Problem 48 - Prealgebra

Question

Find the difference of \(\frac{8}{15}\) and \(\frac{8}{35}\).

Step-by-Step Solution

Answer

The difference is \(\frac{32}{105}\).

Step 1: Find the Least Common Denominator

To subtract fractions, we need to have a common denominator. The denominators here are 15 and 35. The least common multiple of 15 and 35 is 105. Therefore, our common denominator is 105.

Step 2: Rescale the Fractions

Convert both fractions to have this common denominator. For \(\frac{8}{15}\), multiply both the numerator and denominator by 7 to get \(\frac{56}{105}\). For \(\frac{8}{35}\), multiply both the numerator and denominator by 3 to get \(\frac{24}{105}\).

Step 3: Subtract the Fractions

With a common denominator, subtract the numerators: \(56 - 24 = 32\). Thus, the fraction is \(\frac{32}{105}\).

Step 4: Simplify the Result

Check if \(\frac{32}{105}\) can be simplified by finding a common factor. Since 32 and 105 have no common factors other than 1, the simplified form of the difference is \(\frac{32}{105}\).

Key Concepts

Least Common Denominator

When working with fractions, especially when you're subtracting them, finding a common denominator is like finding common ground. The least common denominator (LCD) is crucial because it allows you to compare or combine different fractions easily. Essentially, this means you make the denominators the same. Here's how you can do it:

• Identify the denominators of the fractions you are working with. In our exercise, these are 15 and 35.
• Find the least common multiple (LCM) of these denominators. The LCM is the smallest number that both denominators divide into without any remainder.
• For 15 and 35, listing out the multiples, you'll find that the LCM is 105. This becomes the common denominator.

Once you have the least common denominator, it’s much simpler to rescale the original fractions so they share this denominator, making subtraction possible.

Fraction Simplification

Now, let's talk about making fractions simpler, which is all about reducing them to their most basic form. Fraction simplification involves reducing the numerator and denominator to the smallest numbers possible while still keeping the fraction equivalent. Think of it like cleaning up to keep only the essentials. Here are the steps for simplification:

• Look for a common factor in the numerator and the denominator.
• Divide both the top and bottom of the fraction by this common factor.

In fractions like \( \frac{32}{105} \), you might first check for simple common factors like 2, 3, or 5. However, if there's no common factor other than 1, as in this case, the fraction is already in its simplest form. This ensures the fraction is easier to understand and further operations can be done more straightforwardly.

Multiple Fractions

Handling multiple fractions at once might seem daunting, but it’s manageable when broken down. Understanding how to subtract them involves a few clear steps. The key factor is ensuring that all fractions share a common denominator. This way, you can operate directly on the numerators, making calculations more direct. Here's how you break it down:

• Find the least common denominator for all the fractions involved, as explained earlier.
• Convert each fraction to equivalent fractions that have this least common denominator.
• Perform the operation (like subtraction or addition) on the numerators, keeping the denominator constant.

In our problem, once we changed \( \frac{8}{15} \) to \( \frac{56}{105} \) and \( \frac{8}{35} \) to \( \frac{24}{105} \), we simply subtract the numerators: \( 56 - 24 \), leading to \( \frac{32}{105} \). It's a process that becomes easier with practice, giving you confidence in managing any number of fractions efficiently.