Problem 25 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{2}+\frac{1}{3}}{\frac{1}{3}+\frac{1}{5}} $$

Step-by-Step Solution

Answer

\frac{25}{16}

Step 1 - Simplify the numerator

First, we need to find a common denominator for the fractions in the numerator. For \(\frac{1}{2} + \frac{1}{3}\), the least common denominator (LCD) is 6. Rewrite each fraction with the LCD: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\). Now add them: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Step 2 - Simplify the denominator

Next, find a common denominator for the fractions in the denominator. For \(\frac{1}{3} + \frac{1}{5}\), the LCD is 15. Rewrite each fraction with the LCD: \(\frac{1}{3} = \frac{5}{15}\) and \(\frac{1}{5} = \frac{3}{15}\). Now add them: \(\frac{5}{15} + \frac{3}{15} = \frac{8}{15}\).

Step 3 - Divide the fractions

Now that we have simplified both the numerator and the denominator, we need to divide the two fractions we obtained: \(\frac{\frac{5}{6}}{\frac{8}{15}}\). To divide fractions, multiply the first fraction by the reciprocal of the second fraction: \(\frac{5}{6} \times \frac{15}{8}\).

Step 4 - Simplify the result

Multiply the fractions: \(\frac{5 \times 15}{6 \times 8} = \frac{75}{48}\). Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (which is 3): \(\frac{75 \div 3}{48 \div 3} = \frac{25}{16}\).

Key Concepts

Least Common Denominator

When adding fractions, it's crucial to find a common denominator for all the terms to ensure they share a common base for addition. This common base is called the 'Least Common Denominator' (LCD). In our example, to add fractions \(\frac{1}{2} + \frac{1}{3}\), we need the smallest number that both denominators (2 and 3) can divide without a remainder. The LCD for 2 and 3 is 6. So, we rewrite each fraction with this denominator: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\). This makes it easy to add them up: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Fraction Addition

Once we have our common denominator in place, adding fractions becomes straightforward. We keep the denominator the same and add the numerators. Using our example with the numerator fractions \(\frac{1}{2} + \frac{1}{3}\), after converting to the common denominator of 6, we add \(\frac{3}{6} + \frac{2}{6}\). The result is \(\frac{5}{6}\). Similarly, in the denominator of our original expression, \(\frac{1}{3} + \frac{1}{5}\), the smallest common denominator for 3 and 5 is 15. We rewrite the fractions: \(\frac{1}{3} = \frac{5}{15}\) and \(\frac{1}{5} = \frac{3}{15}\). Then, we add both fractions: \(\frac{5}{15} + \frac{3}{15} = \frac{8}{15}\). Now we have simplified both the numerator and the denominator of our original expression.

Fraction Division

Dividing fractions might seem tricky, but it's actually quite simple. The trick is to multiply the first fraction by the reciprocal (flipped version) of the second fraction. For our example, we end up with \(\frac{5}{6} \div \frac{8}{15}\), which we rewrite as \(\frac{5}{6} \times \frac{15}{8}\). Multiplying these fractions directly gives us: \(\frac{5 \times 15}{6 \times 8} = \frac{75}{48}\). This step transforms the division into a straightforward multiplication process.

Greatest Common Divisor

To simplify the product of our fraction multiplication \(\frac{75}{48}\), we can divide both the numerator and the denominator by their Greatest Common Divisor (GCD). The GCD is the largest number that can exactly divide both the numerator and the denominator. For 75 and 48, the GCD is 3. We divide both parts of \(\frac{75}{48}\): numerator: \(\frac{75}{3} = 25\) and denominator: \(\frac{48}{3} = 16\). Therefore, the simplified fraction is \(\frac{25}{16}\). By finding and applying the GCD, we reduce fractions to their simplest form efficiently.