Question

Change the following mixed numbers to improper fractions. \(2 \frac{1}{2}=\) ______

Short answer

\(2 \frac{1}{2} = \frac{5}{2}\) as an improper fraction.

Step-by-step solution

Understand the Mixed Number

A mixed number is made up of a whole number and a fraction. In this exercise, we have the mixed number \(2 \frac{1}{2}\), where 2 is the whole number, and \(\frac{1}{2}\) is the fractional part.

Convert the Whole Number to a Fraction

To convert the whole number to a fraction, multiply it by the denominator of the fractional part. Here, the denominator is 2. So, \(2 = \frac{2 \times 2}{2} = \frac{4}{2}\).

Add the Fractions Together

Now, add the converted whole number fraction to the fractional part. We have \(\frac{4}{2} + \frac{1}{2}\). Since the denominators are the same, you can add the numerators: \(4 + 1 = 5\).

Write the Improper Fraction

Combine the result to form the improper fraction. So, \(2 \frac{1}{2}\) as an improper fraction is \(\frac{5}{2}\).

Key concepts

Mixed Numbers

Mixed numbers combine a whole number with a fraction, offering a concise way to express numbers that are not whole but still more than zero. For example, the fraction \(2 \frac{1}{2}\) is a mixed number, where 2 is the whole number, and \(\frac{1}{2}\) is the fraction. Mixed numbers are useful in everyday situations, such as when measuring quantities, dividing objects, or when using recipes. Understanding mixed numbers is the first step in manipulating fractions effectively, whether you're adding, subtracting, or converting them.

• Whole Number: Represents the complete units.
• Fraction: Represents the part that is less than a whole.

This format helps in visualizing the number as part whole and part fraction, aiding in easier comprehension and manipulation of fractional values.

Improper Fractions

An improper fraction is a type of fraction where the numerator is larger than or equal to the denominator. This is the opposite of a proper fraction, where the numerator is less than the denominator. For instance, \(\frac{5}{2}\) is an improper fraction, as 5 (numerator) is larger than 2 (denominator). Improper fractions are often the result of converting mixed numbers, allowing for easier calculation in mathematical operations such as addition and division.

• They might look "improper" but are actually very useful.
• They help make complex calculations simpler because they remove the whole number part from mixed numbers.

By turning mixed numbers into improper fractions, you align them with other fractions, thus simplifying various mathematical processes.

Whole Number Conversion

Converting a whole number into a fraction is an essential skill in handling mixed numbers. To accomplish this, multiply the whole number by the denominator of the fractional part. This creates a fraction that represents the original whole number.
For example, with the mixed number \(2 \frac{1}{2}\), the whole number 2 will be converted by multiplying it with 2. Thus, it becomes \(\frac{4}{2}\). This step is crucial as it sets the stage for adding the fractional parts together seamlessly.

• Multiply the whole number by the denominator.
• Express the result as a fraction with the original denominator.

This conversion allows for easy addition with the existing fraction, as the denominators will already be equal.

Addition of Fractions

Adding fractions is a common operation in mathematics that requires common denominators. For fractions like \(\frac{4}{2}\) and \(\frac{1}{2}\), the process becomes easier as they already share the same denominator of 2.
Follow these simple steps:

• Ensure the denominators are the same; if not, find the common denominator.
• Add the numerators while keeping the denominator constant.
• Simplify the fraction if necessary.

In our example, adding \(\frac{4}{2} + \frac{1}{2}\) results in \(\frac{5}{2}\). This sum is straightforward because the denominators align, allowing direct addition of numerators. Practicing the addition of fractions with like denominators reinforces your understanding of fractions, making it a valuable skill in handling diverse mathematical problems.