Problem 62 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(\frac{2}{5}\)

Step 1: Identify the Denominators

Both fractions have the same denominator, which is 35. This means you can add the numerators directly.

Step 2: Add the Numerators

Add the numerators of the two fractions. The numerators are 11 and 3. Thus, \(11 + 3 = 14\).

Step 3: Write the Sum as a Fraction

Now, write the sum of the numerators over the common denominator. This gives \(\frac{14}{35}\).

Step 4: Simplify the Fraction to Lowest Terms

Find the greatest common divisor (GCD) of 14 and 35. The GCD is 7. Divide both the numerator and the denominator by 7: \(\frac{14}{7} = 2\) and \(\frac{35}{7} = 5\). The fraction in lowest terms is \(\frac{2}{5}\).

Key Concepts

Addition of Fractions

Adding fractions may seem challenging at first, but it can be simple with a bit of practice. When fractions have the same denominator, it's a straightforward process.

• First, ensure both fractions you're dealing with have the same denominator, as in this case, they both have 35. The denominator represents the total number of equal parts.
• When denominators are the same, add the numerators while keeping the denominator unchanged. The numerator represents the number of parts we have.

For example, when you add \( \frac{11}{35} \) and \( \frac{3}{35} \), their numerators are added: \( 11 + 3 = 14 \). The result is \( \frac{14}{35} \). This is the sum of the two fractions before simplification.

Simplifying Fractions

A fraction is in its simplest form when the numerator and the denominator have no common divisor other than one. Simplifying involves reducing the fraction to its lowest terms.

• Once you have your fraction, look for any common factor between the numerator and the denominator.
• In our example, the fraction \( \frac{14}{35} \) can be simplified by dividing both numbers by their greatest common factor.

Reducing fractions helps maintain simplicity in calculations and gives a more comprehensible result. After simplification, the fraction might look entirely different, yet it remains equivalent. Always ensure each part of the fraction is reduced as much as possible to streamline further arithmetic calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is a useful tool to simplify fractions. It's the largest number that divides two numbers without leaving a remainder. This is essential when reducing fractions.
To find the GCD:

• List the divisors of each number, which means all numbers that divide the number exactly without leaving a remainder.
• Identify the largest divisor common to both numbers.

For instance, let's find the GCD of 14 and 35. The divisors of 14 are 1, 2, 7, and 14, and those of 35 are 1, 5, 7, and 35. The largest common divisor is 7, which means the GCD is 7.
Using the GCD simplifies the fraction \( \frac{14}{35} \) to \( \frac{2}{5} \), as dividing both the numerator and denominator by 7 gives this results. This ensures the simplest form of the fraction, making further calculations easier.