Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{3}{4} \div \frac{3}{8}=\) ______

Short answer

The result is 2.

Step-by-step solution

Understand Division of Fractions

When dividing two fractions, instead of dividing directly, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

Find the Reciprocal of the Second Fraction

The second fraction is \(\frac{3}{8}\). To find its reciprocal, we swap the numerator and the denominator, resulting in \(\frac{8}{3}\).

Multiply by the Reciprocal

Replace the division with multiplication and use the reciprocal found in Step 2. So, \(\frac{3}{4} \div \frac{3}{8} = \frac{3}{4} \times \frac{8}{3}\).

Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together. This gives \(\frac{3 \times 8}{4 \times 3} = \frac{24}{12}\).

Simplify the Fraction

Simplify \(\frac{24}{12}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 12. Thus, \(\frac{24}{12} = 2\).

Key concepts

Reciprocal of Fractions

When you hear the term "reciprocal," think of flipping a fraction upside down. It's a simple concept with important uses, especially in fraction division.
For any fraction, the numerator (the top number) and the denominator (the bottom number) swap places to form the reciprocal.
Let's consider the fraction \( \frac{3}{8} \). Its reciprocal would be \( \frac{8}{3} \).
Using reciprocals makes dividing fractions much easier, turning division into multiplication.

• Flip the fraction: Swap the numerator and the denominator.
• Remember: The product of a fraction and its reciprocal is always 1.
• Every number has a reciprocal, even whole numbers (like 3, which has a reciprocal of \( \frac{1}{3} \)).

Multiplication of Fractions

Unlike adding and subtracting fractions, multiplying them is quite straightforward. You don't need to make the denominators the same.
Here’s how it works: Take the numerators of the fractions and multiply them together. Then, do the same with the denominators.
For example, if we have \( \frac{3}{4} \times \frac{8}{3} \), we multiply 3 by 8 and 4 by 3. This gives:

• Numerator: \( 3 \times 8 = 24 \)
• Denominator: \( 4 \times 3 = 12 \)

Thus, the result of \( \frac{3}{4} \times \frac{8}{3} \) is \( \frac{24}{12} \).
Multiplying fractions may feel easier after a bit of practice, just remember, it's all about multiplying the top numbers together, and the bottom numbers together.

• Multiply the numerators.
• Multiply the denominators.
• Simplify the result if needed.

Simplifying Fractions

Simplifying fractions involves writing them in their simplest form. It makes them easier to understand and use.
To simplify a fraction, divide the numerator and the denominator by their greatest common divisor (GCD). This is the largest number that divides both evenly.
Let’s simplify \( \frac{24}{12} \):

• Find the GCD of 24 and 12, which is 12.
• Divide the numerator and the denominator by the GCD: \( \frac{24 \div 12}{12 \div 12} = \frac{2}{1} \).
• \( \frac{2}{1} \) simplifies to 2 since any number over 1 is itself.

Simplifying can often make your final answer look cleaner and be easier to interpret for any mathematical problem.

• Find the GCD.
• Divide the numerator and denominator by the GCD.
• Write in simplest form.

In many cases, especially complex math problems, being able to simplify fractions effectively is a valuable skill.