Problem 9 - Prealgebra

Question

Find each sum or difference. Write in simplest form. \(\frac{4}{12}-\frac{10}{12}\)

Step-by-Step Solution

Answer

The result of the subtraction \( \frac{4}{12} - \frac{10}{12} \) is \( \frac{-1}{2} \) in simplest form.

Step 1: Identify Common Denominators

Both fractions have a common denominator of 12. This means we do not need to find a new common denominator and can proceed by directly subtracting the numerators.

Step 2: Subtract the Numerators

Subtract the numerator of the second fraction from the numerator of the first: \(4 - 10 = -6\).

Step 3: Write the New Fraction

Combine the result from the previous step over the common denominator to get \(\frac{-6}{12}\).

Step 4: Simplify the Fraction

Identify the greatest common factor of the numerator and the denominator. Both 6 and 12 can be divided by 6. Simplify the fraction by dividing both the numerator and the denominator by 6 to get \(\frac{-1}{2}\).

Key Concepts

Understanding Common Denominator

Before you can add or subtract fractions, it's important that they share the same denominator, which is also known as the common denominator. The denominator of a fraction is the number at the bottom that tells you into how many parts the whole is divided. In this exercise, both fractions already have a common denominator of 12, which makes the process straightforward. Otherwise, you would need to find a common denominator by finding the least common multiple of the two denominators. This often requires listing multiples of each denominator and identifying the smallest multiple they share. Having a common denominator means you can safely work with the numerators without worrying about unequal parts, leading to an accurate result.

Subtraction of Numerators

Once you identify a common denominator in fractions, subtracting is much like dealing with whole numbers. You simply subtract the numerator of the second fraction from the numerator of the first fraction. In this problem, you subtract 10 from 4, which gives you -6. This results in creating a new fraction:

• The new numerator becomes -6
• The denominator remains 12

This step ensures that the differences between the quantities are accurately represented. It's crucial to keep the denominator unchanged during this process, as it guarantees that the parts you are subtracting are of identical size.

Steps to Simplifying Fractions

After you've performed operations on the fraction, the next step is to express it in its simplest form. Simplifying a fraction involves reducing it to its smallest possible equivalent fraction. For this, both the numerator and the denominator must be divided by their greatest common factor (GCF). In our fraction \(rac{-6}{12}\), both 6 and 12 are divisible by their GCF, which is 6.

• Divide the numerator -6 by 6 to get -1
• Divide the denominator 12 by 6 to get 2

Thus, the simplified fraction is \(rac{-1}{2}\). Simplifying helps in better understanding and comparing sizes of fractions, especially in further mathematical operations.

Finding the Greatest Common Factor

The greatest common factor (GCF) is the largest number that evenly divides both the numerator and the denominator. Finding the GCF is key to simplifying fractions. To calculate the GCF, list the factors of each number and find the greatest number that appears on both lists. For 6 and 12:

• Factors of 6 are 1, 2, 3, 6
• Factors of 12 are 1, 2, 3, 4, 6, 12

The GCF is 6 since it's the largest number that divides both the numerator and the denominator evenly. Using the GCF not only simplifies the fraction to its lowest terms but also makes mathematical procedures more efficient and solutions more understandable.