Problem 42 - Algebra A Combined Function

Question

Solve the following. If 15 times the reciprocal of a number is added to the ratio of 9 times the number minus 7 and the number plus \(2,\) the result is \(9 .\) What is the number?

Step-by-Step Solution

Answer

The number is 3.

Step 1: Define the Problem

Let's denote the unknown number as \( x \). Our task is to form an equation based on the information given and solve for \( x \).

Step 2: Understanding the Recursive Component

The problem involves the reciprocal of the number. The reciprocal of \( x \) is \( \frac{1}{x} \). According to the problem, 15 times the reciprocal of the number is part of the expression. So, this becomes \( 15 \times \frac{1}{x} \).

Step 3: Formulate the Equation

The problem states: "15 times the reciprocal of a number is added to the ratio of 9 times the number minus 7 and the number plus 2." We can express this mathematically as follows:\[ 15 \times \frac{1}{x} + \frac{9x - 7}{x + 2} = 9 \].

Step 4: Eliminate Fractions by Finding a Common Denominator

To simplify and solve the equation, eliminate fractions by multiplying the entire equation by \( x(x + 2) \), the common denominator:\[ 15(x + 2) + (9x - 7)x = 9x(x + 2) \].

Step 5: Simplify and Rearrange the Equation

Simplify the equation from Step 4:- Distribute: \[ 15x + 30 + 9x^2 - 7x = 9x^2 + 18x \]- Combine like terms: \[ 15x - 7x + 30 = 18x \]So, the expanded equation is \( 9x^2 + 8x + 30 = 9x^2 + 18x \).

Step 6: Solve for \( x \)

Subtract \( 9x^2 \) from both sides:\[ 8x + 30 = 18x \]Subtract \( 8x \) from both sides:\[ 30 = 10x \]Divide both sides by 10:\[ x = 3 \].

Step 7: Validate the Solution

Substitute \( x = 3 \) back into the original equation to ensure correctness:- Calculate the reciprocal part: \( 15 \times \frac{1}{3} = 5 \).- Calculate the ratio part: \( \frac{9(3) - 7}{3 + 2} = \frac{27 - 7}{5} = 4 \).- Sum: \( 5 + 4 = 9 \).Since the original expression equals 9, the solution \( x = 3 \) is correct.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are a fundamental aspect of mathematics, used to find unknown values with the help of known quantities. In this particular exercise, the unknown is represented by the variable \( x \). The goal is to use the given information to form an equation that allows us to solve for \( x \). To do this, we start by defining expressions or terms based on the problem statement.
For example, we have the reciprocal of \( x \), which is expressed as \( \frac{1}{x} \). Another term, "15 times the reciprocal of the number," becomes \( 15 \times \frac{1}{x} \). These expressions are then combined into a full equation, as shown:
\[ 15 \times \frac{1}{x} + \frac{9x - 7}{x + 2} = 9 \].
The challenge of solving algebraic equations lies in translating the language of the problem into a mathematical format and then manipulating the equation using algebraic principles such as addition, subtraction, multiplication, and division to find the value of the unknown.

Exploring Ratio and Proportion

Ratios are simply comparisons between two quantities. Proportions, on the other hand, assert that two ratios are equivalent. In solving problems, understanding these concepts allows us to establish relationships between numbers.
In the provided problem, we see a ratio expressed as a fraction: \( \frac{9x - 7}{x + 2} \). This fraction represents what is referred to as the ratio of "9 times the number minus 7" to "the number plus 2." By inserting the known value or known expression into the ratio, you establish a clear mathematical relationship.
In proportion problems, you often work to find a common relationship or missing value that makes the ratios equal. Although this problem does not explicitly require solving for a proportional relationship, understanding ratios helps in interpreting complex algebraic problems.
By analyzing ratios properly, you transform the words and comparisons into expressions that can be used with other parts of the algebraic equation, paving the way for a solution.

Simplifying Fractions in Equations

Simplifying fractions is a critical step in making equations less complex and more manageable. The key aspect of simplifying fractions is finding a common denominator to eliminate the fractional components. In our problem, the equation involves two fractions: \( 15 \times \frac{1}{x} \) and \( \frac{9x - 7}{x + 2} \). To simplify these fractions:

• Identify a common denominator: Here, \( x(x + 2) \) serves this purpose, encompassing both denominators \( x \) and \( x + 2 \).
• Multiply every term in the equation by this common denominator, strategically removing the fractions: \[ 15(x + 2) + (9x - 7)x = 9x(x + 2) \].

This process is essential in solving the equation, as it turns a potentially unwieldy formula into a simpler form, making it easier to isolate \( x \) and find its value. Simplifying fractions requires attention to detail to ensure that each term is treated correctly, facilitating the path to the solution.