Problem 39 - Beginning and Intermediate Algebra

Question

Add or subcract as indicared. Write the answer in lower terms. See Erample 5 $$ \frac{23}{105}+\frac{4}{105} $$

Step-by-Step Solution

Answer

The simplified sum is \(\frac{9}{35}\).

Step 1: Identify the Denominator

Notice that both fractions have a common denominator, which is 105. This means that you can directly add the numerators.

Step 2: Add the Numerators

Add the numerators of the two fractions: \[23 + 4 = 27\]Place the sum over the common denominator: \[\frac{27}{105}\]

Step 3: Simplify the Fraction

To simplify \(\frac{27}{105}\), find the greatest common divisor (GCD) of 27 and 105. The GCD is 3. Divide both numerator and denominator by their GCD:\[\frac{27 \div 3}{105 \div 3} = \frac{9}{35}\]

Step 4: Verify the Simplification

Check that 9 and 35 have no common divisors other than 1, confirming \(\frac{9}{35}\) is in its simplest form.

Key Concepts

Common Denominator

When adding or subtracting fractions, a common denominator is essential. A common denominator is simply the shared number at the bottom of the fractions you're dealing with.
This number represents the same-sized parts in each fraction, allowing for easy addition or subtraction of the numerators (top numbers).

• For example, in the exercise, both fractions already share a common denominator, which is 105.
• When the denominators are the same, you can proceed directly to adding or subtracting the numerators.

Having a common denominator simplifies the process because you're not changing the size of the pieces, just counting how many there are in total. If fractions do not have the same denominator, find a common multiple of the two denominators first, and then change each fraction to equivalent fractions with this new number as the denominator.

Simplifying Fractions

Once you've performed an operation on fractions (like addition or subtraction), simplifying the result is often the next step. Simplifying a fraction means to reduce it to its simplest form.
This occurs when the numerator and the denominator have no common factors other than 1.

• In our example, after adding, we get \(\frac{27}{105}\).
• The task is to simplify this fraction.

Simplifying involves finding a number that divides both the numerator and the denominator evenly and dividing each by this number. After simplification, you end up with a fraction in its simplest form as \(\frac{9}{35}\). This makes the fraction easier to interpret and work with.

Greatest Common Divisor

The greatest common divisor (GCD) is a key player in simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder.
Finding the GCD allows you to accurately simplify a fraction to its lowest terms.

• In our example, the GCD of 27 and 105 is 3.
• To simplify the fraction \(\frac{27}{105}\), you divide both the numerator and the denominator by 3, which is the GCD.

To find the GCD of two numbers, you can use the Euclidean algorithm or simply list out the factors of each and identify the largest one they share.
Understanding GCD helps ensure that fractions are as reduced as possible, making further calculations or comparisons less cumbersome.