Problem 21 - Elementary Algebra

Question

For exercises 13-22, rewrite the improper fraction as a mixed number. $$ \frac{41}{6} $$

Step-by-Step Solution

Answer

The improper fraction \(\frac{41}{6}\) is \(6\ \frac{5}{6}\).

Step 1: Divide the numerator by the denominator

Perform the division of the numerator (41) by the denominator (6). The quotient will be the whole number part of the mixed number, and the remainder will be used for the fractional part.

Step 2: Find the whole number component

Divide 41 by 6: 41 ÷ 6 = 6 with a remainder of 5. So, the whole number part of the mixed number is 6.

Step 3: Find the remainder

The remainder from the division is 5. This remainder becomes the numerator of the fractional part of the mixed number.

Step 4: Form the mixed number

Combine the whole number part (6) with the fractional part \(\frac{5}{6}\). Thus, \(\frac{41}{6} \) can be rewritten as the mixed number \(6\ \frac{5}{6}\).

Key Concepts

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means the fraction represents a value equal to or greater than one. For example, \(\frac{41}{6}\) is an improper fraction because 41 is greater than 6.

Improper fractions are often inconvenient to work with directly since the large numerator can make it challenging to visualize the value. Hence, these fractions are usually converted into mixed numbers for ease.

Mixed Numbers

A mixed number combines a whole number and a proper fraction (a fraction where the numerator is less than the denominator). For instance, when we convert \(\frac{41}{6}\) into a mixed number, we get \(6\frac{5}{6}\).

Mixed numbers make it easier to understand the value of a fraction greater than one because they clearly show how many whole parts and additional fractional parts exist. Use mixed numbers when you want a simpler representation of an improper fraction.

Division of Fractions

To convert an improper fraction into a mixed number, you first need to divide the numerator by the denominator. This step gives two important pieces of information: the quotient and the remainder.

In our example, dividing 41 by 6:

• The quotient is 6, which becomes the whole number part of the mixed number.
• The remainder is 5, which will be the numerator of the fractional part.

The remainder tells us what is left over after creating whole parts from the numerator.

Fractional Remainder

After finding the quotient (whole number part), the remainder from the division becomes the numerator of the fractional part of the mixed number.

Using our example, the division of 41 by 6 gives a remainder of 5. This means that after making 6 whole parts, we still have 5 parts out of 6 left.

Therefore, the final step in forming the mixed number is to combine the whole number with the fractional part:

So, \(\frac{41}{6}\) becomes \(6\frac{5}{6}\).

Converting properly and understanding the fractional remainder is crucial for accurately expressing improper fractions as mixed numbers.