Problem 27 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. Ten times a quantity increased by two is nine.

Step-by-Step Solution

Answer

The equation is \( 10x + 2 = 9 \) and solving it gives \( x = 0.7 \).

Step 1: Identify the Unknown Quantity

Let's begin by identifying the unknown quantity. In the given problem, the unknown quantity is not specified explicitly; therefore, we will use a variable to represent it. Let this unknown quantity be represented by the variable \( x \).

Step 2: Break Down the Phrase

The phrase "ten times a quantity" implies that we need to multiply the unknown quantity \( x \) by 10. This can be represented as \( 10x \).

Step 3: Incorporate the Additional Term

The phrase "increased by two" indicates that we need to add 2 to the expression from Step 2. Therefore, the expression becomes \( 10x + 2 \).

Step 4: Set Up the Equation

The phrase "is nine" tells us that the expression from Step 3 equals 9. We can now form the equation: \( 10x + 2 = 9 \).

Step 5: Solve the Equation

To find the value of \( x \), solve the equation \( 10x + 2 = 9 \). Start by subtracting 2 from both sides to get \( 10x = 7 \). Then, divide both sides by 10 to isolate \( x \), giving us \( x = 0.7 \).

Key Concepts

Understanding the 'Unknown Quantity'

When working with mathematical phrases or equations, it's common to come across the term 'unknown quantity.' This simply refers to an element in the equation that we do not yet know or have to determine. In algebra, we usually represent this unknown with a variable, commonly chosen from letters like \( x \), \( y \), or \( z \).

Identifying the unknown quantity is often the first crucial step in solving algebra problems. It allows us to set up a framework to understand the relationship described in the problem. For instance, in the phrase "ten times a quantity increased by two is nine," the unknown quantity here is the number we are multiplying by ten, which is unknown at the start.

Using a variable such as \( x \) to represent this unknown makes it easier to translate word problems into mathematical expressions. This variable becomes the placeholder for the value to be found, guiding us through to the solution.

Algebraic Equations and Their Formation

Algebraic equations are like a balanced scale and are fundamental in mathematics for describing relationships between quantities. They consist of two expressions set equal to each other, often involving variables, constants, and arithmetic operations such as addition or multiplication.

In translating English phrases into algebraic equations, the key is to look for words that indicate operations or comparisons:

• "Times" usually suggests multiplication.
• "Increased by" points to addition.
• "Is" or "equals" implies an equation or balance.

For the example, "Ten times a quantity increased by two is nine," each part of the phrase contributes to forming the complete equation:

• "Ten times a quantity" becomes \( 10x \).
• "Increased by two" turns into \( 10x + 2 \).
• Finally, "is nine" establishes the equation as \( 10x + 2 = 9 \).

Writing equations from phrases helps us represent complex ideas concisely, making them easier to solve and understand.

Solving Equations Step by Step

Solving an equation means finding the value of the unknown quantity that makes the equation true. For the given equation \( 10x + 2 = 9 \), we follow systematic steps to find \( x \).

Here’s how you can solve it:

• Isolate the term with the variable: Start by removing the constant term (or number) from one side to maintain the balance. By subtracting 2 from both sides of the equation, we get \( 10x = 7 \).
• Divide to isolate the variable: Next, divide both sides by the coefficient of the variable (10) to solve for \( x \). This step gives us \( x = 0.7 \).

Solving equations helps fine-tune your logical thinking and problem-solving skills, as you work through breaking down complex relationships into simple, logical steps. Always remember to check your solution by plugging it back into the original equation to ensure it satisfies the equation.