Problem 61 - Algebra for College Students

Question

Use a calculator to evaluate each complex fraction. Round answers to four decimal places. If your calculator does fractions, then also find the fractional answer. $$\frac{\frac{5}{3}-\frac{4}{5}}{\frac{1}{3}-\frac{5}{6}}$$

Step-by-Step Solution

Answer

-1.7333

Step 1 - Simplify the Numerator

To simplify the numerator, calculate \ \( \frac{5}{3} - \frac{4}{5} \). First, find a common denominator, which is 15. Convert each fraction: \( \frac{5}{3} = \frac{25}{15} \) and \( \frac{4}{5} = \frac{12}{15} \). Now, subtract: \( \frac{25}{15} - \frac{12}{15} = \frac{13}{15} \).

Step 2 - Simplify the Denominator

To simplify the denominator, calculate \ \( \frac{1}{3} - \frac{5}{6} \). Find a common denominator, which is 6. Convert each fraction: \( \frac{1}{3} = \frac{2}{6} \) and \( \frac{5}{6} \) remains the same. Now, subtract: \( \frac{2}{6} - \frac{5}{6} = \frac{-3}{6} = \frac{-1}{2} \).

Step 3 - Divide the Numerator by the Denominator

Now, divide the simplified numerator by the simplified denominator: \( \frac{\frac{13}{15}}{\frac{-1}{2}} \). When dividing fractions, multiply by the reciprocal: \( \frac{13}{15} \cdot \frac{-2}{1} = \frac{13 \cdot -2}{15 \cdot 1} = \frac{-26}{15} \).

Step 4 - Convert to Decimal and Round

Convert the fraction \( \frac{-26}{15} \) to a decimal. Using a calculator, \( \frac{-26}{15} \approx -1.7333 \). Round to four decimal places to confirm: \( -1.7333 \).

Key Concepts

Numerator

The numerator is the top part of a fraction. It represents how many parts of a whole are being considered. For example, in the fraction \( \frac{5}{3} \), the numerator is 5. When simplifying complex fractions, you need to handle the numerators separately. In our exercise, we first need to work with the numerator \( \frac{5}{3} - \frac{4}{5} \). We find a common denominator (15 in this case) and perform the subtraction: \( \frac{25}{15} - \frac{12}{15} = \frac{13}{15} \). This process ensures that we work with the correct value for the top part of the fraction.

Denominator

The denominator is the bottom part of a fraction, displaying the total number of equal parts the whole is divided into. For instance, in the fraction \( \frac{4}{5} \), the denominator is 5, meaning the whole is divided into 5 equal parts. The denominator is crucial for the integrity of the fraction. In our given complex fraction, the denominator is \( \frac{1}{3} - \frac{5}{6} \). To simplify, we need a common denominator. The lowest common denominator between 3 and 6 is 6. We convert each fraction: \( \frac{1}{3} = \frac{2}{6} \) and \( \frac{5}{6} \) remains the same. Then we perform the subtraction: \( \frac{2}{6} - \frac{5}{6} = \frac{-3}{6} = \frac{-1}{2} \).

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. To subtract or add fractions, they must have the same denominator. For example, in the fractions \( \frac{5}{3} \) and \( \frac{4}{5} \), the denominators are 3 and 5, respectively. To find a common denominator, identify the least common multiple (LCM). Here, the LCM of 3 and 5 is 15. Thus, we convert \( \frac{5}{3} \) to \( \frac{25}{15} \) and \( \frac{4}{5} \) to \( \frac{12}{15} \). Now, we can easily subtract them: \( \frac{25}{15} - \frac{12}{15} \).

Fraction Subtraction

Subtracting fractions involves two steps: ensuring a common denominator and then performing the subtraction. First, identify the common denominator, then convert each fraction to have this denominator. For example, subtracting \( \frac{5}{3} \) and \( \frac{4}{5} \) involves converting them to \( \frac{25}{15} \) and \( \frac{12}{15} \) respectively. Subtract the numerators: \( 25 - 12 = 13 \), resulting in \( \frac{13}{15} \). Always ensure the fractions have a common denominator before subtracting.

Reciprocal Multiplication

When dealing with division of fractions, we use the concept of reciprocal multiplication. Multiply by the reciprocal of the divisor, which essentially means flipping the divisor fraction upside down. In our problem, we have \( \frac{\frac{13}{15}}{\frac{-1}{2}} \). To solve this, multiply \( \frac{13}{15} \) by the reciprocal of \( \frac{-1}{2} \), which is \( \frac{-2}{1} \). The multiplication is straightforward: \( \frac{13}{15} \times \frac{-2}{1} = \frac{13 \times -2}{15 \times 1} = \frac{-26}{15} \). This step is vital when simplifying complex fractions.