Finding the Unknown Fraction by Subtraction

The question asks us to find a fraction that, when subtracted from \(\frac{1}{3}\), results in \(\frac{1}{12}\). Let the unknown fraction be represented by \(x\).

Setting up the Equation

We can translate the problem into a mathematical equation:

\[ \frac{1}{3} - x = \frac{1}{12} \]

Solving for the Unknown Fraction

To find the value of \(x\), we need to isolate it. We can do this by rearranging the equation. Subtract \( \frac{1}{12} \) from both sides and add \( x \) to both sides, or simply move \( x \) to one side and \( \frac{1}{12} \) to the other:

\[ x = \frac{1}{3} - \frac{1}{12} \]

Now, we need to subtract the fractions \(\frac{1}{3}\) and \(\frac{1}{12}\). To subtract fractions, they must have a common denominator.

Finding a Common Denominator

The denominators are 3 and 12. The least common multiple (LCM) of 3 and 12 is 12.

We need to convert \(\frac{1}{3}\) into an equivalent fraction with a denominator of 12. We multiply both the numerator and the denominator by 4 (since \(3 \times 4 = 12\)):

\[ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]

Now the equation becomes:

\[ x = \frac{4}{12} - \frac{1}{12} \]

Performing the Subtraction

With a common denominator, we can subtract the numerators:

\[ x = \frac{4 - 1}{12} = \frac{3}{12} \]

Simplifying the Result

The fraction \(\frac{3}{12}\) can be simplified. Both 3 and 12 are divisible by 3. We divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

\[ x = \frac{3 \div 3}{12 \div 3} = \frac{1}{4} \]

So, the unknown fraction is \(\frac{1}{4}\).

Verification

Let's check our answer by plugging \(\frac{1}{4}\) back into the original equation:

\[ \frac{1}{3} - \frac{1}{4} \]

Find a common denominator for 3 and 4, which is 12.

\[ \frac{1 \times 4}{3 \times 4} - \frac{1 \times 3}{4 \times 3} = \frac{4}{12} - \frac{3}{12} = \frac{4 - 3}{12} = \frac{1}{12} \]

This matches the result given in the question, so our answer \(\frac{1}{4}\) is correct.

Conclusion

The fraction that, when taken away from \(\frac{1}{3}\), gives \(\frac{1}{12}\) is \(\frac{1}{4}\).

Revision Table: Fraction Subtraction Concepts

Additional Information on Fraction Operations

Understanding how to perform operations with fractions is fundamental in mathematics. Here's a bit more about related concepts:

• Adding Fractions: Similar to subtraction, fractions must have a common denominator before adding. You add the numerators and keep the common denominator.
• Multiplying Fractions: Multiply the numerators together and multiply the denominators together. Simplify the result if possible.
• Dividing Fractions: To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is obtained by flipping the numerator and denominator (e.g., the reciprocal of \( \frac{1}{4} \) is \( \frac{4}{1} \) or 4).
• Mixed Numbers and Improper Fractions: Mixed numbers contain a whole number part and a fraction part (like \( 1 \frac{1}{2} \)). Improper fractions have a numerator greater than or equal to the denominator (like \( \frac{3}{2} \)). You often convert between these forms when performing calculations.

Mastering these basic fraction operations is key to tackling more complex algebraic problems.