Problem 66 - Algebra 1: Concepts and Skills

Question

SUBTRACTING FRACTIONS Subtract. Write the answer as a fraction or as a mixed number in simplest form. $$ 18 \frac{9}{14}-3 \frac{3}{14} $$

Step-by-Step Solution

Answer

The answer is \(15 \frac{3}{7}\)

Step 1: Subtract Whole Numbers

Subtract the whole numbers first. That is, subtract the 3 from the 18, which equals 15.

Step 2: Subtract Fractions

Next, subtract the fractions. Since the denominators are already identical (14), simply subtract the numerator of the second fraction from the numerator of the first fraction. This is \(9 - 3 = 6\) . Thus, the fraction part of the result is \(6/14\).

Step 3: Simplify the Fraction

In order to simplify \(6/14\), find the greatest common divisor (GCD) of 6 and 14. The GCD is 2. Divide both the numerator and the denominator by 2 to get the simplest form of the fraction: \(6 /2\) = 3 and \(14 / 2\) = 7. The simplified fraction is \(3/7\).

Step 4: Combine the Whole Number and Fraction

To write the final answer as a mixed number, combine the results from step 1 and step 3. This gives us \(15 \frac{3}{7}\)

Key Concepts

Mixed Numbers

When dealing with fractions, especially in subtraction problems, you will often encounter mixed numbers. A mixed number combines a whole number and a fraction like this: \(18 \frac{9}{14}\).
This simply means 18 whole parts and an additional \(\frac{9}{14}\) of a part.
To work with mixed numbers, it is important to separate the whole number from the fractional part.

• First, deal with the whole numbers: subtract them, add them, whatever the operation requires.
• Then, handle the fractions separately.

This approach simplifies the process, making it less overwhelming. In our example, we first subtract the whole numbers 18 and 3, and then we deal separately with the fractions \(\frac{9}{14}\) and \(\frac{3}{14}\). This method allows us to tackle subtraction in easy, manageable steps.

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is a key concept in simplifying fractions. It is the largest integer that can divide both the numerator and the denominator without any remainder.
To find the GCD of two numbers, consider using tools such as:

• Prime Factorization: Break down both numbers into their prime factors and multiply the common ones.
• Euclidean Algorithm: A more advanced method involving division and remainders.

In our exercise, we were tasked with simplifying \(\frac{6}{14}\). The GCD for 6 and 14 turns out to be 2.
By dividing both the numerator and the denominator by their GCD, \(\frac{6}{2} = 3\) and \(\frac{14}{2} = 7\), we transform the fraction into its simplest form, \(\frac{3}{7}\). Finding the GCD and simplifying fractions is crucial for achieving the correct and simplest result.

Simplest Form

Arriving at the simplest form of a fraction means reducing it as much as possible, so that no common divisors other than 1 can divide both numerator and denominator.
This process not only makes the fraction look neat but also ensures further mathematical operations with it become easier. Let's see how you can consistently achieve this:

• Identify the GCD of the numerator and the denominator.
• Divide both by the GCD.
• Re-examine to ensure it truly is simplified.

For the fraction \(\frac{6}{14}\), previously mentioned, reducing it involves dividing by their GCD, found to be 2, resulting in \(\frac{3}{7}\).
When you subtract or add mixed numbers and fractions, always check if the result can be simplified. Simplification not only follows good mathematical practice but also usually makes the result more pleasant to work with! The ability to quickly reduce to simplest form is an excellent skill in all mathematics.