Problem 81 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer in simplest form. \begin{equation} \frac{41}{44}-\frac{19}{44} \end{equation}

Step-by-Step Solution

Answer

The result in simplest form is \( \frac{1}{2} \)

Step 1: Identify the common denominator

In this case, the denominator for both fractions is 44, so we don't need to find a least common denominator or do any conversions.

Step 2: Subtract the numerators

Next, you subtract the numerators: 41 - 19 = 22.

Step 3: Write the result

Write the result as a fraction with the denominator 44. So the answer is \( \frac{22}{44} \)

Step 4: Simplify the fraction

This fraction can be simplified by dividing both the numerator and denominator by their greatest common factor which is 22. The simplified fraction is \( \frac{1}{2} \)

Key Concepts

Subtraction

Subtraction is a basic yet important arithmetic operation where you take away one number from another. In the context of fractions, it's just like subtracting whole numbers, but you focus on the numerators while keeping an eye on the common denominators.

When you subtract fractions with a common denominator, you only subtract the numerators. In our exercise example, we subtract the numerators of the fractions \( \frac{41}{44} - \frac{19}{44} \). This means, you focus on 41 and 19. Subtracting these gives you 22. So, after the subtraction step, you'll have \( \frac{22}{44} \). The denominator remains the same throughout, as it's the common base for both fractions.

In more complicated cases, where the denominators aren't the same, you'll need to find a common denominator before you can perform the subtraction. But here, since 44 is already the denominator for both, it simplifies the process significantly.

Common Denominator

A common denominator makes working with fractions seamless and easy. It is crucial when you need to add or subtract fractions, ensuring a solid ground to keep the fraction balanced.

In our exercise, both fractions have the same denominator, 44. This means they already share a common denominator, so we can skip the usual process of finding a least common denominator (LCD).

• The denominator is the bottom number in a fraction, and it tells you how many parts the whole is divided into.
• You need a common denominator to ensure that the fractions are divided into the same-sized parts when adding or subtracting.

If the denominators weren't the same, you'd have to find an LCD. This often involves finding the smallest number that is a multiple of both denominators. Once you have that, you'd adjust the fractions so they match this common denominator. But in our example, the easy part is we’ve already got it!

Simplifying Fractions

After executing operations with fractions, it’s common practice to simplify the result. Simplifying a fraction means writing it in its smallest form, making the numbers involved as manageable as possible.

To simplify the fraction \( \frac{22}{44} \), you look for the greatest common factor between the numerator (22) and the denominator (44). The greatest common factor here is 22.

• Divide the numerator by this factor: 22 ÷ 22 = 1.
• Divide the denominator by the same factor: 44 ÷ 22 = 2.

The simplified form of \( \frac{22}{44} \) is \( \frac{1}{2} \). Simplification is helpful because it makes the fraction easier to interpret and understand at a glance, revealing its most basic proportion without altering its value. Always check if a fraction can be simplified to ensure it's in its simplest form.