Problem 120 - Introductory Algebra for College Students

Question

Perform the indicated operations. Begin by performing operations in parentheses. $$\left(\frac{1}{2}+\frac{1}{4}\right) \div\left(\frac{1}{2}+\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The short answer is \( \frac{9}{10} \).

Step 1: Solve Numerator

Begin by solving the operation within parentheses in the numerator: \( \frac{1}{2} + \frac{1}{4} \). These are fractional numbers with different denominators. To add them, you need to find the least common multiple of 2 and 4, which is 4. Convert \(\frac{1}{2}\) to the equivalent fraction that has 4 as denominator by multiplying both the numerator and the denominator by 2. So, the equivalent fraction for \(\frac{1}{2}\) will be \(\frac{2}{4}\). Now both \(\frac{2}{4}\) and \(\frac{1}{4}\) have the same denominator and their addition would yield \(\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\).

Step 2: Solve Denominator

Now solve the operation within parentheses in the denominator: \( \frac{1}{2} + \frac{1}{3}\). Here, the denominators are 2 and 3, so to add these fractions, you need to find the least common multiple of 2 and 3 which is 6. Convert both fractions to have the denominators as 6. So, \(\frac{3}{6}\) is the equivalent fraction for \( \frac{1}{2}\) and \(\frac{2}{6}\) is the equivalent one for \(\frac{1}{3}\). Thus, \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Step 3: Peform Division

As per the problem statement, you need to perform the division \( \frac{3}{4} \div \frac{5}{6} \). In fractional division, change the division to multiplication and flip (find the reciprocal of) the second fraction. Then, the operation becomes \( \frac{3}{4} * \frac{6}{5} \). Multiply the numerators together for new numerator and denominators together for new denominator. \( \frac{3*6}{4*5} = \frac{18}{20} \). This can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, \( \frac{18}{20} = \frac{9}{10}\).