Problem 33 - Prealgebra

Question

What is the quotient of \(2 \frac{3}{4}\) and \(3 \frac{1}{4} ?\)

Step-by-Step Solution

Answer

The quotient is \(\frac{11}{13}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(2 \frac{3}{4}\), multiply the whole number by the denominator and add the numerator: \(2 \times 4 + 3 = 8 + 3 = 11\). So, \(2 \frac{3}{4} = \frac{11}{4}\). Similarly, for \(3 \frac{1}{4}\), multiply the whole number by the denominator and add the numerator: \(3 \times 4 + 1 = 12 + 1 = 13\). So, \(3 \frac{1}{4} = \frac{13}{4}\).

Step 2: Set Up the Division of Fractions

Next, set up the division problem using the improper fractions: \(\frac{11}{4} \div \frac{13}{4}\).

Step 3: Apply the Division Rule for Fractions

To divide by a fraction, multiply by its reciprocal. So, \(\frac{11}{4} \div \frac{13}{4}\) becomes \(\frac{11}{4} \times \frac{4}{13}\).

Step 4: Multiply the Fractions

Multiply the numerators and denominators: \(\frac{11}{4} \times \frac{4}{13} = \frac{11 \times 4}{4 \times 13} = \frac{44}{52}\).

Step 5: Simplify the Fraction

Simplify \(\frac{44}{52}\) by finding the greatest common divisor of 44 and 52, which is 4: \(\frac{44 \div 4}{52 \div 4} = \frac{11}{13}\).

Key Concepts

Improper Fractions

Converting mixed numbers to improper fractions is a key step when dividing them. A mixed number combines a whole number with a fraction. However, when performing operations like division, it's easier to work with improper fractions. An improper fraction has a numerator larger than the denominator, which simplifies arithmetic operations. For example, to convert the mixed number \(2 \frac{3}{4}\) into an improper fraction, you:

• Multiply the whole number by the denominator: \(2 \times 4 = 8\).
• Add the numerator: \(8 + 3 = 11\).
• Place the sum over the original denominator: \(\frac{11}{4}\).

This process allows you to handle division and multiplication seamlessly, as improper fractions are ready to go directly into arithmetic operations.

Simplifying Fractions

After performing arithmetic operations on fractions, it's important to simplify the result. Simplifying fractions makes them easier to understand and compare. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). For \(\frac{44}{52}\), determining the GCD:

• List the factors of 44 and 52: 44 is \(1, 2, 4, 11, 22, 44\); 52 is \(1, 2, 4, 13, 26, 52\).
• The largest common factor is 4.

Divide both terms by 4:\[\frac{44 \div 4}{52 \div 4} = \frac{11}{13}\]Simplifying helps in presenting the answer in its most reduced form, which can often be necessary for precise mathematical communication.

Reciprocal of a Fraction

Understanding and using reciprocals is integral when dividing fractions. A reciprocal flips the numerator and denominator of a fraction. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). When you divide by a fraction, you multiply by its reciprocal. This is because division by a fraction is equivalent to multiplying by its inverse.In the division of \(\frac{11}{4} \div \frac{13}{4}\), you:

• Find the reciprocal of \(\frac{13}{4}\): \(\frac{4}{13}\).
• Multiply: \(\frac{11}{4} \times \frac{4}{13}\).

By knowing how to find and use reciprocals, dividing fractions becomes intuitive and efficient.