Problem 23 - Beginning and Intermediate Algebra

Question

Find each sum. $$ -\frac{1}{3}+\left(-\frac{4}{15}\right) $$

Step-by-Step Solution

Answer

The sum is -\(\frac{3}{5}\).

Step 1: Find a common denominator

To add the fractions \(-\frac{1}{3}+\left(-\frac{4}{15}\right)\), first find a common denominator. The denominators are 3 and 15. The least common denominator (LCD) of 3 and 15 is 15.

Step 2: Convert fractions to common denominator

Convert -\(\frac{1}{3}\) to a fraction with the denominator 15. Multiply both the numerator and the denominator by 5: \( -\frac{1}{3} = -\frac{1 \cdot 5}{3 \cdot 5} = -\frac{5}{15} \).

Step 3: Add the fractions

Now add the fractions with the common denominator: \( -\frac{5}{15} + \left(-\frac{4}{15}\right) = \frac{-5 - 4}{15} = \frac{-9}{15} \).

Step 4: Simplify the fraction

Simplify \(\frac{-9}{15}\) by dividing the numerator and the denominator by their greatest common divisor (GCD). The GCD of 9 and 15 is 3: \( \frac{-9 \div 3}{15 \div 3} = \frac{-3}{5} \).

Key Concepts

common denominator

When adding fractions, the first thing to do is to find a common denominator. A common denominator is a shared multiple of the denominators of the fractions you are working with. This allows you to combine the fractions into a single fraction.
For example, if you are adding \(-\frac{1}{3} + (-\frac{4}{15})\), the denominators are 3 and 15. You need to determine the least common denominator (LCD) of these two numbers. The LCD is the smallest number that both denominators can divide into evenly.
In this case, 15 is the smallest number that both 3 and 15 can divide into without leaving a remainder. So, 15 is the common denominator.
Always remember to find this common base before proceeding to add fractions.

simplify fractions

After finding a common denominator and performing the addition, you might end up with a fraction that can be simplified. Simplifying a fraction means reducing it to its lowest terms.
This typically involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
For instance, after adding \(-\frac{5}{15} + (-\frac{4}{15})\), you get \(-\frac{9}{15}\).
To simplify \(-\frac{9}{15}\), divide both the top and bottom by their GCD (which is 3): \(-\frac{9 \div 3}{15 \div 3} = -\frac{3}{5}\).
The simplified fraction is easier to understand and work with.

greatest common divisor

The Greatest Common Divisor (GCD) of two numbers is the highest number that divides both of them without leaving a remainder.
Finding the GCD is essential when you need to simplify fractions. Use the GCD to reduce both the numerator and the denominator by their shared highest factor.
In our example, after calculating \(-\frac{9}{15}\), we need to simplify this fraction. To do this, we identify that the GCD of 9 and 15 is 3.
Divide both the numerator and the denominator by this GCD to get the simplified form: \(-\frac{9 \div 3}{15 \div 3} = -\frac{3}{5}\).
Understanding the GCD concept helps in making fractions clearer and easier to manage.