Finding the Unknown Fraction

The question asks us to find a specific fraction. We are told that when this fraction is subtracted from 1/2, the result is 2/3. We can represent this problem using an equation.

Let the unknown fraction be denoted by $x$.

According to the problem statement, we have:

When the fraction $x$ is taken away from $\frac{1}{2}$, the result is $\frac{2}{3}$.

This can be written as the equation:

$\frac{1}{2} - x = \frac{2}{3}$

Our goal is to solve this equation for $x$.

Solving the Equation for the Fraction

To find the value of the unknown fraction $x$, we need to isolate $x$ on one side of the equation. We can do this by rearranging the terms:

Start with the equation:

$\frac{1}{2} - x = \frac{2}{3}$

Subtract $\frac{1}{2}$ from both sides of the equation:

$-x = \frac{2}{3} - \frac{1}{2}$

Now, we need to subtract the fractions on the right-hand side. To subtract fractions, they must have a common denominator. The least common multiple of 3 and 2 is 6. We convert both fractions to have a denominator of 6:

• $\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$
• $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

Now substitute these equivalent fractions back into the equation:

$-x = \frac{4}{6} - \frac{3}{6}$

Perform the subtraction of the fractions:

$-x = \frac{4 - 3}{6}$

$-x = \frac{1}{6}$

To find $x$, we multiply both sides of the equation by -1:

$(-1) \times (-x) = (-1) \times \frac{1}{6}$

$x = -\frac{1}{6}$

So, the unknown fraction is $-\frac{1}{6}$.

Verifying the Fraction

We can check our answer by substituting $x = -\frac{1}{6}$ back into the original equation:

$\frac{1}{2} - \left(-\frac{1}{6}\right) = \frac{1}{2} + \frac{1}{6}$

To add these fractions, we find a common denominator, which is 6:

$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

So, the expression becomes:

$\frac{3}{6} + \frac{1}{6} = \frac{3 + 1}{6} = \frac{4}{6}$

The fraction $\frac{4}{6}$ can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 2:

$\frac{4 \div 2}{6 \div 2} = \frac{2}{3}$

The result $\frac{2}{3}$ matches the right-hand side of the original equation. Therefore, our calculated fraction $-\frac{1}{6}$ is correct.

Summary of the Solution Process

We followed these steps to find the fraction:

1. Set up an equation based on the word problem.
2. Represent the unknown fraction with a variable ($x$).
3. Isolate the variable in the equation.
4. Perform fraction subtraction using a common denominator.
5. Solve for the variable.

Comparing our result $x = -\frac{1}{6}$ with the given options, we find that it matches one of the options.

Revision Table: Fraction Subtraction and Equations

Additional Information: Working with Fractions and Algebra

Solving problems involving fractions often requires a good understanding of fraction operations (addition, subtraction, multiplication, division) and algebraic techniques for solving equations. When dealing with equations involving fractions, it's often helpful to clear the denominators by multiplying the entire equation by the least common multiple (LCM) of all denominators. However, in this case, directly performing fraction subtraction was also straightforward.

For example, to solve $\frac{1}{2} - x = \frac{2}{3}$ by clearing denominators, you would multiply everything by 6 (the LCM of 2 and 3):

$6 \times \left(\frac{1}{2} - x\right) = 6 \times \frac{2}{3}$

$6 \times \frac{1}{2} - 6 \times x = 6 \times \frac{2}{3}$

$3 - 6x = 4$

Then, solve this simpler linear equation:

$-6x = 4 - 3$

$-6x = 1$

$x = \frac{1}{-6} = -\frac{1}{6}$

Both methods lead to the same correct answer. Choosing the method depends on personal preference and the complexity of the equation.