Problem 87 - Prealgebra

Question

Find each quotient. (Divide) $$4 \div \frac{2}{3}$$

Step-by-Step Solution

Answer

The quotient is 6.

Step 1: Understand Division by a Fraction

To solve the division of a whole number by a fraction, remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is created by flipping the numerator and the denominator. Here, the fraction given is \( \frac{2}{3} \), and its reciprocal is \( \frac{3}{2} \).

Step 2: Multiply by the Reciprocal

Convert the division problem into a multiplication problem using the reciprocal. Thus, instead of \(4 \div \frac{2}{3}\), you should calculate \(4 \times \frac{3}{2}\).

Step 3: Perform the Multiplication

Multiply 4 by \(\frac{3}{2}\). The integer 4 can be written as \(\frac{4}{1}\) for simplification.\[4 \times \frac{3}{2} = \frac{4}{1} \times \frac{3}{2} = \frac{4 \times 3}{1 \times 2} = \frac{12}{2}\]

Step 4: Simplify the Result

Simplify the fraction obtained from the multiplication. \(\frac{12}{2}\) simplifies to 6 because 12 divided by 2 equals 6.

Key Concepts

Understanding Reciprocals

The concept of a reciprocal might sound intimidating at first, but it's actually quite simple. The reciprocal of a fraction is essentially what you get when you flip the fraction upside-down. This means the numerator becomes the denominator and the denominator becomes the numerator. For example, if you have a fraction like \( \frac{2}{3} \), its reciprocal will be \( \frac{3}{2} \). Combining this concept with division is both powerful and handy.

When you divide by a fraction, what you're actually doing is multiplying by its reciprocal. This is a crucial step in converting division problems into multiplication problems. Why is this so helpful? Because multiplying is generally more straightforward and easier to compute, especially when dealing with fractions. Using reciprocals helps simplify the arithmetic operation from division to multiplication, streamlining the process and opening the door to easier calculations in mathematics.

So, keep in mind: the reciprocal is your friend in division.

Multiplying Fractions Made Easy

When it comes to multiplying fractions, the process is much simpler than it seems. Multiplication of fractions doesn't involve cross-multiplying or any complicated steps. You simply multiply the numerators together and then multiply the denominators together.

For example, consider multiplying 4 by \( \frac{3}{2} \). First, express the whole number 4 as a fraction: \( \frac{4}{1} \). This transformation is important because it allows the multiplication to proceed smoothly.

Now, perform the multiplication using this conversion:

• Multiply the numerators: \( 4 \times 3 = 12 \)
• Multiply the denominators: \( 1 \times 2 = 2 \)

The result of the multiplication is \( \frac{12}{2} \). This step-by-step approach ensures accuracy and clarity when dealing with fraction multiplication, paving the way towards our next step, which is simplification.

Simplification of Fractions Explained

Simplifying fractions is all about making fractions as simple as possible. A fraction is considered 'simplified' when the numerator and denominator have no common factors other than 1. This means you've reduced the fraction to its most basic form.

For our example \( \frac{12}{2} \), the simplification process is straightforward. Divide the numerator (12) by the denominator (2) to get the simplified result. The calculation:

• 12 divided by 2 equals 6.

Thus, \( \frac{12}{2} \) simplifies to 6, which is a whole number and is as simple as it gets.

When simplifying fractions, always check for the greatest common divisor (GCD) and divide both the numerator and the denominator by it. This method ensures you're getting the fraction in its simplest form every time. Simplification is a key skill, making complex results accessible and easy to understand, which is particularly useful in solving mathematical problems efficiently.