Question

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

Short answer

The answer is 1.

Step-by-step solution

Convert the Division to Multiplication

To divide fractions, we multiply the first fraction by the reciprocal (inverse) of the second fraction. The reciprocal of \( \frac{8}{3} \) is \( \frac{3}{8} \). Thus, the division problem becomes \( \frac{8}{3} \times \frac{3}{8} \).

Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together. \[\frac{8}{3} \times \frac{3}{8} = \frac{8 \times 3}{3 \times 8} = \frac{24}{24}\]

Simplify the Fraction

The fraction \(\frac{24}{24}\) equals 1 because any number divided by itself is 1.

Key concepts

Reciprocal of a Fraction

When dividing fractions, the first step is finding the reciprocal of the fraction you are dividing by. The reciprocal of a fraction flips its numerator and denominator. This means turning a fraction of the form \( \frac{a}{b} \) into \( \frac{b}{a} \). The reciprocal comes into play because dividing by a fraction is the same as multiplying by its reciprocal. For example, if you have the fraction \( \frac{8}{3} \), its reciprocal is \( \frac{3}{8} \). This process is crucial for converting a division problem into a multiplication problem.

• The reciprocal of \( \frac{8}{3} \) is \( \frac{3}{8} \).
• Finding the reciprocal helps in transforming division into a simpler multiplication task.
• This step simplifies further calculations when dealing with fractions.

Simplifying Fractions

Simplifying, or reducing fractions, means to make the fraction as simple as possible. A fraction is said to be in its simplest form when the greatest common divisor (GCD) of the numerator and the denominator is 1. This process involves dividing both the top and bottom numbers by their GCD. In the case of our exercise, the fraction \( \frac{24}{24} \) results from multiplication and needs to be simplified.

• If numerator equals denominator, divide both by themselves to get 1.
• Ensure each fraction is reduced during calculations for simplicity.
• Simplification often makes it easier to interpret the answer.

Multiplying Fractions

Multiplying fractions is quite straightforward. Once you've converted a division problem into multiplication using the reciprocal, as mentioned previously, the next step is to multiply the fractions. Multiplication with fractions involves:- Multiplying the numerators, which are the top numbers.- Multiplying the denominators, which are the bottom numbers.
For example, in the exercise given, after finding the reciprocal and changing the operation to multiplication, you align the fractions as \( \frac{8}{3} \times \frac{3}{8} \). Multiply across:

• Numerators: \(8 \times 3 = 24\)
• Denominators: \(3 \times 8 = 24\)

Thus, the result is \( \frac{24}{24} \), which simplifies to 1. This step-by-step approach of handling numerators and denominators separately ensures clarity and accuracy in solving fraction problems.