Problem 23 - Fundamentals of Mathematics

Question

Convert each improper fraction to a mixed number. $$\frac{51}{8}$$

Step-by-Step Solution

Answer

\(\frac{51}{8} = 6\frac{3}{8}\).

Step 1: Understand the Improper Fraction

An improper fraction is one where the numerator is larger than the denominator. Here, \(\frac{51}{8}\) is improper since 51 > 8.

Step 2: Divide the Numerator by the Denominator

To convert \(\frac{51}{8}\) into a mixed number, divide 51 by 8. Performing the division gives 6 as the quotient and 3 as the remainder.

Step 3: Write the Mixed Number

The quotient from the division (6) becomes the whole number, and the remainder (3) is the new numerator over the original denominator (8). Thus, \(\frac{51}{8}\) can be written as the mixed number \(6\frac{3}{8}\).

Key Concepts

Mixed Numbers

A mixed number is a way of expressing an improper fraction. It combines a whole number with a fraction. This blend is useful because it can make understanding large improper fractions easier. Consider the improper fraction \( \frac{51}{8} \). By converting it to a mixed number, it becomes clear that there are 6 full parts and a small fractional part remaining.

Here is how you find the mixed number:

• First, divide the numerator by the denominator.
• The quotient you get is the whole number part.
• The remainder becomes the new numerator.
• Keep the original denominator as it is.

In our example, \( \frac{51}{8} \) is converted to \( 6\frac{3}{8} \), meaning there are 6 full sections and 3 parts of 8 in the leftover fraction.

Numerator and Denominator

Fractions are made up of two essential parts: the numerator and the denominator.
These terms can help describe a fraction proportionately. For example, in the fraction \( \frac{51}{8} \):

• The numerator is 51; it tells us the number of parts we have.
• The denominator is 8; it tells us how many parts make up a whole.

A numerator larger than the denominator means we have more than one whole, resulting in an improper fraction.
When converting such a fraction to a mixed number, the numerator plays a key role in performing the division operation to find the whole number and the new numerator for the fractional part.

Division in Fractions

Division in fractions is crucial for transforming improper fractions into mixed numbers.
This involves straightforward arithmetic using the numerator and the denominator.
Here's how you can perform this operation effectively:

• Perform the division of the numerator by the denominator.
• The result will be a quotient and a remainder.
• The quotient represents the whole number in the mixed number.
• The remainder becomes the numerator of the fraction part.

For \( \frac{51}{8} \):
When you divide 51 by 8, you get a quotient of 6 and a remainder of 3. These results construct the mixed number \( 6\frac{3}{8} \), clearly showing the division process' importance.

Understanding division in fractions simplifies dealing with improper fractions in your everyday calculations.