Problem 64 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{11}{7}-\frac{3}{35} $$

Step-by-Step Solution

Answer

The result of the subtraction is \(\frac{52}{35}\), in simplest form.

Step 1: Find a Common Denominator

To subtract fractions, they must have a common denominator. The denominators are 7 and 35. The least common multiple of these numbers is 35.

Step 2: Convert the First Fraction

Convert \(\frac{11}{7}\) to an equivalent fraction with a denominator of 35. Multiply both the numerator and the denominator by 5: \(\frac{11 \times 5}{7 \times 5} = \frac{55}{35}\).

Step 3: Perform the Subtraction

Now, subtract the fractions with the same denominator: \(\frac{55}{35} - \frac{3}{35} = \frac{52}{35}\).

Step 4: Simplify the Result

Check if \(\frac{52}{35}\) can be simplified further. Since 52 and 35 have no common factors other than 1, the fraction \(\frac{52}{35}\) is already in its simplest form.

Key Concepts

Common Denominator

Finding a common denominator is the foundation of adding or subtracting fractions. When fractions have different denominators, they don't "speak the same language," so we need to adjust them to make calculations easier:

• The denominators in our example are 7 and 35.
• To find a common denominator, look for the Least Common Multiple (LCM) of both denominators.

The LCM is the smallest number that is a multiple of both denominators. In this case, it is 35 because 35 is a multiple of 7 and 35 itself. Once we establish 35 as the common denominator, we can proceed to convert each fraction to have this common denominator before performing any operations.

Equivalent Fractions

Equivalent fractions represent the same value, even though they may look different. To achieve this, we adjust both the numerator and the denominator by the same multiplier, which does not change the fraction's value.

• Our example involves converting \(\frac{11}{7}\) to have a denominator of 35.
• We achieve this by multiplying the numerator and the denominator by 5, resulting in \(\frac{55}{35}\).

This process doesn't change the value of the fraction but lets us work with a consistent denominator. Keep in mind that altering both parts of a fraction equally maintains its original ratio.

Simplifying Fractions

Simplifying fractions means making them as simple as possible, eliminating any unnecessary factors. After doing calculations with fractions, it's common to simplify the result:

• In our subtraction result, \(\frac{52}{35}\), we check if there are any common factors for the numerator and the denominator other than 1.
• If no such common factors exist, the fraction is already in its simplest form.

For \(\frac{52}{35}\), both 52 and 35 are relative primes, meaning they don't share any common factors besides 1. Therefore, the fraction remains unchanged after simplification, ensuring it's presented in the simplest way possible.