Problem 28 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{3}}{\frac{1}{2}+\frac{1}{7}} $$

Step-by-Step Solution

Answer

\frac{14}{27}

Step 1 - Simplify the Denominator

Start by simplifying the expression in the denominator, which is \(\frac{1}{2} + \frac{1}{7}\). To do this, find a common denominator for the fractions. The common denominator of 2 and 7 is 14. Rewrite each fraction with the common denominator: \(\frac{1}{2} = \frac{7}{14}\) and \(\frac{1}{7} = \frac{2}{14}\). Then, add the fractions together: \(\frac{7}{14} + \frac{2}{14} = \frac{9}{14}\).

Step 2 - Simplify the Expression

Now that the denominator \(\frac{1}{2} + \frac{1}{7}\) is simplified to \(\frac{9}{14}\), substitute it back into the original expression: \(\frac{\frac{1}{3}}{\frac{9}{14}}\).

Step 3 - Apply Division of Fractions

To divide the fractions \(\frac{1}{3}\) by \(\frac{9}{14}\), multiply by the reciprocal of the denominator: \(\frac{1}{3} \times \frac{14}{9}\).

Step 4 - Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \(\frac{1 \times 14}{3 \times 9} = \frac{14}{27}\).

Step 5 - Simplify the Fraction (if necessary)

Check if the fraction \(\frac{14}{27}\) can be simplified. Since 14 and 27 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Understanding Common Denominator

When adding or subtracting fractions, it’s important to find a common denominator. A common denominator is a shared multiple of the denominators in the fractions you are working with. For example, in the exercise we looked at two fractions: \(\frac{1}{2}\) and \(\frac{1}{7}\). The denominators here are 2 and 7. To add these fractions, you need a common multiple of these denominators. The lowest common multiple of 2 and 7 is 14.

To rewrite each fraction with this common denominator, we adjust the numerators: \(\frac{1}{2} = \frac{7}{14}\) and \(\frac{1}{7}= \frac{2}{14}\). This way, you keep the value of the fractions the same but with a unified denominator, making addition straightforward: \(\frac{7}{14} + \frac{2}{14} = \frac{9}{14}\).

Grasping The Reciprocal

The reciprocal of a number is simply the number flipped upside down. For example, the reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1} \) (or just 3) and vice versa. In the given problem, you encountered the fraction \(\frac{9}{14}\). Its reciprocal is \(\frac{14}{9}\).

Finding the reciprocal is crucial when dividing fractions. For example, dividing by a fraction is the same as multiplying by its reciprocal. In our exercise, this step turned \(\frac{1}{3}\frac{9}{14} \) into \(\frac{1}{3} \times \frac{14}{9}\), simplifying the process.

Simplifying Fraction Addition

Fraction addition requires matching up denominators. Once a common denominator is found (as shown in the first section), you can add the numerators directly: \(\frac{7}{14} + \frac{2}{14} = \frac{9}{14}\).

This is much like adding whole numbers but arranged over a common base. Always remember, the denominator stays the same after addition, only the numerators sum up.

Numerator and Denominator Basics

A fraction has two parts: the numerator and the denominator. The numerator (top part) shows how many parts you have, while the denominator (bottom part) shows how many equal parts the whole is divided into.

For example, in \(\frac{3}{4} \), 3 is the numerator, and 4 is the denominator. In the exercise, starting with \( \frac{1}{2}\) and \( \frac{1}{7} \), getting a common denominator of 14 meant converting them to \( \frac{7}{14} \) and \( \frac{2}{14} \) respectively. This standardizes the fractions, simplifying their addition to \( \frac{9}{14}. \) Finally, the division step transformed the operation by using reciprocals, showing the power of numerators and denominators in fraction operations.