Problem 30 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -\frac{1}{10}-\frac{7}{8} $$

Step-by-Step Solution

Answer

The result of the subtraction is -\(\frac{39}{40}\).

Step 1: Make Fractions Like Denominators

Convert both fractions to have a common denominator. In this case, the lowest common denominator (LCD) of 10 and 8 is 40. Convert \(-\frac{1}{10}\) to \(-\frac{4}{40}\) and \(-\frac{7}{8}\) to \(-\frac{35}{40}\).

Step 2: Subtract Fractions

Now that both fractions have the same denominator, subtract the numerators: \(-4\) - (35) = -39. The result is \(-\frac{39}{40}\).

Step 3: Simplify the Result

Check if the resulting fraction can be simplified. Since there are no common factors for 39 and 40, \(-\frac{39}{40}\) is already in its simplest form.

Key Concepts

Common Denominator

When dealing with fractions, especially in subtraction or addition, finding a common denominator is crucial. This ensures that the fractions have the same bottom number, making them easier to work with. To find a common denominator:

• Identify the denominators of each fraction you are dealing with, in this case, 10 and 8.
• Find the least common multiple (LCM) of these denominators. The LCM is the smallest number that both denominators can divide into without a remainder. Here, the LCM of 10 and 8 is 40.

Once the common denominator is found, adjust the numerators accordingly so that the value of the fractions remains unchanged. For example, \(-\frac{1}{10}\) becomes \(-\frac{4}{40}\) and \(-\frac{7}{8}\) becomes \(-\frac{35}{40}\). Now, with a common denominator of 40, you can easily perform operations like subtraction.

Simplify Fractions

After performing any operations with fractions, it is essential to check if the resulting fraction can be simplified. Simplification means reducing the fraction to its simplest form, where the numerator and the denominator no longer have any common factors except 1. For instance:

• Examine the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD to simplify.

In the solution \(-\frac{39}{40}\), there are no common factors between 39 and 40 besides 1. Thus, this fraction is already in its simplest form. Remembering to simplify ensures your final answer is clear and concise.

Like Denominators

The concept of like denominators aligns closely with having a common denominator. By converting fractions to have like denominators, you simplify operations between them, such as addition or subtraction. "Like denominators" merely means that each of the fractions shares the same denominator:

• This uniformity contrasts the fractions only by their numerators, making computations straightforward.
• Once you have transformed fractions into having like denominators, proceed with direct operations on the numerators.

For example, in the problem \(-\frac{1}{10}-\frac{7}{8}\), using a common denominator of 40 achieved like denominators: \(-\frac{4}{40}\) and \(-\frac{35}{40}\), allowing you to subtract the numerators easily, resulting in \(-\frac{39}{40}\). Understanding and applying this makes fraction operations more efficient and less prone to errors.