Problem 45 - Algebra 1: Concepts and Skills

Question

Write the improper fraction as a mixed number. $$ \frac{20}{3} $$

Step-by-Step Solution

Answer

The mixed number equivalent of the improper fraction \(\frac{20}{3}\) is 6 \(\frac{2}{3}\).

Step 1: Understand the Improper Fraction

An improper fraction is given, which is \(\frac{20}{3}\). The numerator is 20 and the denominator is 3. This fraction is known as an improper fraction because the numerator is greater than the denominator.

Step 2: Divide the Numerator by the Denominator

In order to convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient forms the whole part, and the remainder forms the numerator of the fraction part. So, divide 20 by 3.

Step 3: Write the Mixed Number

The division in the previous step yields a quotient of 6 and a remainder of 2. This means that the mixed number will have a whole part of 6 and a fraction part of \(\frac{2}{3}\). So, the mixed number equivalent of the improper fraction \(\frac{20}{3}\) is 6 \(\frac{2}{3}\).

Key Concepts

Improper Fraction

When you come across a fraction where the numerator (top number) is larger than the denominator (bottom number), you are looking at an improper fraction. These fractions are intriguing because they represent a value greater than one whole. Imagine slicing a pizza into three pieces, then suddenly having 20 slices out of the three that make up a whole pizza - it seems like you've got extra slices, right? That's the gist of an improper fraction; it's as if you have more than one 'whole' of something.

For example, \( \frac{20}{3} \) is an improper fraction, where the 20 (more slices than we have in one whole pizza) is the numerator, and 3 (the original pizza slices making up one whole) is the denominator.

Mixed Number

Now, what if we want to express that 'extra' in terms of whole pizzas and leftover slices? This is where the mixed number comes into play. A mixed number combines a whole number and a proper fraction (where the numerator is smaller than the denominator). It's like saying, 'I have 2 whole pizzas and a half of another one'.

A real-life example of a mixed number could be if you've eaten 6 whole apples and half of another one; you'd say you had 6 and 1/2 apples. In mathematical language, our previous improper fraction \( \frac{20}{3} \) can be converted into the mixed number 6 \( \frac{2}{3} \) where '6' represents the whole pizzas and \( \frac{2}{3} \) signifies the leftover slices.

Converting Fractions

The process of changing an improper fraction to a mixed number is like breaking down how many 'wholes' you can make from the parts you have. Here's the breakdown of the conversion steps:

• Divide the numerator by the denominator to find how many 'whole' units you have
• The quotient (result of the division) is your whole number.
• The remainder becomes the new numerator of the fractional part.
• Keep the original denominator as the denominator of the fractional part.

To use our previous example, dividing 20 (numerator) by 3 (denominator) gives us 6 whole units and a remainder of 2. Those remainders are the key to constructing the proper fraction for the mixed number, making our final mixed number 6 \( \frac{2}{3} \) in this case.

Numerators and Denominators

To confidently manipulate fractions, understanding the roles of numerators and denominators is crucial. The numerator, which is the space above the fraction bar, tells us how many parts we have. Think of it as the count of things you've got. On the other hand, the denominator, the figure below the fraction bar, tells us what to divide the whole into, like cutting a cake into pieces. The size of these pieces (or 'slices') is determined by the denominator.

If we increase the denominator while keeping the numerator constant, each piece and thus the total fraction gets smaller. Conversely, if we increase the numerator with a constant denominator, the fraction's value climbs up, potentially leading to an improper fraction when the numerator exceeds the denominator.