Problem 205 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{2}{\frac{1}{3}+\frac{1}{5}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{15}{4} \).

Step 1: Identify the inner fractions

Recognize the fractions inside the denominator: \ \( \frac{1}{3} \ \) and \ \( \frac{1}{5} \ \).

Step 2: Find a common denominator

To add the fractions, find a common denominator. The least common multiple (LCM) of 3 and 5 is 15. Convert the fractions: \[ \frac{1}{3} = \frac{5}{15} \] and \[ \frac{1}{5} = \frac{3}{15} \].

Step 3: Add the fractions

Now, add the fractions with the common denominator: \[ \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \].

Step 4: Divide by a fraction

Simplify the fraction \( \frac{2}{\frac{8}{15}} \) by multiplying by the reciprocal of the denominator: \[ \frac{2}{1} \times \frac{15}{8} = \frac{2 \times 15}{1 \times 8} = \frac{30}{8} \].

Step 5: Simplify the result

Reduce \[ \frac{30}{8} \] to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \[ \frac{30 \div 2}{8 \div 2} = \frac{15}{4} \].

Key Concepts

inner fractions identification

Before you can start working on simplifying a complex fraction, you need to identify the inner fractions. In our exercise, we need to simplify \[\frac{2}{\frac{1}{3}+\frac{1}{5}}\]. Notice that there are two fractions in the denominator: \(\frac{1}{3}\) and \(\frac{1}{5}\). These are the inner fractions we need to work with first. By spotting these fractions early, you can create a clear plan to simplify the overall fraction step-by-step.

common denominator

After identifying the inner fractions, the next step is to find a common denominator. A common denominator allows us to add the fractions easily. For the fractions \(\frac{1}{3}\) and \(\frac{1}{5}\), the smallest number that both denominators can divide evenly into, or the least common multiple (LCM), is 15.
Convert each fraction so that they have 15 as their denominator. This involves multiplying the numerator and the denominator by whatever number makes the denominator 15:

• \(\frac{1}{3} \rightarrow \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\)
• \(\frac{1}{5} \rightarrow \frac{1 \times 3}{5 \times 3} = \frac{3}{15}\)

These new fractions with a common denominator can now be added together.

adding fractions

Once the fractions have a common denominator, they can be easily added. We now have \(\frac{5}{15}\) and \(\frac{3}{15}\). To add these fractions, simply add their numerators and keep the common denominator: \[ \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \] Adding fractions becomes straightforward when they share the same denominator. This step helps pave the way for the subsequent processes like dividing and simplifying the overall fraction.

dividing by a fraction

Now we need to simplify the complex fraction \(\frac{2}{\frac{8}{15}}\). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{8}{15}\) is \(\frac{15}{8}\). Therefore, we multiply \(\frac{2}{1}\) by \(\frac{15}{8}\):
\[\frac{2}{1} \times \frac{15}{8} = \frac{2 \times 15}{1 \times 8} = \frac{30}{8}\]
This approach simplifies the problem, making the next steps of reduction or further simplification more manageable.

simplifying fractions

Finally, it's essential to simplify the fraction to its simplest form. From our previous result, we have \( \frac{30}{8} \). To simplify, find the greatest common divisor (GCD) of 30 and 8, which is 2. Divide both the numerator and the denominator by 2:
\[\frac{30 \rightarrow 30 \text{ ÷ } 2}{8 \rightarrow 8 \text{ ÷ } 2} = \frac{15}{4}\] The final simplified form is \( \frac{15}{4} \). Simplifying fractions makes them easier to understand and work with, especially for more complex mathematical problems.