Problem 19 - Basic College Mathematics

Question

Divide and simplify. \(\frac{12}{7} \div 4\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{3}{7}\).

Step 1 - Understand Division of Fractions

To divide by a number, you can multiply by its reciprocal. The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 4 is \(\frac{1}{4}\).

Step 2 - Rewrite the Division as Multiplication

Rewrite the division problem \(\frac{12}{7} \div 4\) as a multiplication problem using the reciprocal of 4: \(\frac{12}{7} \times \frac{1}{4}\).

Step 3 - Multiply the Fractions

Multiply the numerators and the denominators: \(\frac{12 \times 1}{7 \times 4} = \frac{12}{28}\).

Step 4 - Simplify the Fraction

Simplify \(\frac{12}{28}\) by finding the greatest common divisor (GCD) of 12 and 28, which is 4. Divide both the numerator and the denominator by 4: \(\frac{12 \div 4}{28 \div 4} = \frac{3}{7}\).

Key Concepts

reciprocal

When dealing with fractions, the concept of the reciprocal is essential. A reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 4 is \(\frac{1}{4}\). This is particularly useful when you're dividing fractions. Instead of directly dividing, we can flip the divisor (make it a reciprocal) and turn the division problem into a multiplication one.
Here are a few steps to find reciprocals:

• For a whole number like 4, its reciprocal is 1 divided by 4, which is \(\frac{1}{4}\).
• For a fraction like \(\frac{3}{5}\), its reciprocal is simply flipping the numerator and the denominator, which gives us \(\frac{5}{3}\).

Knowing how to find and use reciprocals can greatly simplify solving many math problems, especially those involving division.

multiplication of fractions

Multiplying fractions may seem challenging, but it's straightforward if you break it down. When you multiply two fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, let's multiply: \(\frac{3}{7} \times \frac{2}{5}\).

• Multiply the numerators: 3 and 2, giving 6.
• Multiply the denominators: 7 and 5, giving 35.

This results in \(\frac{6}{35}\). There's no need to find a common denominator like in addition or subtraction of fractions. Simply focus on the top and bottom separately. Also, if possible, simplify your fraction at the end to make it as simple as possible.

simplifying fractions

Simplifying fractions makes them easier to understand and work with. A fraction is simplified when the numerator and denominator have no common factors other than 1. To simplify a fraction, follow these steps:

• Identify the Greatest Common Divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.

For instance, if we have \(\frac{12}{28}\), we can simplify it by:

• Finding the GCD of 12 and 28, which is 4.
• Dividing 12 by 4 to get 3.
• Dividing 28 by 4 to get 7.

This means \(\frac{12}{28}\) can be simplified to \(\frac{3}{7}\). Simplified fractions are easier to read and understand, making them fundamental to mastering fractions.