Problem 179 - Intermediate Algebra

Question

In the following exercises, add or subtract. $$ -\frac{39}{56}-\frac{22}{35} $$

Step-by-Step Solution

Answer

-\frac{371}{280}

Step 1: Find a Common Denominator

To add or subtract fractions, first find a common denominator. The denominators are 56 and 35. The least common multiple (LCM) of 56 and 35 can be found by listing the multiples or using prime factorization. The LCM of 56 and 35 is 280.

Step 2: Convert Fractions to Equivalent Fractions with the Common Denominator

Convert \(-\frac{39}{56}\) and \(-\frac{22}{35}\) to have the common denominator of 280. Multiply the numerator and denominator of each fraction to achieve this:\[ -\frac{39}{56} = -\frac{39 \times 5}{56 \times 5} = -\frac{195}{280} \]\[ -\frac{22}{35} = -\frac{22 \times 8}{35 \times 8} = -\frac{176}{280} \]

Step 3: Subtract the Fractions

Now subtract the numerators of the fractions because the denominators are the same:\[ -\frac{195}{280} - \frac{176}{280} = -\frac{195 + 176}{280} = -\frac{371}{280} \]

Step 4: Simplify the Fraction

The fraction \(-\frac{371}{280}\) cannot be simplified further because 371 and 280 have no common factors other than 1. Hence, the final answer remains \(-\frac{371}{280}\).

Key Concepts

common denominator

To add or subtract fractions, it is crucial to have the same denominators. This shared denominator is referred to as the common denominator. Without it, directly performing operations on fractions is impossible. For example, with the fractions \(-\frac{39}{56}\) and \(-\frac{22}{35}\), it is necessary to express them with the same denominator. This involves finding a number that both denominators can divide into evenly. Once the common denominator is established, you can convert each fraction to this common denominator, thereby simplifying the process of addition or subtraction.

least common multiple

The least common multiple (LCM) is vital in finding the common denominator. It is the smallest number that is a multiple of two or more numbers. For numerators 56 and 35, we look for the LCM. By listing multiples or using prime factorization, we find that the LCM of 56 and 35 is 280. This means 280 is the smallest number both 56 and 35 can divide into without leaving a remainder.
Here's how to convert the original fractions using the LCM:

• For \(-\frac{39}{56}\), multiply the numerator and the denominator by 5 to make the denominator 280. This gives us \(-\frac{195}{280}\).


• For \(-\frac{22}{35}\), multiply the numerator and the denominator by 8 to make the denominator 280. This gives us \(-\frac{176}{280}\).

Now, both fractions have a common denominator of 280, allowing us to subtract them easily.

fraction simplification

After converting fractions to a common denominator and performing operations on them, it's typically necessary to simplify the result. Fraction simplification involves reducing the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common factor (GCF).
In our exercise, after subtracting the numerators, we get \(-\frac{371}{280}\). Since there are no common factors between 371 and 280 other than 1, the fraction cannot be simplified further. Therefore, the simplest form of the result is \(-\frac{371}{280}\).
Remember, simplification ensures the fraction is in its easiest, most understandable form. Sometimes fractions can't be simplified, as is the case here.