Problem 177 - Intermediate Algebra

Question

In the following exercises, add or subtract. $$ -\frac{13}{30}+\frac{25}{42} $$

Step-by-Step Solution

Answer

\frac{17}{105}

Step 1: Find the Least Common Denominator (LCD)

To add or subtract fractions, the first step is to find the least common denominator (LCD). For the fractions \(\frac{-13}{30} \) and \( \frac{25}{42} \), we need the LCD of 30 and 42. Find the prime factorization: \( 30 = 2 \times 3 \times 5 \) and \( 42 = 2 \times 3 \times 7 \). The LCD is \( 2 \times 3 \times 5 \times 7 = 210 \).

Step 2: Convert Fractions to the Same Denominator

Now we need to express each fraction with the common denominator, which is 210. For \(\frac{-13}{30}\), we multiply the numerator and the denominator by 7: \(\frac{-13 \times 7}{30 \times 7} = \frac{-91}{210}\). For \( \frac{25}{42}\), we multiply the numerator and the denominator by 5: \(\frac{25 \times 5}{42 \times 5} = \frac{125}{210}\).

Step 3: Add the Fractions

With both fractions having the same denominator, we can add the numerators: \(\frac{-91}{210} + \frac{125}{210} = \frac{-91 + 125}{210} = \frac{34}{210}\).

Step 4: Simplify the Result

Finally, simplify the fraction if possible. The greatest common divisor (GCD) of 34 and 210 is 2. So, we divide both the numerator and the denominator by 2: \( \frac{34 \times 2}{210 \times 2} = \frac{17}{105} \).

Key Concepts

least common denominator

When adding or subtracting fractions, the first step is finding the Least Common Denominator (LCD). The LCD is the smallest number that both denominators evenly divide into. This helps in combining fractions with different denominators.
To find the LCD for \(\frac{-13}{30}\) and \(\frac{25}{42}\), we start with their prime factorization:

• 30 = 2 × 3 × 5
• 42 = 2 × 3 × 7

The LCD is found by taking all prime factors to their highest powers. For these fractions, the primes are 2, 3, 5, and 7. Hence,
\[LCD = 2 \times 3 \times 5 \times 7 = 210 \]
This means 210 is the smallest number both 30 and 42 divide into without leaving a remainder.

fraction simplification

After obtaining the LCD, the next step is to simplify each fraction. Simplification involves expressing each faction with the common denominator. This makes it easier to add or subtract them. For \(\frac{-13}{30}\):
Multiply the numerator and denominator by 7 to get the common denominator of 210:
\(\frac{-13 \times 7}{30 \times 7} = \frac{-91}{210}\)
For \(\frac{25}{42}\):
Multiply the numerator and denominator by 5 to also get the common denominator of 210:
\(\frac{25 \times 5}{42 \times 5} = \frac{125}{210}\)

prime factorization

Prime factorization is the process of breaking down a number into the product of its prime numbers. These prime numbers are the building blocks for integers and helpful in operations such as finding the LCD or GCD.
For the numbers in our fractions:

• 30 can be factored as: 30 = 2 × 3 × 5
• 42 can be factored as: 42 = 2 × 3 × 7

By understanding the prime composition, we can easily find the least common multiples or greatest common divisors, simplifying our fractions and operations further.
Prime factorization is integral for handling any arithmetic involving fractions or integers.

greatest common divisor

The Greatest Common Divisor (GCD) is the largest number that divides two numbers without leaving a remainder. When simplifying fractions, it's essential to find the GCD of the numerator and denominator.
For the fraction \(\frac{34}{210}\), the numerators and denominators share common factors. Here:
\(\frac{34}{210}\) has prime factors:

• 34 = 2 × 17
• 210 = 2 × 3 × 5 × 7

The common factor here is 2. So, the GCD of 34 and 210 is 2.
Simplify the fraction by dividing both numerator and denominator by the GCD:
\(\frac{34 \times 2}{210 \times 2} = \frac{17}{105}\)
This results in the simplest form of the fraction.