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A mixed number is a combination of a whole number and a fraction. It is often used to represent quantities that are more than a whole but less than the next whole number. For example, if you have 1 and an additional fraction of 9/20, it is written as the mixed number \(1 \frac{9}{20}\). In mixed numbers, the whole number shows how many complete units you have, and the fraction accounts for the part that is less than a whole.

Mixed numbers can be positive or negative. A negative mixed number is represented with a negative sign before the whole number, for example, \(-1 \frac{9}{20}\). When working with mixed numbers, it is often necessary to convert them into improper fractions, especially in mathematical operations like multiplication or division. Doing so simplifies the calculation by using only fractions.

The numerator is the top part of a fraction and represents how many parts of the whole are being considered. In the fraction \(\frac{9}{20}\), the numerator is 9. It tells us that, out of a total of 20 equal parts, we are considering 9 parts.

When converting a mixed number into an improper fraction, the numerator is recalculated. You do this by taking the whole number, multiplying it by the denominator, and then adding the original numerator. This new value becomes the numerator of the improper fraction. Using the exercise example, for \(-1 \frac{9}{20}\), the math to find the new numerator is:

• Multiply: \(-1 \times 20 = -20\)
• Add the original numerator: \(-20 + 9 = -11\)

Hence, the improper fraction becomes \(\frac{-11}{20}\), with -11 as the numerator.

The denominator is the bottom part of a fraction, showing into how many equal parts the whole is divided. In the fraction \(\frac{9}{20}\), 20 is the denominator. It tells us that each whole is divided into 20 parts.

Denominators remain unchanged when converting a mixed number into an improper fraction. Regardless of what changes in the numerator during conversion, the denominator continues to represent the total number of parts the whole is divided into.

• For example, in both \(-1 \frac{9}{20}\) and its improper fraction form \(\frac{-11}{20}\), the denominator is 20.

This consistency helps maintain the proportion of the fraction throughout conversions.

Whole numbers are the numbers without fractional or decimal parts, including all integers and zero. They are straightforward and are often paired with fractions to make mixed numbers.

In the context of mixed numbers, the whole number indicates the complete units you have on top of the fractional part. In \(-1 \frac{9}{20}\), the whole number is -1, suggesting that we have one complete unit in the negative.

To convert a mixed number into an improper fraction, the whole number plays a critical role, as it is multiplied by the denominator and combined with the numerator:

• Whole number \(-1: -1 \times 20 = -20\)
• Adding numerator 9 gives you \(-20 + 9 = -11\)

This transformation is necessary for operations requiring fractions alone, like adding fractions with the same denominator.