Problem 1 - Prealgebra

Question

Write each number as a fraction. $$-2 \frac{1}{3}$$

Step-by-Step Solution

Answer

The number \(-2 \frac{1}{3}\) written as a fraction is \(-\frac{5}{3}\).

Step 1: Understand Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. In our exercise, we are given \(-2 \frac{1}{3} \) which represents a whole number \(-2\) and a fraction \(\frac{1}{3}\). Our task is to convert this into an improper fraction.

Step 2: Convert Whole Number to Fraction

The whole number part \(-2\) can be converted to a fraction by writing it as \(-\frac{2}{1}\). This step will help in adding fractions later on.

Step 3: Make Denominator Same

Now, to easily add the fractions, rewrite \(-\frac{2}{1}\) with the same denominator as \(\frac{1}{3}\). Multiply both the numerator and denominator by 3 to get \(-\frac{6}{3}\). This way, the denominators are consistent.

Step 4: Combine Fractions

Now combine \(-\frac{6}{3}\) and \(\frac{1}{3}\). Add the numerators: \(-6 + 1 = -5\). Keep the denominator the same (3). Thus, the fraction becomes \(-\frac{5}{3}\).

Step 5: Verify the Result

Check to ensure that \(-\frac{5}{3}\) corresponds to \(-2 \frac{1}{3}\). Dividing \(-5\) by 3 gives \(-1\) with a remainder of 2, which verifies that the improper fraction correctly represents the original mixed number.

Key Concepts

Mixed Numbers

Mixed numbers are a way to express a value that has both a whole number and a fractional component. Picture them as a combination of an integer part and a smaller fraction part. For instance, in the mixed number \(-2 \frac{1}{3}\), \(-2\) is the whole number and \(\frac{1}{3}\) is the fraction. These numbers are useful in everyday situations like cooking or measuring, as they provide a clear representation of whole items and leftover parts. To work with mixed numbers mathematically, such as in addition or subtraction, it's helpful to convert them into improper fractions.

Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This format is crucial in simplifying mathematical operations involving fractions. For example, if we have the mixed number \(-2 \frac{1}{3}\), we convert it to an improper fraction. By multiplying the whole number by the denominator and adding the numerator, we transform \(-2\) into \(-\frac{6}{3}\) by ensuring the denominators match first. Then, combining it with \(\frac{1}{3}\), results in the improper fraction \(-\frac{5}{3}\). Improper fractions are especially handy because they allow us to straightforwardly execute fraction arithmetic without juggling multiple parts.

Fraction Addition

Adding fractions seems daunting at first, but becomes straightforward with practice. The key is to ensure the fractions share a common denominator before addition. This common denominator allows us "connect" fractions, just like adding apples to apples rather than apples to oranges. When we wanted to add \(-\frac{6}{3}\) and \(\frac{1}{3}\) in our exercise, both fractions already had the denominator 3.

• Add the numerators (-6 + 1 = -5)
• Keep the denominator the same (3)

This helps simplify the process, giving you a single fraction, \(-\frac{5}{3}\), which represents the summed value of the original fractions.

Denominator

The denominator of a fraction is the bottom number, which indicates into how many parts the whole is divided. Understanding its role is essential for operations such as addition, subtraction, and comparison of fractions. In any fraction operation, having common denominators means we are dealing equivalent parts.
In our exercise, making sure that \(-\frac{2}{1}\) and \(\frac{1}{3}\) have the same denominator simplifies our task.
Multiply anywhere a denominator mismatch occurs, like turning \(-\frac{2}{1}\) into \(-\frac{6}{3}\) by multiplying by 3, ensures consistency, simplifies calculations, and makes it hassle-free to combine fractions.