Problem 78 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 15 \div 3 \frac{1}{3} $$

Step-by-Step Solution

Answer

The simplified result of the operation is \( \frac{9}{2} \).

Step 1: Convert Mixed Number to Improper Fraction

The divisor, 3 \( \frac{1}{3} \), is a mixed number which we can convert into an improper fraction. Multiply the whole number by the denominator and add the numerator: \( 3 \times 3 + 1 = 10 \), so the improper fraction is \( \frac{10}{3} \).

Step 2: Perform Division by Multiplication

To divide by a fraction, multiply by its reciprocal. Therefore, divide 15 by \( \frac{10}{3} \) by multiplying 15 by \( \frac{3}{10} \): \[ 15 \times \frac{3}{10} \].

Step 3: Simplify the Expression

Multiply the whole number by the fraction: \[ 15 \times \frac{3}{10} = \frac{15 \times 3}{10} = \frac{45}{10} \].

Step 4: Simplify the Fraction

Simplify \( \frac{45}{10} \) by dividing the numerator and the denominator by their greatest common divisor, which is 5: \[ \frac{45 \div 5}{10 \div 5} = \frac{9}{2} \].

Key Concepts

Converting Mixed Numbers

Mixed numbers consist of a whole number and a fraction, like 3 \( \frac{1}{3} \). Converting them to improper fractions makes mathematical operations, like division, easier to handle. Here's how you do it:

• Multiply the whole number by the denominator of the fraction. In our example: \( 3 \times 3 = 9 \).
• Add the numerator to this product. For 3 \( \frac{1}{3} \): \( 9 + 1 = 10 \).
• Write this sum over the original denominator. So, 3 \( \frac{1}{3} \) becomes \( \frac{10}{3} \).

This process helps to manage calculations more efficiently. Converting mixed numbers to improper fractions simplifies them, making division more manageable.

Reciprocal of Fractions

A reciprocal flips a fraction upside down. It's crucial when dividing by fractions because dividing by a fraction is equivalent to multiplying by its reciprocal. To find a reciprocal:

• Swap the numerator and the denominator. For \( \frac{10}{3} \), the reciprocal is \( \frac{3}{10} \).

Why do we use reciprocals? This technique changes division into multiplication, which is generally easier to compute:

• Instead of dividing 15 by \( \frac{10}{3} \), we multiply: \( 15 \times \frac{3}{10} \).

This change makes the division process much simpler, turning a complex division into a straightforward multiplication task.

Simplifying Fractions

Simplifying fractions involves reducing them to their most basic form. This means ensuring the numbers in the numerator and the denominator have no common factors other than one. To simplify a fraction like \( \frac{45}{10} \):

• First, find the greatest common divisor (GCD) of both numbers. Here, the GCD of 45 and 10 is 5.
• Divide both the top and bottom numbers by this GCD. For \( \frac{45}{10} \), this is \( \frac{45 \div 5}{10 \div 5} = \frac{9}{2} \).

Now, the fraction is simplified, which not only makes it easier to understand but also helps if further calculations are involved. With practice, you can quickly spot and reduce fractions for clearer, more concise results.