Problem 39 - Algebra 1: Concepts and Skills

Question

Write the improper fraction as a mixed number. $$ \frac{21}{9} $$

Step-by-Step Solution

Answer

The mixed number representation of \( \frac{21}{9} \) is 2 \( \frac{3}{9} \).

Step 1: Division

Divide the numerator by the denominator, \( \frac{21}{9} \). This returns a quotient and possible remainder. In this case, the quotient is 2 (because 9 goes into 21 two times) and the remainder is 3 (because after subtracting 18 (2 multiplied by 9) from 21, 3 is left over).

Step 2: The Whole Number Part

The quotient from the division in step 1 represents the whole number part of the mixed number. Here, it is 2.

Step 3: The Fractional Part

The remainder from the division in step 1 gets placed over the original denominator to form the fractional part of the mixed number. Therefore, the fractional part here is \( \frac{3}{9} \).

Key Concepts

Simplifying Fractions

When working with fractions, it's often helpful to express them in their simplest form. Simplifying fractions means reducing them to their lowest terms. To do this, you find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number. For example, consider the fraction \( \frac{3}{9} \) from the exercise. The GCD of 3 and 9 is 3. So, you divide both the numerator and the denominator by 3 to simplify the fraction:
\[ \frac{3 \div 3}{9 \div 3} = \frac{1}{3} \]
By simplifying, the fraction becomes easier to understand and work with in future arithmetic operations.

Division of Fractions

When dividing one fraction by another, you apply the rule of multiplying by the reciprocal. However, in the context of converting improper fractions into mixed numbers, you're effectively dividing the numerator by the denominator. This is a straightforward process where you determine how many times the denominator can fit into the numerator. It's akin to traditional division in basic arithmetic. For the fraction \( \frac{21}{9} \), you calculate how many groups of 9 can be made out of 21. This is equivalent to \( \frac{21 \div 9}{1} \) or 2 with a remainder.

Mixed Number Conversion

Improper fractions are fractions where the numerator is larger than the denominator. Converting an improper fraction to a mixed number involves a mix of division and fractions. The whole number part is obtained by dividing the numerator by the denominator. In our example, \( \frac{21}{9} \) becomes 2 because 21 divided by 9 equals 2 whole groups.

Finding the Fractional Part

After that, the remainder from the division becomes the numerator of the fractional part. Here, the remainder is 3, so the fractional part is \( \frac{3}{9} \). Remember to simplify the fractional part as outlined in the simplifying fractions section to achieve the final mixed number: 2 \( \frac{1}{3} \).

Fractional Remainders

A fractional remainder is what's left over after performing the division to convert an improper fraction to a mixed number. This remainder can't be divided further without yielding another fraction. It becomes the numerator of the fractional part of the mixed number, with the denominator remaining unchanged. For example, in the improper fraction \( \frac{21}{9} \), the division yields a remainder of 3. This remainder is placed over the original denominator to form the fractional part of \( \frac{3}{9} \) which simplifies to \( \frac{1}{3} \) as part of the final mixed number.