6 1/2 Divided By 3/2

Mastering Fraction Division: Solving 6 1/2 ÷ 3/2

At first glance, the expression 6 1/2 divided by 3/2 might look like a simple arithmetic problem from a school textbook. However, this calculation sits at the crossroads of fundamental mathematical concepts—mixed numbers, improper fractions, and the critical operation of division by a fraction. Understanding how to solve it is not just about getting an answer; it’s about grasping a foundational skill that empowers you to handle real-world scenarios, from adjusting recipes to calculating material requirements in construction. This article will deconstruct this specific problem into a comprehensive lesson on fraction division, ensuring you not only solve 6 1/2 ÷ 3/2 but also understand the "why" behind every step, building confidence for any similar challenge.

Detailed Explanation: The Components of the Problem

Before we can divide, we must clearly understand the numbers we are working with. The problem presents us with two distinct types of fractional quantities. The first number, 6 1/2, is a mixed number. It combines a whole number (6) with a proper fraction (1/2). This represents a value greater than 6 but less than 7. In practical terms, if you had 6 whole pizzas and an additional half pizza, you would have 6 1/2 pizzas in total.

The second number, 3/2, is an improper fraction. Its numerator (3) is larger than its denominator (2), meaning its value is greater than 1. Specifically, 3/2 is equivalent to 1 1/2, or one and a half. The operation we need to perform is division, which in this context asks a fundamental question: "How many groups of size 3/2 can be made from a total of 6 1/2?" To answer this, we must transform our mixed number into a format compatible with the fraction we are dividing by.

The key principle to internalize is that division by a fraction is equivalent to multiplication by its reciprocal. The reciprocal of a fraction is simply that fraction flipped upside down. For 3/2, the reciprocal is 2/3. This rule is the cornerstone of solving all problems of this nature and stems from the inverse relationship between multiplication and division.

Step-by-Step Breakdown: From Problem to Solution

Let’s walk through the process methodically, treating 6 1/2 ÷ 3/2 as a sequence of logical, manageable steps.

Step 1: Convert All Numbers to Improper Fractions. Our first task is to eliminate the mixed number. To convert 6 1/2 to an improper fraction, we use this formula: (whole number × denominator) + numerator / denominator.

• Multiply the whole number (6) by the denominator (2): 6 × 2 = 12.
• Add this product to the numerator (1): 12 + 1 = 13.
• Place this sum over the original denominator (2). Therefore, 6 1/2 = 13/2. Our problem is now 13/2 ÷ 3/2.

Step 2: Apply the "Keep, Change, Flip" Rule. This memorable mnemonic simplifies the process of dividing fractions.

• Keep the first fraction exactly as it is: 13/2.
• Change the division sign (÷) to a multiplication sign (×).
• Flip the second fraction (3/2) to find its reciprocal (2/3). Our transformed problem is now: 13/2 × 2/3.

Step 3: Multiply the Fractions. Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together.

• Numerator: 13 × 2 = 26
• Denominator: 2 × 3 = 6 This gives us the intermediate product: 26/6.

Step 4: Simplify the Result. The fraction 26/6 is not in its simplest form. We must reduce it by finding the greatest common divisor (GCD) of the numerator and denominator. Both 26 and 6 are divisible by 2.

• 26 ÷ 2 = 13
• 6 ÷ 2 = 3 The simplified result is the improper fraction 13/3.

Step 5: Convert Back to a Mixed Number (Optional but Often Preferred). While 13/3 is a correct answer, it is often more intuitive to express it as a mixed number.

• Divide the numerator (13) by the denominator (3): 13 ÷ 3 = 4 with a remainder of 1.
• The quotient (4) becomes the whole number.
• The remainder (1) becomes the new numerator, placed over the original denominator (3). Thus, the final, most-readable answer is 4 1/3.

Real-World Examples: Why This Calculation Matters

The abstract process becomes tangible through application. Imagine you are a baker. Your recipe for a large batch of cookies calls for 6 1/2 cups of flour. However, you want to scale the recipe down to make only 2/3 of the original batch (since 3/2 of a batch would be more, but let's adjust the example: if you had 6 1/2 cups and your new recipe requires 3/2 cups per batch, how many full batches can you make?). Actually, a better fit: You have 6 1/2 liters of a concentrated cleaning solution. The standard dilution requires 3/2 liters of concentrate per bucket of cleaning mix. How many full buckets can you prepare? The answer, 4 1/3 buckets, tells you you can make 4 complete buckets and will have enough concentrate left for one-third of another bucket. In construction, if a beam requires a 6 1/2-foot long cut from a stock, and your saw blade removes 3/2 inches per pass (a different unit, but the proportional thinking is