Problem 20 - Prealgebra

Question

Change each improper fraction to a mixed number. $$\frac{41}{15}$$

Step-by-Step Solution

Answer

The mixed number is \( 2 \frac{11}{15} \).

Step 1: Understand the Problem

An improper fraction is where the numerator (top number) is greater than the denominator (bottom number). A mixed number has a whole number part and a proper fraction part. We need to express \( \frac{41}{15} \) as a mixed number.

Step 2: Divide the Numerator by the Denominator

Divide \( 41 \) by \( 15 \) to find out how many whole numbers can be taken from the fraction. Perform the division: \( 41 \div 15 = 2 \) with a remainder of \( 11 \). This means \( 15 \times 2 = 30 \) and there are \( 41 - 30 = 11 \) remaining.

Step 3: Construct the Mixed Number

The quotient from the division, \( 2 \), becomes the whole number part of the mixed number. The remainder, \( 11 \), becomes the numerator of the fractional part, and the original denominator \( 15 \) remains the same. Therefore, the mixed number is \( 2 \frac{11}{15} \).

Key Concepts

Mixed Numbers

A mixed number is a special way of expressing an improper fraction, specifically one where the numerator (the number on top) is larger than the denominator (the bottom number).
It consists of two parts:

• A whole number.
• A proper fraction, which is a fraction where the numerator is smaller than the denominator.

To convert an improper fraction to a mixed number, divide the numerator by the denominator to find out how many whole parts fit into the fraction.
The remainder becomes the numerator of the fractional part, keeping the same denominator. For example, in the fraction \(\frac{41}{15}\), when we divide 41 by 15, we get 2 as a quotient and 11 as a remainder.
Thus, the mixed number is \(2 \frac{11}{15}\). This representation makes it easier to understand and work with when solving math problems.

Division

Division plays a key role in converting an improper fraction to a mixed number. When you divide the numerator by the denominator, you determine the whole number part of the mixed number.
Here's how division is used in this process: The numerator of 41 is divided by the denominator, 15. The quotient from this division gives us the whole number part of the mixed number.
In our example, this division gives a result of 2. The remainder from this division becomes the numerator of the fractional part. Therefore, division helps break down improper fractions into easier parts, assisting in simplification and understanding.

Numerator and Denominator Concepts

The terms numerator and denominator are foundational concepts in fractions.

• Numerator: This is the top number of a fraction, indicating how many parts we have.
• Denominator: This is the bottom number, showing into how many equal parts something is divided.

In an improper fraction like \(\frac{41}{15}\), the numerator (41) is larger than the denominator (15).
When converting it to a mixed number, the numerator tells you how many parts you have, and the denominator tells you the size of each part. By dividing the numerator by the denominator, you find out how many whole pieces can be made, and any leftover pieces are represented by the remainder. Thus, understanding these concepts is vital for correctly performing fractional conversions.