Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{11}{5}=\) ______

Short answer

\(2\frac{1}{5}\)

Step-by-step solution

Divide the Numerator by the Denominator

To convert an improper fraction to a mixed number, start by dividing the numerator (11) by the denominator (5). In this case, 5 goes into 11 twice. So, the quotient is 2.

Determine the Remainder

After dividing, find the remainder from the division. 5 times 2 is 10, and subtracting this from 11 gives a remainder of 1.

Write the Mixed Number

Use the quotient and the remainder to form the mixed number. The quotient (2) becomes the whole number part, and the remainder (1) over the original denominator (5) becomes the fractional part. Therefore, the mixed number is \(2\frac{1}{5}\).

Simplify the Fraction

Check to see if the fractional part of the mixed number can be simplified. Since 1 and 5 have no common factors other than 1, the fraction \(\frac{1}{5}\) is already in its simplest form.

Key concepts

Understanding Mixed Numbers

When we talk about mixed numbers, we are essentially dealing with a combination of a whole number and a fraction together. This happens when a fraction (which we call an improper fraction) has a numerator larger than its denominator. To convert such a fraction into a mixed number:

• First, divide the numerator by the denominator to find the whole number part.
• The remainder becomes the new numerator of the fractional part, while the original denominator remains unchanged.

For example, with \(\frac{11}{5}\), dividing 11 by 5 gives a quotient of 2, meaning the whole number is 2. The remainder is 1, forming the fractional part \(\frac{1}{5}\). Therefore, the mixed number is \(2\frac{1}{5}\). Mixed numbers are useful because they offer a clearer picture of quantities that exceed a whole number, making some math problems easier to understand and manage.

Simplifying Fractions Made Easy

Simplifying fractions is all about making the fraction as simple as possible by ensuring the numerator and the denominator have no common factors other than 1. This results in the fraction's simplest or lowest terms. Here’s how you can simplify any fraction:

• First, find the greatest common divisor (GCD) of both the numerator and the denominator.
• Next, divide both parts of the fraction by this GCD.

In the case of our example \(\frac{1}{5}\), the numbers 1 and 5 have no common factor other than 1, meaning the fraction is already in its simplest form. Simplifying fractions helps in making calculations easier, understanding fractions better, and achieving a neater, clearer result in your math work.

The Role of Division with Remainders

Division with remainders is a crucial concept when converting improper fractions to mixed numbers. This division process not only gives us the whole number but also provides the remainder needed for the fractional part. Here's a brief overview of how it works:

• Perform the division normally as you would with whole numbers.
• The quotient from the division becomes the whole number part of the mixed number.
• The remainder becomes the numerator of the fraction, while the denominator remains the same as the original fraction.

Taking our example, dividing 11 by 5 gives a quotient of 2, which is the whole part. The remainder is 1, so the fractional part is \(\frac{1}{5}\). This simple process is invaluable as it forms a bridge between fractions and mixed numbers, making it easier to comprehend and work with quantities that aren't whole numbers.