Problem 105 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{7}{30}+\frac{1}{50}-\frac{19}{75} $$

Step-by-Step Solution

Answer

The result of the operation is 0.

Step 1: Find a Common Denominator

To perform the addition and subtraction of fractions, we first need to find a common denominator. The denominators we have are 30, 50, and 75. The least common multiple (LCM) of these numbers will be the common denominator. The prime factorization of 30 is \(2 \times 3 \times 5\), of 50 is \(2 \times 5^2\), and of 75 is \(3 \times 5^2\). The LCM is \(2 \times 3 \times 5^2 = 150\).

Step 2: Convert Fractions to Equivalent Fractions

Now that we have a common denominator of 150, convert each fraction to an equivalent fraction with this denominator: \[ \frac{7}{30} = \frac{7 \times 5}{30 \times 5} = \frac{35}{150} \] \[ \frac{1}{50} = \frac{1 \times 3}{50 \times 3} = \frac{3}{150} \] \[ \frac{19}{75} = \frac{19 \times 2}{75 \times 2} = \frac{38}{150} \]

Step 3: Add and Subtract Fractions

With all fractions having the same denominator, we can now add and subtract the numerators: \[ \frac{35}{150} + \frac{3}{150} - \frac{38}{150} = \frac{35 + 3 - 38}{150} = \frac{0}{150} \]

Step 4: Simplify the Result

The fraction \(\frac{0}{150}\) simplifies to 0, since the numerator is 0.

Key Concepts

Least Common Denominator

The least common denominator is crucial when you want to add or subtract fractions with different denominators. Without it, you can't easily compare or combine fractions. The least common denominator (LCD) is essentially the smallest number that all denominators can divide without a remainder. To find it, list the prime factors of each denominator:

• 30 is composed of prime factors: \(2 \times 3 \times 5\)
• 50 breaks down into: \(2 \times 5^2\)
• 75 is factored as: \(3 \times 5^2\)

To determine the LCD, you take the highest power of each prime number present:- Two is raised to the first power and present in the factors of 30 and 50.- Three appears simple once in the factors of 30 and 75.- Five, however, appears as \(5^2\) twice in both 50 and 75.Thus, the LCD becomes \(2 \times 3 \times 5^2 = 150\). By knowing this number, you can convert each fraction to be expressed with the same denominator.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. To simplify a fraction means to reduce it to its simplest form, where the numerator and denominator have no common factors other than 1.In the process of adding or subtracting fractions, simplification usually occurs at the end after operations are performed. For example, if your result is \(\frac{0}{150}\), it simplifies to 0 because any fraction with 0 as its numerator is equal to 0.To simplify any fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by the GCD.

Simplifying helps by providing the most efficient and effective perspective on the answer, allowing you and others to understand the numerical relationship clearly and quickly.

Adding and Subtracting Fractions

Adding and subtracting fractions primarily boils down to ensuring all fractions involved have the same denominator. Only with this shared denominator can you directly perform the arithmetic operations on the numerators.Here’s a walkthrough using the given example:1. Find the least common denominator for all fractions, which is 150.2. Convert each fraction into an equivalent fraction with this denominator: - \(\frac{7}{30}\) becomes \(\frac{35}{150}\). - \(\frac{1}{50}\) converts to \(\frac{3}{150}\). - \(\frac{19}{75}\) translates to \(\frac{38}{150}\).3. Now add and subtract just the numerators: \(\frac{35 + 3 - 38}{150} = \frac{0}{150}\)By doing this, you maintain the integrity of the fractions while effectively combining them. This method ensures your final result represents the same values as the initial fractions but in a comprehensible manner.