Problem 15 - Elementary Algebra

Question

For exercises 13-22, rewrite the improper fraction as a mixed number. $$ \frac{21}{2} $$

Step-by-Step Solution

Answer

The mixed number is 10\(\frac{1}{2}\).

Step 1 - Divide the numerator by the denominator

To convert an improper fraction to a mixed number, divide the numerator (21) by the denominator (2). Perform the division to find the quotient and the remainder. \[21 \div 2 = 10 \text{ R } 1\]

Step 2 - Write down the quotient

The quotient (10) becomes the whole number part of the mixed number.

Step 3 - Write down the remainder as the new numerator

The remainder (1) becomes the numerator of the fraction part of the mixed number. Keep the same denominator (2).

Step 4 - Combine the whole number and the fraction

Put the whole number and the fraction together to form the mixed number: \[10 \frac{1}{2} \]

Key Concepts

division

When converting an improper fraction to a mixed number, the first step involves dividing the numerator by the denominator. This division helps us to break down the fraction into two parts: a whole number and a smaller fraction. For example, to convert \(\frac{21}{2}\) into a mixed number, we divide 21 by 2. This process helps us understand how many times the denominator fits into the numerator wholly. The quotient tells us the whole number part, while what’s left is the remainder, which will be used in the fraction part of the mixed number.

quotient

In the division process, the quotient is the result of dividing the numerator by the denominator. It represents the whole number part of the mixed number. For example, when dividing 21 by 2, we get a quotient of 10. This means that 2 goes into 21 ten whole times.

The quotient forms the first part of the mixed number, indicating how many full units we have.

remainder

After finding the quotient, there might still be a part of the numerator left over, which we call the remainder. This remainder is what’s left after the full division of the numerator by the denominator. For instance, in our example of dividing 21 by 2, after having 21 divided by 2 to give a quotient of 10, we are left with a remainder of 1.

This remainder becomes the numerator of the fractional part of our mixed number. It shows the part that is not fully divisible by the denominator.

mixed number

A mixed number combines a whole number with a proper fraction. We use the quotient as the whole number and the remainder over the original denominator as the fractional part. After performing the division of 21 by 2, we get:

• Quotient (whole number): 10
• Remainder (fractional numerator): 1

Combining these, we write the improper fraction \(\frac{21}{2}\) as the mixed number \(\text{10} \frac{1}{2}\). This provides a clearer and more intuitive representation of the fraction by separating the whole number part from the fractional part.