Problem 17 - Algebra A Combined Function

Question

Solve the following. Three times the reciprocal of a number equals 9 times the reciprocal of \(6 .\) Find the number.

Step-by-Step Solution

Answer

The number is 2.

Step 1: Understanding the problem

We are given that three times the reciprocal of a number equals nine times the reciprocal of 6. We need to find the number. Reciprocal of a number, say \( x \), is \( \frac{1}{x} \).

Step 2: Write the equation

Let the unknown number be \( x \). The problem states that three times the reciprocal of the number equals nine times the reciprocal of 6. Therefore, the equation is: \( 3 \times \frac{1}{x} = 9 \times \frac{1}{6} \).

Step 3: Simplify the right side

Calculate \( 9 \times \frac{1}{6} \) which simplifies to \( \frac{9}{6} = \frac{3}{2} \).

Step 4: Set up the equation

Now, our equation is \( \frac{3}{x} = \frac{3}{2} \).

Step 5: Solve for \( x \)

Cross-multiply to solve for \( x \): \( 3 \times 2 = x \times 3 \). Simplifying, we get \( 6 = 3x \).

Step 6: Divide to isolate \( x \)

Divide both sides by 3 to get \( x = \frac{6}{3} = 2 \).

Key Concepts

Reciprocal

In mathematics, a reciprocal is a number that, when multiplied by the original number, results in the product being 1. The concept of reciprocals is crucial in algebra, especially when dealing with equations involving fractions. If you have a non-zero number, say \( x \), the reciprocal of this number is \( \frac{1}{x} \). This means that when \( x \) is multiplied by \( \frac{1}{x} \), the result is \( \frac{x}{x} = 1 \).

Reciprocals are useful for simplifying division problems. Instead of dividing by a number, you can multiply by its reciprocal. For instance, rather than computing \( \frac{a}{b} \), you can calculate \( a \times \frac{1}{b} \). This makes operations simpler and often more intuitive in algebraic manipulations.

Equation Solving

Solving an equation involves finding the value of the unknown variable that makes the equation true. In our exercise, the equation was set up from the problem statement: \( 3 \times \frac{1}{x} = 9 \times \frac{1}{6} \). This gives us an equation with the variable \( x \) to be solved.

Equation solving involves several systematic steps:

• First, understand the problem and translate it into a mathematical statement or equation.
• Simplify any complex components in the equation. This often involves carrying out basic arithmetic operations or simplifying fractions.
• Finally, isolate the variable (in this case \( x \)), to find its value.

Equation solving is foundational in algebra and many mathematical concepts because it helps us find unknown information from given data.

Cross-Multiplication

Cross-multiplication is a technique used to solve equations involving two fractions. When you have an equation like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication simplifies it by multiplying across the equation diagonally, leading to \( a \cdot d = b \cdot c \).

This technique is particularly useful because it clears the fractions, making the equation easier to handle. In our problem, we reached the step where we had \( \frac{3}{x} = \frac{3}{2} \). By cross-multiplying, we convert this into \( 3 \times 2 = x \times 3 \). This simplifies directly to a basic equation: \( 6 = 3x \), making it easy to solve for \( x \).

Cross-multiplication is widely used in solving proportion-related problems and is a reliable method for students to practice and polish their algebra skills.