Problem 72 - Beginning and Intermediate Algebra

Question

Find each difference. $$ \frac{9}{10}-\left(\frac{1}{8}-\frac{3}{10}\right) $$

Step-by-Step Solution

Answer

\(\frac{43}{40}\)

Step 1 - Evaluate the inner expression

Start by calculating the inner expression \(\frac{1}{8}-\frac{3}{10}\). To do that, you need a common denominator. The least common denominator (LCD) for 8 and 10 is 40.

Step 2 - Convert fractions to have a common denominator

Convert \(\frac{1}{8}\) and \(\frac{3}{10}\) to fractions with the common denominator 40: \(\frac{1}{8}=\frac{5}{40}\) and \(\frac{3}{10}=\frac{12}{40}\).

Step 3 - Subtract the fractions

Now subtract the two fractions: \(\frac{5}{40}-\frac{12}{40}\rightarrow \frac{5-12}{40}=\frac{-7}{40}\). So, \(\frac{1}{8}-\frac{3}{10}=\frac{-7}{40}\).

Step 4 - Substitute back into the main expression

Substitute \(\frac{-7}{40}\) back into the original expression: \(\frac{9}{10}-\frac{-7}{40}\).

Step 5 - Find a common denominator

Convert \(\frac{9}{10}\) and \(\frac{-7}{40}\) to fractions with the common denominator 40: \(\frac{9}{10}=\frac{36}{40}\).

Step 6 - Perform the subtraction

Now subtract the fractions: \(\frac{36}{40}-\frac{-7}{40}\rightarrow \frac{36+7}{40}=\frac{43}{40}\).

Step 7 - Simplify if needed

\(\frac{43}{40}\) is already in its simplest form. Thus, the final answer is \(\frac{43}{40}\).

Key Concepts

common denominators

When subtracting fractions, the first step is to find common denominators. A common denominator is a shared multiple of both denominators of the fractions you are working with. It allows you to convert the fractions so they have the same base, making the arithmetic operation much simpler.

To find a common denominator, identify the least common multiple (LCM) of the two denominators. This will be your common denominator. For example, in the exercise, converting \(\frac{1}{8}\) and \(\frac{3}{10}\), the LCM of 8 and 10 is 40. This is the least common denominator (LCD) and is necessary to proceed with the subtraction.

fraction subtraction

Subtracting fractions becomes straightforward once they have common denominators.

Here’s a step-by-step rundown:

• First, convert the fractions so that they have the same denominator.
• Next, subtract the numerators while keeping the denominator the same.

For instance, converting \(\frac{1}{8}\) and \(\frac{3}{10}\) to have the common denominator 40, they become \(\frac{5}{40}\) and \(\frac{12}{40}\). You then subtract the numerators: 5 - 12 to get \(\frac{-7}{40}\).

Repeat these steps as needed to perform multi-step operations. As shown in the exercise, after calculating \(\frac{-7}{40}\), it’s essential to substitute back and perform similar steps if necessary, ensuring all subtractions and additions involve fractions with common denominators.

least common denominator

The least common denominator (LCD) plays a critical role in the process of fraction subtraction.

The LCD of two fractions is the smallest common multiple of their denominators. By using the LCD, you can ensure that the fractions are compatible (i.e., they have the same denominator), simplifying the arithmetic process.

For example in the given exercise, to subtract \(\frac{1}{8}\) from \(\frac{3}{10}\), we found the LCD of 8 and 10, which is 40.

• Converting \(\frac{1}{8}\) to \(\frac{5}{40}\)
• Converting \(\frac{3}{10}\) to \(\frac{12}{40}\)

The LCD makes it possible to subtract the fractions easily, and doing this helps in avoiding complex arithmetic steps while maintaining accuracy.

Understanding and finding the least common denominator is vital for subtracting fractions correctly and efficiently.