Problem 24 - Fundamentals of Mathematics

Question

Convert each improper fraction to a mixed number. $$\frac{121}{15}$$

Step-by-Step Solution

Answer

\(8\frac{1}{15}\)

Step 1: Divide the Numerator by the Denominator

First, divide the numerator (121) by the denominator (15) to find how many whole numbers fit into the fraction. Perform the division: 121 divided by 15 equals 8 remainder 1. So, 15 fits into 121 a total of 8 times with a remainder of 1.

Step 2: Write the Whole Number

The result from the division, 8, will be the whole number part of our mixed number.

Step 3: Formulate the Fractional Part

Use the remainder obtained in Step 1 (which is 1) as the new numerator. The denominator remains the same as the original fraction, which is 15. Thus, the fractional part is \(\frac{1}{15}\).

Step 4: Combine Whole Number and Fraction

Combine the whole number and the fractional part to form the mixed number, which is \(8\frac{1}{15}\).

Key Concepts

Mixed Numbers

Mixed numbers are a helpful way to present improper fractions in a more understandable format by separating the whole number part from the fraction.
Think of them as a blend of a whole number and a proper fraction.
For example, when you have an improper fraction like \(\frac{121}{15}\), we can convert it to a mixed number.

• The division of the numerator by the denominator reveals how many times the whole denominator fits into the numerator.
• The result becomes the whole number part of the mixed number.
• In our example, dividing 121 by 15 gives 8, which is our whole number.

Now, the remainder of this division (1 in our example) doesn't disappear. It's used to form the fractional part of the mixed number. The mixed number hence becomes \(8\frac{1}{15}\).This representation makes it easier to visualize and work with when solving problems.

Numerator and Denominator

Understanding the terms numerator and denominator is crucial for working with fractions, including when converting them to mixed numbers.
The numerator is the top part of the fraction which indicates how many parts we are considering.
The denominator, on the other hand, is the bottom part showing into how many equal parts the whole is divided.

• In \(\frac{121}{15}\), 121 is the numerator, and 15 is the denominator.
• The numerator (121) tells us how many total parts are being considered.
• The denominator (15) shows each whole is divided into 15 equal parts.

When converting to a mixed number, the process involves dividing the numerator by the denominator to find out how many complete sets (or wholes) of the denominator fit into the numerator.The result of this division gives us both the whole number and the remainder for the mixed number.

Division in Fractions

Division is a fundamental process when dealing with fractions, especially improper ones.
It’s through division that we determine the mixed number form of an improper fraction.
Here's how the process works through our example of \(\frac{121}{15}\):- **Perform the division**: Divide the numerator (121) by the denominator (15).- **Identify the quotient**: This is 8 in our case, which forms the whole number part of the mixed number.- **Find the remainder**: The remainder here is 1, forming the numerator of the fractional part of our mixed number.The division essentially breaks down how many full denominators (or wholes) can be taken from the numerator.This is key for conversion and also for understanding relationships between numbers in mathematics.