Problem 22 - Prealgebra

Question

Find the following quotients. $$12 \div 3 \frac{1}{6}$$

Step-by-Step Solution

Answer

The quotient of \( 12 \div 3 \frac{1}{6} \) is \( \frac{72}{19} \).

Step 1: Convert Mixed Number to Improper Fraction

The divisor in the expression is a mixed number: \[ 3 \frac{1}{6} \]First, convert it into an improper fraction. Multiply the whole number (3) by the denominator (6) and add the numerator (1):\[ 3 \times 6 + 1 = 19 \]So, \( 3 \frac{1}{6} \) becomes \( \frac{19}{6} \).

Step 2: Divide by a Fraction

When dividing by a fraction, instead multiply by its reciprocal. The reciprocal of \( \frac{19}{6} \) is \( \frac{6}{19} \). Therefore, \( 12 \div \frac{19}{6} \) becomes \( 12 \times \frac{6}{19} \).

Step 3: Multiply and Simplify

Multiply the whole number by the fraction:\[ 12 \times \frac{6}{19} = \frac{12 \times 6}{19} = \frac{72}{19} \]This fraction cannot be simplified further as 72 and 19 have no common factors other than 1.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. Proper fractions have numerators smaller than their denominators. In the example of \(3 \frac{1}{6}\), 3 is the whole number, and \(\frac{1}{6}\) is the fraction. Mixed numbers are often used in everyday math to express quantities like 3 and a fraction more, such as in recipes or measurements. To perform operations like addition, subtraction, or division, it’s typically easier to work with improper fractions. This is why, when dividing by a mixed number, the first step is converting it into an improper fraction.

• Whole numbers are those without fractions or decimals.
• A proper fraction has a numerator smaller than its denominator.
• Mixed numbers can be easily converted for calculations.

Improper Fractions

Improper fractions have numerators that are equal to or larger than their denominators. This is helpful in calculations because they represent a number greater than or equal to 1, which simplifies mathematical operations. Converting a mixed number to an improper fraction involves multiplying the whole number by the denominator, adding the numerator, and putting the result over the original denominator. This conversion provides a straightforward form for performing arithmetic operations such as division or multiplication.
📏 Example: \(3 \frac{1}{6}\) converts to \(\frac{19}{6}\).

• This method assists in performing calculations without misunderstandings.
• Improper fractions efficiently express larger quantities.
• They are easy to manipulate in equations.

Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. When you need to divide by a fraction, you multiply by its reciprocal. This method leverages the concept that dividing by a number is the same as multiplying by its reciprocal, effectively transforming a division problem into a multiplication one. It turns a potentially complex operation into a simpler one, making it easier to solve. In the exercise given, the reciprocal of \(\frac{19}{6}\) is \(\frac{6}{19}\), which changes the division \(12 \div \frac{19}{6}\) into the multiplication \(12 \times \frac{6}{19}\).

• Reciprocal operation simplifies division into multiplication.
• Turns complex division into straightforward multiplication.
• Essential in solving fractions-involved equations.