Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(12 \div \frac{2}{3}=\) ______

Short answer

The result of the division is 18.

Step-by-step solution

Write the Division as a Multiplication

To divide by a fraction, you multiply by its reciprocal. The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). So, we rewrite the division problem as a multiplication: \[12 \div \frac{2}{3} = 12 \times \frac{3}{2}\]

Convert 12 into a Fraction

Before multiplying, convert 12 into a fraction by expressing it as \(\frac{12}{1}\). This makes it easier to multiply. \[12 = \frac{12}{1}\]

Multiply the Numerators

Multiply the numerators together: \[12 \times 3 = 36\]

Multiply the Denominators

Multiply the denominators together: \[1 \times 2 = 2\]

Write the Resulting Fraction

The product of the numerators over the product of the denominators gives the new fraction: \[\frac{36}{2}\]

Simplify the Fraction

Divide the numerator and the denominator by their greatest common divisor, which is 2 in this case:\[\frac{36}{2} = 18\]So, the fraction simplifies to 18.

Key concepts

Reciprocals

Understanding reciprocals is essential when you're dealing with the division of fractions. A reciprocal of a fraction is essentially what you multiply a number by to get 1. For a given fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). This means you simply switch the numerator and the denominator.

When you need to divide by a fraction, like from the problem \( 12 \div \frac{2}{3} \), you don’t actually do the division directly. Instead, you multiply by the reciprocal of the fraction. Thus, you convert the operation to: \( 12 \times \frac{3}{2} \).

This process makes it much easier to handle fractions because it turns the problem into a multiplication one, which you already know!

Fraction Multiplication

Multiplying fractions might initially seem tricky, but it's straightforward once you understand the process. After converting a division problem into multiplication by using reciprocals, the next step is to perform the multiplication.

Here's how to multiply fractions:

• Multiply the numerators together: In our exercise, \( 12 \times 3 = 36 \).
• Multiply the denominators together: For our problem, \( 1 \times 2 = 2 \).

These two simple steps give you a new fraction, \( \frac{36}{2} \).

It's a straightforward operation, making it a favorite method when working with divided fractions. Also, using this method helps you keep the fractions in line and prevent mistakes with complex calculations.

Simplifying Fractions

Simplifying, or reducing, fractions is an important final step that often makes the result easier to understand and work with. After you multiply, you might end up with a fraction that's not in the simplest form.

To simplify, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number. In our example:

• The fraction \( \frac{36}{2} \) has a GCD of 2.
• Divide both the numerator and denominator by 2.
• This gives you \( \frac{36 \div 2}{2 \div 2} = \frac{18}{1} \).

At this point, \( \frac{18}{1} \) can further be expressed as just 18. Remember, always look to simplify where possible, so your answers are neat, concise, and clear.