Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{11}{4}=\) ______

Short answer

\(2 \frac{3}{4}\)

Step-by-step solution

Divide the numerator by the denominator

Start by dividing the numerator 11 by the denominator 4. 11 divided by 4 is equal to 2 with a remainder of 3. This means that 11 divided by 4 goes 2 whole times.

Write the whole number

The whole number is 2, as calculated from the division. This will be the whole part of the mixed number.

Identify the remainder

The remainder from the division is 3. This remainder becomes the new numerator of the fraction part of the mixed number.

Combine to form the mixed number

The remainder (numerator) is 3 and the original denominator remains 4. Thus, the fraction part is \(\frac{3}{4}\). So, the mixed number is 2 \(\frac{3}{4}\).

Reduce to lowest terms (if necessary)

Check if \(\frac{3}{4}\) can be reduced to lowest terms. Since 3 and 4 have no common factors other than 1, it is already in its simplest form.

Key concepts

Mixed Numbers

Mixed numbers are numbers that contain both a whole number and a fraction combined together. They make it easier to visualize quantities greater than a single whole but not quite reaching the next whole number. For example, when you have an improper fraction like \( \frac{11}{4} \), you can convert it into a mixed number to simplify its representation.
Here's how you convert an improper fraction to a mixed number:

• Divide the numerator (the top number) by the denominator (the bottom number) to find out how many whole numbers can be formed.
• The quotient becomes the whole number in the mixed number, and the remainder becomes the numerator of the fractional part.
• The original denominator remains as the denominator of the fraction part of the mixed number.

In our example, dividing 11 by 4 gives a quotient of 2 with a remainder of 3, resulting in the mixed number 2 \( \frac{3}{4} \). Mixed numbers provide a more intuitive way to understand excess values beyond full units.

Numerator and Denominator

In the realm of fractions, the numerator and the denominator hold pivotal roles. The numerator is the top number in a fraction and it indicates how many parts of the whole you are considering. The denominator, on the other hand, is the bottom number, and it tells you into how many equal parts the whole is divided.
For example, in the fraction \( \frac{3}{4} \),

• The numerator is 3, which means you are considering 3 parts of the whole.
• The denominator is 4, which means the whole is divided into 4 equal parts.

Understanding these terms is essential because they form the building blocks of dividing quantities. When you dealt with \( \frac{11}{4} \), the 11 indicates how many parts we have, and 4 indicates into how many parts a whole is divided. After converting this to a mixed number, the fractional part \( \frac{3}{4} \) maintains the same denominator, showing continuity of the division of wholes.

Fraction Reduction

Fraction reduction involves simplifying a fraction to its smallest equivalent form. This means adjusting the numerator and the denominator such that they have no common factors other than 1. This process ensures that the fraction is as compact and understandable as possible.
Here's how to reduce a fraction:

• Determine the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.

For instance, when simplifying \( \frac{3}{4} \), we observe that the only common factor between 3 and 4 is 1. Therefore, this fraction is already in its simplest form. Reducing fractions is a crucial step as it helps maintain clarity in mathematical expressions and ensures that fractions are easy to interpret at a glance.