Problem 25 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$\frac{7}{8}-\frac{2}{5}$$

Step-by-Step Solution

Answer

The difference is \(\frac{19}{40}\).

Step 1: Find the Least Common Denominator (LCD)

To subtract fractions, we need a common denominator. The two denominators are 8 and 5. The least common multiple of 8 and 5 is 40. Thus, 40 will be our least common denominator.

Step 2: Convert Fractions to Equivalent Fractions

Convert \(\frac{7}{8}\) and \(\frac{2}{5}\) to equivalent fractions with a denominator of 40. For \(\frac{7}{8}\), multiply numerator and denominator by 5: \(\frac{7}{8} = \frac{7\times 5}{8\times 5} = \frac{35}{40}\).For \(\frac{2}{5}\), multiply numerator and denominator by 8: \(\frac{2}{5} = \frac{2\times 8}{5\times 8} = \frac{16}{40}\).

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, subtract their numerators:\(\frac{35}{40} - \frac{16}{40} = \frac{35 - 16}{40} = \frac{19}{40}\).

Step 4: Simplify the Result

The fraction \(\frac{19}{40}\) is already in its simplest form because 19 is a prime number and it doesn't share any factors with 40.

Key Concepts

Least Common Denominator

When working with fractions, it's crucial to find a common denominator before adding or subtracting them. The least common denominator (LCD) is the smallest number that both denominators can divide into without a remainder. In our example of subtracting \(\frac{7}{8}\) from \(\frac{2}{5}\), the denominators are 8 and 5.

To find the LCD:

• Identify the denominators of your fractions—in this case, 8 and 5.
• Find the least common multiple (LCM) of these numbers. The LCM of 8 and 5 is 40.

This means that 40 is the smallest shared multiple, making it the least common denominator. Once you have this number, you can convert the original fractions so that they both have this common denominator. This step is vital for accurate addition or subtraction of fractions.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. Simplifying makes fractions easier to understand and work with. After you find the LCD and perform the subtraction, you might need to simplify the resulting fraction.

In our exercise, after calculating \(\frac{7}{8} - \frac{2}{5}\), we end up with \(\frac{19}{40}\).

• Check if 19 and 40 have any common factors. If none exist, the fraction is in its simplest form.
• Recognize that 19 is a prime number, meaning it only has one and itself as factors, which makes it even easier to identify as simplified.

This confirms that \(\frac{19}{40}\) is already simplified since 19 does not divide into 40.

Subtracting Fractions

Subtracting fractions involves several steps, primarily around ensuring you have a common denominator.

• First, as we've discussed, find the least common denominator of the fractions involved.
• Next, adjust each fraction so it has this common denominator.
• Perform the subtraction on the new fractions.

For our specific problem, converting \(\frac{7}{8}\) and \(\frac{2}{5}\) into fractions with a denominator of 40 gives us \(\frac{35}{40}\) and \(\frac{16}{40}\) respectively. We then subtract the numerators: \[\frac{35}{40} - \frac{16}{40} = \frac{35 - 16}{40} = \frac{19}{40}\]Remember, the denominator remains the same during subtraction because it indicates how many parts the whole is divided into, while the subtraction operation is performed only on the numerators.