Problem 29 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

\frac{1}{8}

Step 1 - Find a common denominator for the fractions in the numerator

The fractions in the numerator are \(\frac{2}{5}\) and \(\frac{1}{3}\). To subtract these fractions, convert them to have a common denominator. The least common denominator (LCD) of 5 and 3 is 15. Therefore, \(\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}\) and \(\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\).

Step 2 - Subtract the fractions in the numerator

With the common denominator, we now subtract the fractions: \(\frac{6}{15} - \frac{5}{15} = \frac{6 - 5}{15} = \frac{1}{15}\).

Step 3 - Find a common denominator for the fractions in the denominator

The fractions in the denominator are \(\frac{1}{5}\) and \(\frac{1}{3}\). Convert them to a common denominator, which is 15. Therefore, \(\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}\) and \(\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\).

Step 4 - Add the fractions in the denominator

With the common denominator, we now add the fractions: \(\frac{3}{15} + \frac{5}{15} = \frac{3 + 5}{15} = \frac{8}{15}\).

Step 5 - Divide the results of the numerator by the denominator

We now have \(\frac{\frac{1}{15}}{\frac{8}{15}}\). Dividing these fractions is equivalent to multiplying the numerator by the reciprocal of the denominator: \(\frac{1}{15} \times \frac{15}{8} = \frac{1 \times 15}{15 \times 8} = \frac{15}{120} = \frac{1}{8}\).

Key Concepts

common denominator

A common denominator is needed when adding or subtracting fractions. It is a shared multiple of the denominators of the fractions you are working with. Finding a common denominator makes the fractions compatible for these operations. Typically, the least common denominator (LCD) is used because it simplifies calculations. For example, for the fractions \(\frac{2}{5}\) and \(\frac{1}{3}\), the LCD of 5 and 3 is 15. Thus, we convert \(\frac{2}{5}\) to \(\frac{6}{15}\) and \(\frac{1}{3}\) to \(\frac{5}{15}\). Now, they share a common denominator, making it possible to add or subtract these fractions easily.

subtracting fractions

Subtracting fractions might seem tricky, but it gets easier with a common denominator. Start by converting the fractions so that their denominators are the same. Once this is done, simply subtract the numerators and keep the denominator the same. For the fractions \(\frac{2}{5}\) and \(\frac{1}{3}\) with a common denominator of 15, we get \(\frac{6}{15} - \frac{5}{15} = \frac{1}{15}\). This result, \(\frac{1}{15}\), is what we get after subtracting the original fractions.

adding fractions

Adding fractions also requires a common denominator. Once fractions share a common denominator, you can add their numerators directly. Keep the denominator the same. For instance, if we need to add \(\frac{1}{5}\) and \(\frac{1}{3}\), first convert them to a common denominator of 15. This gives us \(\frac{3}{15}\) and \(\frac{5}{15}\). Adding these together, we get \(\frac{3}{15} + \frac{5}{15} = \frac{8}{15}\).

reciprocal

The reciprocal of a fraction is found by flipping the numerator and the denominator. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). Reciprocals are especially useful in division of fractions. Instead of dividing by a fraction, you multiply by its reciprocal. For example, when dividing \(\frac{\frac{1}{15}}{\frac{8}{15}}\), we multiply \(\frac{1}{15}\) by the reciprocal of \(\frac{8}{15}\), which is \(\frac{15}{8}\). The calculation becomes \(\frac{1}{15} \times \frac{15}{8} = \frac{15}{120} = \frac{1}{8}\).