Problem 43 - Prealgebra and Introductory Algebra

Question

What is \(\frac{7}{12}\) added to \(-\frac{11}{16} ?\)

Step-by-Step Solution

Answer

The sum of \(\frac{7}{12}\) and \(-\frac{11}{16}\) is \(-\frac{5}{48}\).

Step 1: Find a common denominator

For \(\frac{7}{12}\) and \(-\frac{11}{16}\), the least common multiple of the denominators 12 and 16 is 48. So, for the first fraction, multiply the numerator 7 and the denominator 12 each by 4 to get \(\frac{28}{48}\). For the second fraction, multiply the numerator -11 and the denominator 16 each by 3 to get \(-\frac{33}{48}\).

Step 2: Add the fractions

Now that the fractions have the same denominator, add the fractions as: \(\frac{28}{48} - \frac{33}{48}\) which simplifies to \(-\frac{5}{48}\).

Step 3: Simplify the Result if possible

The fraction can't be simplified any further.

Key Concepts

Finding a Common Denominator

When adding fractions, it's crucial to have the same bottom number, known as the denominator. This is because fractions are parts of a whole, and we need like-sized pieces to combine them correctly. A common denominator is essentially the same value you apply to both fractions, allowing them to be combined or compared easily. To find this, we determine a multiple of both denominators involved. Take

• For example, if you have fractions like \(\frac{7}{12}\) and \(-\frac{11}{16}\), we need a common denominator to add them together correctly.
• In the given exercise, combining these fractions requires that you first find this shared denominator, which gives each fraction a uniform division of their respective wholes.
• Once you have equal denominators, adding or subtracting fractions is straightforward as you only combine the numerators.

Understanding Least Common Multiple

The Least Common Multiple (LCM) is a pivotal concept in finding a common denominator while adding fractions. The LCM of two numbers is the smallest number that is both a multiple of each of these numbers. Think of it like finding how many times your two denominators would evenly fit into the same larger pie. This is essential in fraction addition.

• For the example \(\frac{7}{12} + -\frac{11}{16}\), we find the LCM of 12 and 16.
• The LCM of 12 and 16 is 48. This tells us that 48 is the smallest value into which 12 and 16 can both divide evenly.
• Knowing the LCM allows us to convert each fraction to have this common denominator (thus making comparison or addition simple).

After finding 48 as the LCM, both fractions can be rewritten with this new denominator, allowing for the arithmetic to proceed smoothly. This step is crucial for the simplification of adding or subtracting fractions.

Simplifying Fractions

Simplifying fractions is the practice of reducing fractions to their simplest form. This means that the numerator and denominator share no common divisors other than 1, making the fraction as straightforward as possible.

• To simplify a result, examine the numbers to see if both parts of the fraction can be divided by the same number.
• In our exercise involving \(\frac{28}{48} - \frac{33}{48} = -\frac{5}{48}\), we see that \(-\frac{5}{48}\) already is in its simplest form since 5 and 48 share no common factors except 1.
• This step not only makes fractions easier to understand but ensures that all duplicates from both the numerator and the denominator are removed.

Simplifying allows readers to see the fraction clearly without unnecessary complexity, and it provides an accurate representation of the original problem solved. So, always check if there's room to simplify for a neat final answer.