Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{9}{2}=\) ______

Short answer

The mixed number is \(4 \frac{1}{2}\).

Step-by-step solution

Identify the Whole Number

To convert an improper fraction to a mixed number, first divide the numerator by the denominator. For \(\frac{9}{2}\), divide 9 by 2. This gives 4, which is the whole number part of the mixed number since \(9 \div 2 = 4\) with a remainder.

Calculate the Remainder

After calculating \(9 \div 2\), the remainder is 1. This remainder will become the new numerator of the fractional part of the mixed number.

Write the Mixed Number

Combine the whole number from step 1 and the fractional part derived from the remainder. The fraction is \(\frac{1}{2}\), so the mixed number is \(4 \frac{1}{2}\).

Reduce to Lowest Terms

Check if the fractional part \(\frac{1}{2}\) can be reduced further. Since it is already in its simplest form, no further reduction is needed.

Key concepts

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. For example, when you have an improper fraction like \(\frac{9}{2}\), it can be transformed into a mixed number. This involves splitting the fraction into a whole number part and a remaining fraction part.

The key process is identifying how many times the denominator can fit into the numerator completely. In our example, \(2\) fits into \(9\) four times, leaving a remainder. This \(4\) is the whole number.

The leftover part becomes the numerator of the fractional component. In \(\frac{9}{2}\), once \(2\) goes into \(9\) four times, there is a remainder of \(1\). Hence, the mixed number becomes \(4 \frac{1}{2}\).

• Mixed numbers make understanding and utilizing fractions easier in practical situations.
• They are especially useful in measurements and in expressing results that aren’t whole.

Division of Fractions

In mathematics, dividing fractions typically involves flipping the second fraction and multiplying. However, when converting improper fractions to mixed numbers, the division serves a different purpose. Here, it helps determine the whole number part of the mixed number.

For the improper fraction \(\frac{9}{2}\), we divide \(9\) by \(2\). The result of the division gives us the whole number component of the mixed number, which is \(4\). This means that the division directly helps us identify the first part of a mixed number.

• Division by fractions also involves ensuring you understand the quotient and remainder.
• In mixed number conversions, the remainder is crucial as it forms the new fraction's numerator.

Reducing Fractions

Reducing a fraction means simplifying it to its lowest terms. This is often done after converting an improper fraction to a mixed number. Simplifying ensures the fraction part of a mixed number is in its simplest form.

To reduce a fraction, you divide both its numerator and denominator by their greatest common divisor (GCD). For the fraction \(\frac{1}{2}\) from \(4 \frac{1}{2}\), no further simplification is needed since \(1\) and \(2\) are already in the simplest ratio.

This step is essential because:

• Simplified fractions are easier to compare and use in further mathematical operations.
• It validates the correctness and completeness of the original conversion process.

Numerator and Denominator

The numerator and denominator are fundamental parts of a fraction. The numerator is the number above the line in a fraction, indicating the number of parts considered. In contrast, the denominator is below the line, representing the total number of equal parts in a whole.

In the improper fraction \(\frac{9}{2}\), \(9\) is the numerator, meaning nine parts are considered. The denominator is \(2\), showing each whole consists of two parts. Knowing these helps in understanding the division process and the remainder when converting to a mixed number.

• Always ensure you know which number is the numerator and which is the denominator to avoid errors.
• Understanding these terms helps in simplifying, adding, and performing operations with fractions.