Problem 25 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(\frac{1}{12}-\left(-\frac{7}{12}\right)\)

Step-by-Step Solution

Answer

\( \frac{2}{3} \)

Step 1: Identify the Operation

The expression given is \( \frac{1}{12} - \left(-\frac{7}{12}\right) \). Notice that you are subtracting a negative fraction. By mathematical rules, subtracting a negative is equivalent to adding the positive of that number.

Step 2: Transform the Expression

Change the subtraction of a negative number to addition: \( \frac{1}{12} - \left(-\frac{7}{12}\right) = \frac{1}{12} + \frac{7}{12} \).

Step 3: Add Fractions with Same Denominator

Since both fractions \( \frac{1}{12} \) and \( \frac{7}{12} \) have the same denominator of 12, you can add the numerators directly: \( \frac{1 + 7}{12} = \frac{8}{12} \).

Step 4: Simplify the Fraction

The fraction \( \frac{8}{12} \) can be simplified by finding the greatest common divisor (GCD) of 8 and 12, which is 4. Divide both the numerator and the denominator by 4: \( \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \).

Key Concepts

Adding Fractions

When adding fractions, it's important to pay attention to the denominators, which are the numbers at the bottom of each fraction. To add fractions that have the same denominator, like \( \frac{1}{12} \) and \( \frac{7}{12} \), you simply add the numerators, which are the numbers on top of each fraction.
This means you perform the addition: \( 1 + 7 = 8 \). You then place that result over the common denominator, making the sum \( \frac{8}{12} \). If the denominators are different, you must first find a common denominator before adding, but in this case, since they are the same, you can skip this step.

• Ensure denominators are the same.
• Add the numerators together.
• Keep the common denominator.

Remember, it's only the numerators that change, the denominators stay put unless you're working with different ones.

Subtracting Fractions

The concept of subtracting fractions closely connects to adding them, especially when subtracting a negative fraction.
Mathematically, subtracting a negative fraction is the same as adding the positive version of that fraction. For example, subtracting \( -\frac{7}{12} \) becomes adding \( \frac{7}{12} \).
This is due to the double negative rule in mathematics where two negatives make a positive.

• Recognize subtraction of negative fractions as addition.
• Revise the expression to transform subtraction to addition.

Once the subtraction is rephrased as an addition, proceed by following the rules for adding fractions.

Simplifying Fractions

Simplifying a fraction involves reducing it to its simplest form, so it is easier to understand and use.
For instance, after adding \( \frac{1}{12} \) and \( \frac{7}{12} \) to get \( \frac{8}{12} \), you can simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
The GCD of 8 and 12 is 4. You then divide both the numerator and the denominator by this GCD.

• Find the GCD of the numerator and the denominator.
• Divide both parts of the fraction by the GCD.
• Write the simplified fraction.

So, \( \frac{8}{12} \) becomes \( \frac{2}{3} \) once simplified, as \( 8 \div 4 = 2 \) and \( 12 \div 4 = 3 \). Simplification helps to make fractions more manageable and neat.