Problem 47 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{1}{6 m}+\frac{2}{5 m}+\frac{4}{m}$$

Step-by-Step Solution

Answer

\( \frac{137}{30m} \)

Step 1: Identify the problem type

We are given the problem of adding three fractions with a common variable in their denominators: \( \frac{1}{6m} + \frac{2}{5m} + \frac{4}{m} \). We need to find the sum of these fractions.

Step 2: Identify the common denominator

To simplify the addition process, we first need to find a common denominator for the fractions. The denominators are \(6m\), \(5m\), and \(m\). The least common denominator (LCD) here is \(30m\) since \(30\) is the least common multiple of \(6\), \(5\), and \(1\).

Step 3: Rewrite each fraction with the common denominator

Transform each fraction so that they all have the common denominator \(30m\):- \( \frac{1}{6m} = \frac{1 \times 5}{6m \times 5} = \frac{5}{30m} \)- \( \frac{2}{5m} = \frac{2 \times 6}{5m \times 6} = \frac{12}{30m} \)- \( \frac{4}{m} = \frac{4 \times 30}{m \times 30} = \frac{120}{30m} \)

Step 4: Add the fractions

Since all fractions now have the common denominator \(30m\), add them directly:\[ \frac{5}{30m} + \frac{12}{30m} + \frac{120}{30m} = \frac{5 + 12 + 120}{30m} = \frac{137}{30m} \]

Step 5: Verify and simplify the result

The result of \( \frac{137}{30m} \) cannot be simplified further since \(137\) and \(30\) have no common factors other than 1. Therefore, the final answer is simplified.

Key Concepts

Fractions

Fractions are a way of representing a part of a whole. They consist of a numerator and a denominator. The numerator, which is the top number, tells you how many parts we are dealing with, while the denominator, the bottom number, indicates into how many parts the whole is divided.

• If the fraction is \( \frac{3}{4} \), it means you have 3 out of 4 equal parts.
• A fraction like \( \frac{1}{6m} \) goes further by incorporating variables, which can represent different values depending on the context.

Fractions are useful for expressing numbers that aren't whole numbers and are essential in everyday mathematics. They allow us to perform operations like addition, subtraction, multiplication, and division. Understanding how fractions work is vital for solving problems that involve these mathematical concepts.

Common Denominator

When adding or subtracting fractions, it is essential to have a common denominator. A common denominator is a shared multiple of the denominators of each fraction involved. It allows you to combine fractions more easily because it makes the parts comparable.

• For example, if you are working with \( \frac{1}{6m} \) and \( \frac{2}{5m} \), you need to find a number that both 6m and 5m evenly divide into.
• In this case, the least common multiple (LCM) of the numbers 6 and 5 is 30, so the common denominator is \( 30m \).

By converting each fraction so they all have the same denominator, these fractions can then be easily added together. Finding a common denominator is a crucial step for combining fractions, as it aligns their denominators and ensures that each fraction represents the same-sized parts.

Addition of Fractions

Adding fractions might seem tricky at first, but with a common denominator in place, it becomes a straightforward procedure.

• Once each fraction has been rewritten with the common denominator, simply add up all the numerators.
• For example, when adding \( \frac{5}{30m} + \frac{12}{30m} + \frac{120}{30m} \), you only add the top numbers: 5, 12, and 120.

The sum \[\frac{5 + 12 + 120}{30m} = \frac{137}{30m}\]is your result. The denominator remains the same throughout the addition because it already represents the equal parts into which the whole is divided.Finally, it's always wise to check if the fraction can be simplified. In this case, since 137 and 30 have no common factors other than 1, the fraction \( \frac{137}{30m} \) is in its simplest form. Understanding how to add fractions paves the way for more complex mathematical operations and helps in making sense of real-world data and measurements.