Problem 24 - Beginning and Intermediate Algebra

Question

Find each sum. $$ -\frac{1}{4}+\left(-\frac{5}{12}\right) $$

Step-by-Step Solution

Answer

-\frac{2}{3}

Step 1: Identify the common denominator

To add the fractions \(-\frac{1}{4} \) and \(-\frac{5}{12}\), first find a common denominator. The denominators are 4 and 12. The least common multiple (LCM) of 4 and 12 is 12.

Step 2: Convert fractions to the common denominator

Convert the first fraction to have the common denominator of 12. To do this, multiply the numerator and denominator of \(-\frac{1}{4}\) by 3: \(-\frac{1}{4} \times \frac{3}{3} = -\frac{3}{12}\). Now, the two fractions are \(-\frac{3}{12}\) and \(-\frac{5}{12}\).

Step 3: Add the fractions

Now that both fractions have the same denominator, you can add the numerators directly: \(-\frac{3}{12} + -\frac{5}{12} = \frac{-3 + -5}{12} = \frac{-8}{12}\).

Step 4: Simplify the fraction

Simplify \(-\frac{8}{12}\) by dividing the numerator and the denominator by their greatest common divisor (GCD). The GCD of 8 and 12 is 4. Thus, \(-\frac{8}{12} \div 4 = -\frac{2}{3}\).

Key Concepts

Common Denominator

When adding or subtracting fractions, we need a common denominator. The denominator is the bottom number of a fraction.
It's important because it shows how many equal parts something is divided into.
For example, in the problem \(-\frac{1}{4} + \left(-\frac{5}{12}\right)\), the fractions have denominators of 4 and 12.
We can't directly add them because the pieces are different sizes.
To fix this, we find a common denominator.
The easiest way to do this is by finding the least common multiple (LCM) of the denominators.
This turns the fractions into equivalent fractions with the same bottom number.

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that both numbers divide into without leaving a remainder.
In our exercise, the denominators are 4 and 12.
We need to find the LCM of these numbers.
List the multiples of each number:
Multiples of 4: 4, 8, 12, 16, ...
Multiples of 12: 12, 24, 36, ...
The least common multiple is the first number to appear in both lists.
Here, it's 12. Once the common multiple is found, we convert all fractions to have this common denominator.

Simplifying Fractions

After adding the fractions, the final step is simplifying the fraction.
This means making the fraction as simple as possible.
For example, after finding \(-\frac{8}{12}\), we simplify it.
We do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
The GCD is the largest number that divides both the numerator and the denominator evenly.
For 8 and 12, the GCD is 4.
So, \(-\frac{8}{12} \div 4 = -\frac{2}{3}\).
Now the fraction is in its simplest form, which is easy to interpret and use.