Problem 180 - Intermediate Algebra

Question

In the following exercises, add or subtract. $$ -\frac{33}{49}-\frac{18}{35} $$

Step-by-Step Solution

Answer

-\frac{291}{245}

Step 1 - Find a Common Denominator

The denominators of the fractions are 49 and 35. To find a common denominator, find the least common multiple (LCM) of these numbers. The LCM of 49 and 35 is 245.

Step 2 - Convert Fractions to the Common Denominator

Convert \(-\frac{33}{49}\) and \(-\frac{18}{35}\) to fractions with a denominator of 245.For \(-\frac{33}{49}\):\( -\frac{33}{49} \times \frac{5}{5} = -\frac{165}{245} \)For \(-\frac{18}{35}\):\( -\frac{18}{35} \times \frac{7}{7} = -\frac{126}{245} \)

Step 3 - Add the Fractions

Add the fractions with the common denominators: \(-\frac{165}{245} + -\frac{126}{245} = -\frac{291}{245}\).

Step 4 - Simplify the Fraction (if possible)

Check if \(-\frac{291}{245}\) can be simplified. The greatest common divisor (GCD) of 291 and 245 is 1, so the fraction is already in its simplest form.

Key Concepts

Least Common Multiple (LCM)

When working with fractions that have different denominators, the first step is finding a common denominator. This is often done through the Least Common Multiple (LCM). The LCM of two numbers is the smallest multiple that both numbers share. To find the LCM of 49 and 35, list the multiples of each number:
- Multiples of 49: 49, 98, 147, 196, 245, ...
- Multiples of 35: 35, 70, 105, 140, 175, 210, 245, ...
The smallest number that appears in both lists is 245. This makes 245 the LCM of 49 and 35. By finding the LCM, we ensure we can convert both fractions to have the same denominator, making addition or subtraction possible.

Common Denominator

A common denominator is crucial for adding or subtracting fractions. It's the shared denominator used to rewrite each fraction.
Start with the given fractions: \(-\frac{33}{49}\) and \(-\frac{18}{35}\). Convert these to have the common denominator found earlier (245).
For \(-\frac{33}{49}\): Multiply by \(\frac{5}{5}\) to get: \(-\frac{33}{49} \times \frac{5}{5} = -\frac{165}{245}\)
For \(-\frac{18}{35}\): Multiply by \(\frac{7}{7}\) to get: \(-\frac{18}{35} \times \frac{7}{7} = -\frac{126}{245}\)
This step ensures both fractions are expressed in terms of the same denominator, making it easy to add or subtract them directly.

Fraction Simplification

Simplifying a fraction means expressing it in the most reduced form. After performing the addition or subtraction, check if the resulting fraction can be simplified. The fraction we got is: \(-\frac{291}{245}\)
To simplify, find the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD of 291 and 245 is 1, meaning they share no common factors other than 1.
This means \(-\frac{291}{245}\) is already in its simplest form. If there were a larger GCD, we would divide both the numerator and the denominator by that number to simplify the fraction.
Remember, simplifying fractions makes them easier to understand and handle in further calculations.