Problem 32 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. Eleven fifteenths of two more than a number is eight.

Step-by-Step Solution

Answer

Answer: \(\frac{11}{15}(x+2) = 8\)

Step 1: Identify the unknown number

Let's represent the unknown number as "x".

Step 2: Translate 'two more than a number' to an expression

Since 'two more' means adding two, the phrase 'two more than a number' can be represented as (x+2).

Step 3: Translate 'eleven fifteenths' to a fraction

The phrase 'eleven fifteenths' is a fraction, and can be represented as \(\frac{11}{15}\).

Step 4: Translate 'eleven fifteenths of two more than a number' to an expression

Multiply the fraction \(\frac{11}{15}\) by the expression (x+2) to represent 'eleven fifteenths of two more than a number'. So, this can be written as \(\frac{11}{15}(x+2)\).

Step 5: Translate the whole sentence into an equation

Finally, the sentence 'Eleven fifteenths of two more than a number is eight' can be translated to the following equation: \(\frac{11}{15}(x+2) = 8\).

Key Concepts

Mathematical Expressions

Mathematical expressions are a way to convey ideas using numbers, variables, and mathematical symbols. They help us represent complex thoughts in a simple numeric form. For example, if you want to express the idea of adding two to a number, you could write it as \( x + 2 \) where \( x \) stands for the number in question. By translating phrases into these expressions, you effectively convert real-world situations into a format that can be easily manipulated in algebra or calculus.

• Clarity: Mathematical expressions provide a clear and concise way to detail operations such as addition, subtraction, multiplication, or division.
• Universality: They are a universal language understood across various scientific fields, making collaboration more effective.

Understanding and creating mathematical expressions is a fundamental skill for solving word problems and equations.

Algebraic Equations

Algebraic equations are statements that show the equality of two expressions. They form the bedrock of algebraic problem-solving. In our given problem, the phrase 'Eleven fifteenths of two more than a number is eight' becomes an equation: \( \frac{11}{15}(x+2) = 8 \). This equation states that when you perform the operation given in the expression, you should end up with 8.

• Balancing Method: Solving equations often involves maintaining balance, meaning what you do to one side of the equation, you must do to the other.
• Process: The primary goal is to isolate the unknown variable to determine its value.

With algebraic equations, you take abstract problems and make them solvable through logical steps, allowing you to find unknown values.

Fractions

Fractions represent a part of a whole and are a common component of mathematical expressions and equations. In this exercise, 'eleven fifteenths' is written as the fraction \( \frac{11}{15} \). Fractions like this show the ratio of two numbers, where the top number (numerator) is divided by the bottom number (denominator).

• Simplification: Fractions can often be simplified by dividing both the numerator and denominator by their greatest common divisor.
• Multiplication: When multiplying fractions, you multiply the numerators together and the denominators together, simplifying the result if possible.

Fractions can appear complex, but understanding their operations is key in translating word problems into mathematical language and solving them accurately.

Unknown Variables

Unknown variables are symbols like \( x \) used in expressions and equations to represent numbers we do not yet know. They are placeholders that take on specific values once an equation is solved. In our problem, the unknown variable is \( x \), representing the unknown number mentioned in the sentence.

• Purpose: Variables allow us to build mathematical models that describe real-world situations.
• Solution: To solve equations with unknown variables, you manipulate the equation to isolate and find the value of the variable.

Understanding how to work with unknown variables is essential in solving algebraic equations, as they often hold the key to the problem's solution.