Problem 45 - Prealgebra

Question

Add and subtract as indicated. $$\frac{2}{5}-\frac{3}{8}$$

Step-by-Step Solution

Answer

The result is \( \frac{1}{40} \).

Step 1: Find a common denominator

To add or subtract fractions, they must have a common denominator. The denominators here are 5 and 8. The least common multiple (LCM) of 5 and 8 is 40. Therefore, we will use 40 as the common denominator.

Step 2: Convert both fractions

Convert each fraction to an equivalent fraction with the common denominator of 40. For \( \frac{2}{5} \), multiply the numerator and the denominator by 8 to get \( \frac{16}{40} \). For \( \frac{3}{8} \), multiply the numerator and the denominator by 5 to get \( \frac{15}{40} \).

Step 3: Subtract the fractions

Now that both fractions have a common denominator, subtract the numerators: \( 16 - 15 = 1 \). The result is \( \frac{1}{40} \).

Step 4: Simplify the result

The fraction \( \frac{1}{40} \) is already in its simplest form, as the numerator 1 is the smallest possible integer greater than zero.

Key Concepts

Finding a Common Denominator

When you want to add or subtract fractions, a common denominator is essential. Think of the denominator as the bottom section of a fraction, which tells you how many equal parts something is divided into. `Finding a common denominator` means figuring out a way to get the parts from both fractions to match, so that they can be easily combined or subtracted. Here's how to do it:

• Identify the denominators of both fractions. In this case, they are 5 and 8.
• To make the denominators the same, you need to use the Least Common Multiple (LCM). The LCM is the smallest number that both denominators can divide into without leaving any remainder.
• Once you find the LCM, use it as your common denominator. In our exercise, the LCM of 5 and 8 is 40. Both fractions will be converted to have this denominator so that they can be subtracted easily.

Understanding Least Common Multiple

The least common multiple (LCM) is a crucial concept for comparing fractions. It's the smallest number that is a multiple of both denominators. Imagine you're trying to find a common ground to make calculations simpler. Here's the process to find the LCM:

• List the multiples of each denominator. For 5, you get 5, 10, 15, 20, etc. For 8, the multiples are 8, 16, 24, 32, 40, etc.
• Look for the smallest number that appears in both lists. In this case, 40 appears in both lists.
• This number helps standardize the fractions by making the denominators the same, which is your aim when adding or subtracting fractions.

Finding the LCM might require some patience at first, but with practice, it becomes a quick and intuitive step in working with fractions.

Fraction Simplification Insights

Simplifying fractions means breaking them down to their smallest possible form. After performing operations like addition or subtraction, you might end up with a fraction that can be simplified. In the example, after subtracting \(\frac{2}{5} - \frac{3}{8}\), you obtain \(\frac{1}{40}\).Here's how to check if it can be simplified:

• A fraction is simplified when both the numerator and denominator have no common factors other than 1.
• Check if the numerator can divide into the denominator without leaving a remainder and only results in whole numbers—this indicates that simplification is possible.
• If any common factors exist (other than 1), divide both the numerator and the denominator by that common factor.
• In our example, the numerator 1 and the denominator 40 cannot be simplified further because their only common factor is 1.

Thus, the fraction is already as simplified as it can be. It's often helpful to write fractions in this manner, as they are neater and easier to understand. With practice, simplifying fractions can become second nature!