Problem 45 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{5 x}{2} \div 3$$

Step-by-Step Solution

Answer

\[ \frac{5x}{6} \]

Step 1: Convert Division into Multiplication

To divide by a number, convert the division into multiplication by the reciprocal of the number. So, \ \ \[ \frac{5x}{2} \times \frac{1}{3} \]

Step 2: Multiply the Numerators

Multiply the numerators together:\[ 5x \times 1 = 5x \]

Step 3: Multiply the Denominators

Multiply the denominators together:\[ 2 \times 3 = 6 \]

Step 4: Write the Result

Combine the results from Step 2 and Step 3 to give the final fraction:\[ \frac{5x}{6} \]

Key Concepts

Dividing Fractions

Dividing fractions may seem tricky at first, but it's a simple process once you know the steps. When you have a division problem involving fractions, you can convert it into a multiplication problem. This is done by using the reciprocal of the divisor. Here, the divisor is 3, which is the same as \( \frac{3}{1} \). The reciprocal of \( \frac{3}{1} \) is \( \frac{1}{3} \). So, instead of dividing by 3, you multiply by \( \frac{1}{3} \).

For example, in the problem \( \frac{5x}{2} \) divided by 3, you turn it into: \[ \frac{5x}{2} \times \frac{1}{3} \] By changing the problem to multiplication, the process becomes much simpler.

Multiplication of Fractions

Once you've converted the division into multiplication, the next step is straightforward. First, multiply the numerators (the top numbers) of the fractions. Using our example: \[ 5x \times 1 = 5x \]

Then, multiply the denominators (the bottom numbers): \[ 2 \times 3 = 6 \]

When multiplying fractions, always remember:

• Numerator with numerator
• Denominator with denominator

Combine these results to get your final answer. Here, it becomes: \[ \frac{5x}{6} \] This method works for any fractions you need to multiply.

Reciprocal

Understanding the concept of the reciprocal is vital in both division and multiplication of fractions. The reciprocal of a number is simply flipping the numerator and the denominator. For instance:

• The reciprocal of 5 is \( \frac{1}{5} \)
• The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \)
• The reciprocal of 2 (which is \( \frac{2}{1} \)) is \( \frac{1}{2} \)

When dividing fractions, you always use the reciprocal of the second fraction. This allows you to convert the division problem into a multiplication problem, making it easier to solve. Practice finding reciprocals to become more comfortable with these operations.