Problem 30 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{2}{3}-\frac{1}{2}}{\frac{1}{3}+\frac{1}{6}} $$

Step-by-Step Solution

Answer

\(\frac{1}{3}\)

Step 1 - Evaluate the Numerator

To begin, we need to evaluate the numerator of the fraction, which is \(\frac{2}{3} - \frac{1}{2} \). Find a common denominator for \(\frac{2}{3}\) and \(\frac{1}{2}\). The common denominator is 6. Transform each fraction: \(\frac{2}{3} = \frac{4}{6}\) and \(\frac{1}{2} = \frac{3}{6}\). Now subtract: \(\frac{4}{6} - \frac{3}{6} = \frac{1}{6}\).

Step 2 - Evaluate the Denominator

Next, evaluate the denominator, which is \(\frac{1}{3} + \frac{1}{6}\). Again, find a common denominator. The common denominator is 6. Transform each fraction: \(\frac{1}{3} = \frac{2}{6}\) and \(\frac{1}{6}\) stays the same. Now add: \(\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\).

Step 3 - Divide the Numerator by the Denominator

Now, divide the result of the numerator by the result of the denominator: \(\frac{\frac{1}{6}}{\frac{1}{2}}\). This is the same as multiplying by the reciprocal of the denominator: \(\frac{1}{6} \times \frac{2}{1} = \frac{2}{6} = \frac{1}{3}\). So, the simplified form is \(\frac{1}{3}\).

Key Concepts

Common Denominator

When adding or subtracting fractions, finding a common denominator is essential. A common denominator is a number that both denominators can evenly divide into. This shared baseline allows you to easily perform addition or subtraction.

In the given problem, we need to find the common denominators for both parts of the fraction. For \(\frac{2}{3} - \frac{1}{2}\):

• The denominators are 3 and 2.
• The smallest number both 3 and 2 can divide into is 6.
• So, the common denominator is 6.

This transformation helps us rewrite the fractions as \(\frac{4}{6}\) and \(\frac{3}{6}\).

For \(\frac{1}{3} + \frac{1}{6}\):

• The denominators are 3 and 6.
• Again, 6 is a number both denominators can divide into.

So, the fractions transform into \(\frac{2}{6}\) and \(\frac{1}{6}\). Having a common denominator simplifies both subtraction and addition.

Fraction Subtraction

Subtracting fractions requires a common denominator. Once we have the same denominators, we can subtract the numerators directly.

In the numerator of the problem, we have \(\frac{2}{3} - \frac{1}{2}\), which we converted to \(\frac{4}{6} - \frac{3}{6}\). Now, let's subtract:

• Subtract the numerators: 4 - 3 = 1
• Keep the common denominator: 6.

This gives us the result \(\frac{1}{6}\). Subtracting fractions becomes simple once the fractions share a common denominator.

Fraction Addition

Adding fractions also requires a common denominator. With the same denominators, just add the numerators.

In the denominator of the problem, we have \(\frac{1}{3} + \frac{1}{6}\), which transforms to \(\frac{2}{6} + \frac{1}{6}\). Now, let's add:

• Add the numerators: 2 + 1 = 3
• Keep the common denominator: 6.

This gives us \(\frac{3}{6}\), which simplifies to \(\frac{1}{2}\). Addition of fractions is straightforward once they share a common denominator.

Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. In the fractional division, multiplying by the reciprocal is essential.

In the final step, we need to divide \(\frac{1}{6}\) by \(\frac{1}{2}\). Division by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\). So, instead of dividing, we multiply by the reciprocal:
\(\frac{1}{6}\) x \(\frac{2}{1}\) = \(\frac{2}{6} = \frac{1}{3}\).
This turns our final simplified fraction into \(\frac{1}{3}\). The concept of reciprocals is invaluable in solving complex fraction problems.