Problem 91 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or mixed number in simplest form. \(\frac{1}{2}-\frac{3}{4}+\frac{5}{6}-\frac{7}{8}\)

Step-by-Step Solution

Answer

-\(\frac{7}{24}\)

Step 1: Find the Common Denominator

Firstly, in order to subtract or add fractions, they all need to have the same denominator. The easiest way to achieve this is by finding the least common multiple (LCM) of the denominators. In this case, the denominators are 2, 4, 6, and 8. The LCM of these numbers is 24. So, rewrite each fraction with 24 as the denominator.

Step 2: Rewrite the Fractions

Rewrite each fraction with 24 as the denominator. This will involve multiplying both the numerator and denominator by the same number: \(\frac{1}{2}\) becomes \(\frac{12}{24}\), \(\frac{3}{4}\) becomes \(\frac{18}{24}\), \(\frac{5}{6}\) becomes \(\frac{20}{24}\), and \(\frac{7}{8}\) becomes \(\frac{21}{24}\). Now the expression can be rewritten as \(\frac{12}{24}-\frac{18}{24}+\frac{20}{24}-\frac{21}{24}\)

Step 3: Perform the Operations

Now that all the fractions have the same denominator, they can be added or subtracted: \(\frac{12}{24}-\frac{18}{24}+\frac{20}{24}-\frac{21}{24} = -\frac{7}{24}\)

Step 4: Simplify the Result

The resultant fraction is \(\frac{-7}{24}\) which is already in simplest form since 7 and 24 have no common factors other than 1.

Key Concepts

Least Common Multiple

When working with fractions, especially when adding or subtracting them, it's crucial to have a common denominator. The least common multiple, or LCM, helps us find the smallest common denominator for all fractions involved in an equation. For the numbers 2, 4, 6, and 8, their LCM is 24. This means they can all evenly divide into 24 without leaving a remainder. To find the LCM:

• List out the multiples of each number.
• Identify the smallest number that appears in each list.

Using the LCM avoids unnecessary complication and ensures the operations on fractions are straightforward. In this exercise, getting all denominators to 24 simplifies further calculations.

Subtracting Fractions

Subtracting fractions might seem daunting at first, but once all fractions have the same denominator, it becomes much simpler. Here's how you do it:

• Ensure the fractions have the same denominator.
• Subtract the numerators while keeping the denominator the same.

For example, with \[\frac{18}{24}\]subtracted from \[\frac{12}{24},\]the numerators are the only change: \[12 - 18 = -6.\]So, the result is \[\frac{-6}{24}.\]Always pay attention to the numerator during subtraction: the sign matters just as much as the number itself.

Adding Fractions

When adding fractions with a common denominator, the process is similarly simplified as with subtraction. The steps include:

• Make sure denominators are the same (which they should be already, if prepared with the LCM).
• Add the numerators directly, keeping the denominator unchanged.

An example from the exercise is combining \[\frac{12}{24}\]and\[\frac{20}{24}\].Their numerators, 12 and 20, add up directly to 32, giving us\[\frac{32}{24}.\]The concept holds true when adding multiple fractions in sequence—just ensure each operation is accurately calculated.

Simplest Form

After performing operations with fractions, it's always good practice to express the result in its simplest form. To do this, follow these steps:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.

In our expression, we ended up with \[\frac{-7}{24}.\]Since 7 is a prime number and does not share any factors with 24 (besides 1), the fraction is already simplified. Always check if further reduction is possible, but in this case, we reach \[\frac{-7}{24}\] without needing any adjustments.