Problem 76 - Elementary Algebra

Question

\(\frac{1}{3}-\left(\frac{1}{2}\right)^{2}\)

Step-by-Step Solution

Answer

\(\frac{1}{12}\).

Step 1: Calculate the Square

First, calculate the square of \(\frac{1}{2}\). This can be written as \[ \left(\frac{1}{2}\right)^{2} = \frac{1}{4} \].

Step 2: Substitute the Result

Next, substitute the result from Step 1 back into the original expression. The expression becomes: \[ \frac{1}{3} - \frac{1}{4} \].

Step 3: Find a Common Denominator

To subtract the fractions, find a common denominator. The least common denominator (LCD) of 3 and 4 is 12, so convert both fractions: \[ \frac{1}{3} = \frac{4}{12} \] and \[ \frac{1}{4} = \frac{3}{12} \].

Step 4: Subtract the Fractions

Now, subtract the two fractions: \[ \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \].

Key Concepts

Fraction Subtraction

Subtracting fractions is an essential skill in math. It's a bit different from subtracting whole numbers. The key point in fraction subtraction is to make sure that the fractions have the same denominator before you subtract the numerators. For example, in the expression \(\frac{1}{3} - \frac{1}{4}\), you can't directly subtract the fractions because 3 and 4 are different denominators.

Steps to subtract fractions:

• Find a common denominator.
• Convert each fraction to the equivalent fraction with the common denominator.
• Subtract the numerators.
• Simplify the resulting fraction if possible.

This step-by-step method helps ensure that you subtract fractions accurately.

Least Common Denominator

To subtract fractions, finding the least common denominator (LCD) is crucial. The LCD is the smallest number that both denominators can divide into evenly. For example, in \(\frac{1}{3} - \frac{1}{4}\), the denominators are 3 and 4. To find the LCD, list the multiples of each number:

• Multiples of 3: 3, 6, 9, 12, 15...
• Multiples of 4: 4, 8, 12, 16...

The smallest common multiple is 12, so the LCD is 12. Once you have the LCD, you can convert the fractions to \(\frac{4}{12}\) and \(\frac{3}{12}\). It's much easier to work with fractions when they share the same denominator.

Squared Fractions

Sometimes, you might encounter fractions that need to be squared. Squaring a fraction means multiplying the fraction by itself. For instance, \(\frac{1}{2}\) squared is:
\[ \left( \frac{1}{2} \right)^{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

When squaring a fraction, remember to square both the numerator and the denominator. This ensures you get the correct value. In our example exercise, squaring \(\frac{1}{2}\) was an early step to solve the main problem. Once we got that result, we could substitute back into the original expression.

Fraction Conversion

Converting fractions is a crucial skill, especially when dealing with different denominators. In the solved exercise, converting \(\frac{1}{3}\) to \(\frac{4}{12}\) and \(\frac{1}{4}\) to \(\frac{3}{12}\) allowed us to perform subtraction. Here's the general process:

• Identify the LCD for the fractions involved.
• Multiply the numerator and the denominator of each fraction by the necessary factor to get the LCD.
• Perform the required arithmetic operation, such as addition or subtraction.

Converting fractions ensures they share the same base, facilitating easier calculations. This process is fundamental when working with multiple fractions in any context.