Problem 36 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$6 \frac{2}{3} \div 5$$

Step-by-Step Solution

Answer

\(6 \frac{2}{3} \div 5 = \frac{4}{3}\) or as a mixed number, \(1 \frac{1}{3}\).

Step 1: Convert Mixed Number to Improper Fraction

The first step is to convert the mixed number \(6 \frac{2}{3}\) into an improper fraction. To do this, multiply the whole number 6 by the denominator 3 and add the numerator 2. This gives us: \(6 \times 3 + 2 = 18 + 2 = 20\). So, \(6 \frac{2}{3} = \frac{20}{3}\).

Step 2: Division of Fractions

To divide by a fraction, we multiply by its reciprocal. The division expression \(\frac{20}{3} \div 5\) is equivalent to multiplying \(\frac{20}{3}\) by the reciprocal of 5, which is \(\frac{1}{5}\).

Step 3: Multiply Improper Fraction by Reciprocal

Now, multiply the improper fraction by the reciprocal: \(\frac{20}{3} \times \frac{1}{5}\). Multiply the numerators: \(20 \times 1 = 20\), and multiply the denominators: \(3 \times 5 = 15\). This gives us \(\frac{20}{15}\).

Step 4: Simplify the Fraction

Finally, simplify \(\frac{20}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, \(\frac{20 \div 5}{15 \div 5} = \frac{4}{3}\).

Step 5: Write as a Mixed Number (Optional)

If desired, convert the improper fraction \(\frac{4}{3}\) back into a mixed number. Divide 4 by 3: the quotient is 1 with a remainder of 1, so \(\frac{4}{3} = 1 \frac{1}{3}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They come handy in situations where quantities are more than a whole but less than the next whole. For instance, if you have 6 full pizzas and two-thirds of another, you can express this as the mixed number \(6 \frac{2}{3}\). This structure makes it easier to visualize and understand than its improper fraction counterpart.

To convert a mixed number into an improper fraction, multiply the whole number by the denominator of the fraction and add the numerator. For example, with \(6 \frac{2}{3}\):

• Multiply the whole number 6 by the denominator 3: \(6 \times 3 = 18\).
• Add the result to the numerator 2: \(18 + 2 = 20\).
• Therefore, \(6 \frac{2}{3} = \frac{20}{3}\).

By changing it into an improper fraction, calculations become more straightforward.

Getting to Know Improper Fractions

Unlike mixed numbers, improper fractions have numerators larger than their denominators. This might seem odd at first, but they are very useful, especially in mathematical calculations.

Consider the improper fraction \(\frac{20}{3}\) which is derived from the mixed number \(6 \frac{2}{3}\). While it implies "how many thirds there are altogether", the improper fraction is beneficial when performing operations like multiplication, division, or simplifying. Keeping the fraction as improper allows us to use it directly in operations without going back and forth between mixed and proper fractions. This is why they are preferred for continuous mathematical operations.

The Role of Reciprocal Multiplication in Division

Reciprocal multiplication is a fundamental process when dividing fractions. Instead of dividing directly, you multiply by the reciprocal of the divisor. The reciprocal flips the numerator and the denominator.

Consider \(\frac{20}{3} \div 5\), which can be transformed by taking the reciprocal of 5, turning it into \(\frac{1}{5}\). Instead of dividing, we then multiply:

• Multiply the numerators: \(20 \times 1 = 20\).
• Multiply the denominators: \(3 \times 5 = 15\).

This process turns the problem into \(\frac{20}{3} \times \frac{1}{5} = \frac{20}{15}\). Reciprocal multiplication simplifies division to an easily manageable multiplication problem.

Simplifying Fractions for Clarity

Simplifying fractions makes them easier to understand and solve. This involves using the greatest common divisor (GCD) to reduce both the numerator and the denominator.

Take \(\frac{20}{15}\) as an example. Find the GCD of 20 and 15, which is 5.

• Divide the numerator by this number: \(20 \div 5 = 4\).
• Divide the denominator by this number: \(15 \div 5 = 3\).

After simplification, you end up with a simpler fraction: \(\frac{4}{3}\). Sometimes, it might be useful to convert back into a mixed number, which can be expressed as \(1 \frac{1}{3}\). Simplifying ensures that fractions remain clear and concise for further calculations.