Problem 17 - Basic College Mathematics

Question

Divide and simplify. \(\frac{3}{8} \div 3\)

Step-by-Step Solution

Answer

The simplified answer is \(\frac{1}{8}\).

Step 1 - Understand the division of fractions

Remember that dividing by a number is the same as multiplying by its reciprocal. For \(\frac{3}{8} \div 3\), this can be rewritten as \(\frac{3}{8} \times \frac{1}{3}\).

Step 2 - Multiply the fractions

Multiply the numerators together and the denominators together. \(\frac{3}{8} \times \frac{1}{3} = \frac{3 \times 1}{8 \times 3} = \frac{3}{24}\).

Step 3 - Simplify the fraction

Simplify the fraction \(\frac{3}{24}\) by finding the greatest common divisor (GCD) of 3 and 24, which is 3. Divide both numerator and denominator by 3: \(\frac{3 \div 3}{24 \div 3} = \frac{1}{8}\).

Key Concepts

Dividing Fractions

When you come across a division involving fractions, it can seem tricky at first. However, a crucial point to remember is that dividing by a fraction is the same as multiplying by its reciprocal. Here is a step-by-step approach:
First, take the reciprocal of the fraction you are dividing by. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
For example, if you have to divide \(\frac{3}{8} \div 3 \), start by writing it as \(\frac{3}{8} \times \frac{1}{3}\).
This strategic approach of converting division into multiplication simplifies the process.

Key steps in dividing fractions:

• Identify the reciprocal.
• Convert division into multiplication.
• Proceed with multiplication.

Multiplying Fractions

Once you have converted division into multiplication by using the reciprocal, the next step is multiplying the fractions. Multiplying fractions is straightforward. Here’s how:
You simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.

For example, with \(\frac{3}{8} \times \frac{1}{3}\):
Multiply the numerators: \[3 \times 1 = 3\]
Multiply the denominators: \[8 \times 3 = 24\]
So, \(\frac{3}{8} \times \frac{1}{3} = \frac{3}{24}\).
Keep in mind that each step should be methodical to avoid mistakes.

Simplifying Fractions

Finally, to simplify a fraction, you need to find its greatest common divisor (GCD). Simplifying makes the fraction easier to understand and work with.
To simplify \(\frac{3}{24}\), find the GCD of 3 and 24. In this case, the GCD is 3.

Divide both the numerator and the denominator by the GCD:
\[ \frac{3 \div 3}{24 \div 3} = \frac{1}{8} \]
So, \(\frac{3}{24}\) simplifies to \(\frac{1}{8}\).

Key steps in simplifying fractions:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Write down the simplified fraction.

By breaking down these steps, you make fraction division simple and manageable.