanding fractional coefficients is essential in forming, solving, and analyzing equations.
In the presented equation \(11 = \frac{2}{5}(n-13)\), the fractional coefficient \(\frac{2}{5}\) indicates that \(n - 13\) is being scaled by a fraction. This means instead of directly equating the outcomes, they are proportional based on the factor \(\frac{2}{5}\):

• They represent 'part' relationships in problems, like "two fifths of a quantity".
• Require careful manipulation in equations, especially during multiplication and division.
• Often necessitate steps like multiplying both sides of an equation to clear fractions.

Grasping the concept of fractional coefficients empowers students to handle equations systematically, simplifying steps in solving for unknowns or explaining relationships in context.