Problem 30 - Fundamentals of Mathematics

Question

\(\frac{1}{6}\) of what number is \(\frac{1}{6} ?\)

Step-by-Step Solution

Answer

The number is 1.

Step 1: Understand the Problem

We are asked to find a number such that when we take \( \frac{1}{6} \) of that number, we get \( \frac{1}{6} \). This can be expressed as an equation: \( \frac{1}{6} \times x = \frac{1}{6} \), where \( x \) is the number we need to find.

Step 2: Write the Equation

The equation representing the problem is \( \frac{1}{6} \times x = \frac{1}{6} \). Our goal is to solve for \( x \).

Step 3: Solve the Equation

To find \( x \), we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by 6 to eliminate the fraction. This gives us \( x = 6 \times \frac{1}{6} \).

Step 4: Simplify the Equation

Perform the multiplication on the right side: \( 6 \times \frac{1}{6} = 1 \). Thus, \( x = 1 \).

Step 5: Verify the Solution

Verify by plugging \( x = 1 \) back into the original condition: \( \frac{1}{6} \times 1 = \frac{1}{6} \), which holds true. Thus, the solution is correct.

Key Concepts

Understanding Mathematical Equations

Mathematical equations are like puzzles where we solve for unknown values. An equation balances two sides that are joined by an equals sign '='. Think of it as a balance scale, where each side needs to weigh the same. In this exercise, the equation is \( \frac{1}{6} \times x = \frac{1}{6} \).
Here, \( x \) is the unknown number we need to find, and the task is to ensure both sides of the equation remain equal.

• Keep the balance: Both sides of the equation must represent the same value.
• Use operations to move and simplify terms to isolate \( x \).

Understanding these concepts helps make complex problems simple, like finding out what times \( x \) we need to multiply to keep the equation true.

Problem Solving in Math

Solving a math problem is about understanding what is being asked and figuring out a plan to find the solution. For our problem, the goal is to find the number \( x \) such that \( \frac{1}{6} \) of \( x \) gives us \( \frac{1}{6} \). This involves visualizing the problem in steps.

Steps to Solve:

• Interpret the problem statement and translate it into a clear equation.
• Analyze how each part of the equation fits together.
• Apply operations that allow us to solve for the unknown.
• Think critically at each step to ensure math operations are applied correctly.

These steps help simplify complex problems by breaking them down into parts that are easier to manage and solve.

Multiplication in Fractions

Multiplying fractions may seem tricky, but once you understand how, it becomes simple. For fractions like \( \frac{1}{6} \times x = \frac{1}{6} \), following these rules can turn complexity into clarity.

• You multiply the numerators (top numbers) together.
• You multiply the denominators (bottom numbers) together.

Simplifying Fractions:

The key is to simplify by multiplying the reciprocal if needed. In the equation where multiplying by \( 6 \) allowed us to rewrite \( \frac{6}{6} \) as \( 1 \), making the solution \( x = 1 \). This shows that multiplying a number with its inverse yields one, a vital principle of fractions.
Understanding these steps equips you with the tools to approach all fraction multiplication problems efficiently.