Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{22}{6}=\) ______

Short answer

The mixed number is \(3 \frac{2}{3}\).

Step-by-step solution

Divide the Numerator by the Denominator

To convert an improper fraction to a mixed number, first divide the numerator (22) by the denominator (6). So, perform 22 ÷ 6 which equals 3 with a remainder of 4.

Write the Mixed Number

The quotient from Step 1 becomes the whole number part of the mixed number. The remainder becomes the numerator of the new fraction, and the original denominator remains.Thus, \[\frac{22}{6} = 3 \frac{4}{6}\]

Simplify the Fraction

Now simplify the fractional part \(\frac{4}{6}\). Find the greatest common divisor (GCD) of 4 and 6, which is 2. Divide both the numerator and denominator by 2.\[\frac{4}{6} = \frac{4\div 2}{6\div 2} = \frac{2}{3}\]

Write the Mixed Number in Simplest Form

With the simplified fraction from Step 3, rewrite the mixed number.The final answer is:\[3 \frac{2}{3}\]

Key concepts

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means the fraction represents a number greater than or equal to one. Understanding improper fractions is essential for converting them into mixed numbers, which provide a clearer sense of measurement in daily life. For example,

• The improper fraction \( \frac{22}{6} \) indicates that 22 parts are being divided into groups of 6.

To make sense of these, improper fractions can be converted into mixed numbers, which are expressions that combine a whole number and a proper fraction. This not only simplifies comprehension but also brings clarity to operations like addition and subtraction. In our exercise, when we divide the numerator (22) by the denominator (6), we get a quotient of 3 and a remainder of 4. Therefore,

• \( \frac{22}{6} = 3 \frac{4}{6} \)

Rethinking fractions in this mixed form gives context and ease in many mathematical problems.

Simplifying Fractions

Simplifying fractions means reducing the fraction to its simplest form, where the numerator and the denominator are the smallest possible integers that retain the same value of the fraction. Simplifying is crucial because it makes the fraction easier to work with and understand.To simplify, we look for the greatest common divisor (GCD) of both the numerator and the denominator. Once this number is identified, you divide both parts of the fraction by this GCD. For example, in the fraction \( \frac{4}{6} \) from our exercise, the GCD is 2. Thus:

• You divide the numerator, 4, by 2, resulting in 2.
• You divide the denominator, 6, by 2, resulting in 3.

Thus, the fraction simplifies to \( \frac{2}{3} \). This step is significant as it decreases the complexity of the fraction and hence, errors during calculations. A simplified fraction shows the simplest ratio of the two numbers, offering a clear view into the structure of the fraction.

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is the largest positive integer that divides two or more integers without leaving a remainder. Finding the GCD is essential not just in simplifying fractions but in various mathematical processes.To determine the GCD of two numbers such as 4 and 6, you need to find all divisors of each number and identify the largest one they have in common. For practical use in simplifying, you can:

• Write down the factors of each number. For 4, the factors are 1, 2, and 4. For 6, the factors are 1, 2, 3, and 6.
• Identify the common factors, which in this case for 4 and 6, include 1 and 2. The greatest is 2.

Using this GCD, you improve the fraction \( \frac{4}{6} \) to its simplest form \( \frac{2}{3} \), keeping its value but ensuring it's more straightforward for any operation or further use in problem-solving scenarios.