Problem 67 - Algebra 1

Question

Write the verbal sentence as an equation. Twelve less than nine times a number is 60 .

Step-by-Step Solution

Answer

The equation is \(9n - 12 = 60\).

Step 1: Identify the unknown

The problem speaks about 'a number', which is not given. This number will be the unknown. We can denote it by a variable, say 'n' for number.

Step 2: Translate to Mathematical sentence

In the sentence 'Twelve less than nine times a number is 60', 'nine times a number' implies \(9n\). The 'less than' operation implies a subtraction involving a value '12'. So, the first part translates to \(9n - 12\). The term 'is' in mathematics refers to equality. Therefore, we can write the equation as \(9n - 12 = 60\).

Key Concepts

Verbal Expressions

When solving mathematics problems, we often encounter verbal expressions. These are phrases or sentences that describe mathematical operations using words. Understanding verbal expressions is crucial for translating real-world situations or problems into mathematical language.
For instance, in our example, the sentence "Twelve less than nine times a number is 60" is a verbal expression. Here, 'twelve less than' indicates subtraction, while 'nine times a number' suggests multiplication. Finally, the term 'is' signifies an equality or equation setup. Breaking down such expressions helps clearly identify mathematical operations needed for forming equations.

Variables

Variables are essential in mathematics as they represent unknown values in problems. They act as placeholders that allow us to create equations and solve for these unknowns.
In our exercise, the phrase 'nine times a number' implies a multiplication involving an unknown quantity. To work with this unknown, we introduce a variable, commonly represented by a letter like 'n'.
By using the variable 'n', we can effectively handle the unknown, perform calculations, and ultimately find its value. Variables not only make equations simpler but also help apply the same methods to different situations, providing a fundamental tool in algebra and mathematics as a whole.

Translation to Mathematical Sentences

The final step in our exercise involves translating verbal expressions into equations or mathematical sentences. This process helps convert complex, wordy problems into concise, solvable mathematical forms.
In the given problem, 'nine times a number' becomes the mathematical term \( 9n \), where \( n \) represents our variable. The phrase 'twelve less than' signals us to subtract 12 from this product, forming the expression \( 9n - 12 \). Lastly, translating 'is 60' into a mathematical sentence involves setting the expression equal to 60: \( 9n - 12 = 60 \).
Through this translation, we construct a clear equation that can be solved, showing how verbal statements relate directly to mathematical procedures and solutions.