Dividing Fractions: A Step-by-Step Guide With Examples

by Alex Johnson 55 views

Are you struggling with dividing fractions? Don't worry, you're not alone! Many people find fraction division a bit tricky at first, but with the right approach, it becomes quite manageable. This guide will walk you through the process step by step, providing clear explanations and examples to help you master this essential math skill. We'll tackle a specific example: dividing 7/24 by 35/48 and reducing the result to its simplest form. Let’s dive in!

Understanding the Basics of Fraction Division

Before we get into the specific problem, let's cover the fundamental concept of dividing fractions. Dividing fractions might seem daunting, but it's essentially multiplying by the reciprocal. The reciprocal of a fraction is simply the fraction flipped over. For example, the reciprocal of 2/3 is 3/2. When you divide by a fraction, you multiply by its reciprocal. This method works because division is the inverse operation of multiplication. Imagine you have half a pizza (1/2) and you want to divide it among 3 people. Mathematically, this is (1/2) ÷ 3, which is the same as (1/2) ÷ (3/1). To solve it, you multiply (1/2) by the reciprocal of (3/1), which is (1/3). So, (1/2) * (1/3) = 1/6. Each person gets one-sixth of the pizza. This basic concept is key to handling any fraction division problem.

The "Keep, Change, Flip" Method

The most common method for dividing fractions is often remembered by the acronym "KCF," which stands for Keep, Change, Flip. Let’s break down what each of these steps means:

  1. Keep: Keep the first fraction as it is. In our example of 7/24 ÷ 35/48, we keep 7/24.
  2. Change: Change the division sign (÷) to a multiplication sign (×).
  3. Flip: Flip the second fraction (the divisor) to find its reciprocal. This means swapping the numerator and the denominator. For 35/48, flipping it gives us 48/35.

So, the original problem 7/24 ÷ 35/48 becomes 7/24 × 48/35. This transformation makes the problem much easier to solve because multiplication is generally more straightforward to handle than division. This method works consistently because it's grounded in the mathematical principle that dividing by a number is the same as multiplying by its inverse. By understanding this, you’re not just memorizing a trick; you’re applying a fundamental mathematical concept. Knowing why the method works helps you remember it and apply it correctly in various situations. This foundational knowledge will also assist you in tackling more complex math problems involving fractions and other operations.

Step-by-Step Solution: Dividing 7/24 by 35/48

Now, let's apply the "Keep, Change, Flip" method to our specific problem: 7/24 divided by 35/48. This step-by-step solution will make the process clear and easy to follow.

Step 1: Keep the First Fraction

The first step is to keep the first fraction, which is 7/24, exactly as it is. We don’t change anything about this fraction at this stage. So, 7/24 remains 7/24. This might seem simple, but it’s a crucial starting point. Keeping the first fraction the same ensures that we’re maintaining the original proportion in relation to the second fraction, which we will manipulate to perform the division. This step is about preserving the initial value that we are dividing.

Step 2: Change the Division Sign to Multiplication

Next, we change the division sign (÷) to a multiplication sign (×). This is the core of the "Keep, Change, Flip" method, transforming the division problem into a multiplication problem. Mathematically, this is justified because division is the inverse operation of multiplication. By changing the operation, we are setting up the problem to be solved using the more familiar rules of fraction multiplication. This single change makes the problem much more approachable, as multiplying fractions is generally easier than dividing them. So, we replace '÷' with '×', which sets the stage for the next step, where we deal with the second fraction.

Step 3: Flip the Second Fraction (Find the Reciprocal)

This is where we flip the second fraction, 35/48, to find its reciprocal. Flipping a fraction means swapping its numerator (the top number) and its denominator (the bottom number). So, 35/48 becomes 48/35. The reciprocal of a fraction, when multiplied by the original fraction, equals 1. This property is essential for the mathematical validity of the "Keep, Change, Flip" method. By using the reciprocal, we are effectively undoing the division operation and converting it into an equivalent multiplication problem. This step is crucial because it allows us to use the rules of fraction multiplication, which are often more intuitive to apply.

Step 4: Multiply the Fractions

Now that we've kept the first fraction, changed the division to multiplication, and flipped the second fraction, we can multiply the two fractions. Our problem now looks like this: 7/24 × 48/35. To multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we have:

  • Numerator: 7 × 48
  • Denominator: 24 × 35

Calculating these products gives us:

  • 7 × 48 = 336
  • 24 × 35 = 840

So, the result of the multiplication is 336/840. While this is the correct product, it’s not in its simplest form yet. The next step is to reduce this fraction to its lowest terms, which means finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it. This process ensures that we express the fraction in its most concise form, making it easier to understand and work with in future calculations.

Reducing the Fraction to Its Simplest Form

After multiplying the fractions, we obtained 336/840. To reduce this fraction to its simplest form, we need to find the greatest common factor (GCF) of 336 and 840 and divide both the numerator and the denominator by this GCF.

Step 1: Find the Greatest Common Factor (GCF)

The GCF is the largest number that divides both 336 and 840 without leaving a remainder. One way to find the GCF is by listing the factors of each number and identifying the largest one they have in common. However, a more efficient method is to use prime factorization.

  1. Prime Factorization of 336:
    • 336 = 2 × 168
    • 168 = 2 × 84
    • 84 = 2 × 42
    • 42 = 2 × 21
    • 21 = 3 × 7
    • So, 336 = 2 × 2 × 2 × 2 × 3 × 7 = 2⁴ × 3 × 7
  2. Prime Factorization of 840:
    • 840 = 2 × 420
    • 420 = 2 × 210
    • 210 = 2 × 105
    • 105 = 3 × 35
    • 35 = 5 × 7
    • So, 840 = 2 × 2 × 2 × 3 × 5 × 7 = 2³ × 3 × 5 × 7

Now, we identify the common prime factors and their lowest powers:

  • 2 is a common factor; the lowest power is 2³
  • 3 is a common factor; the lowest power is 3¹
  • 7 is a common factor; the lowest power is 7¹

Multiply these together to find the GCF: GCF = 2³ × 3 × 7 = 8 × 3 × 7 = 168.

Step 2: Divide the Numerator and Denominator by the GCF

Now that we’ve found the GCF to be 168, we divide both the numerator (336) and the denominator (840) by 168:

  • 336 ÷ 168 = 2
  • 840 ÷ 168 = 5

So, the reduced fraction is 2/5. This is the simplest form of the fraction, as 2 and 5 have no common factors other than 1. Reducing fractions to their simplest form is important because it makes them easier to understand and compare. It also simplifies further calculations involving these fractions. By dividing both the numerator and the denominator by their GCF, we ensure that the fraction is expressed in its most concise and manageable form.

Final Answer and Conclusion

After dividing 7/24 by 35/48 and reducing the result to its simplest form, we arrive at the fraction 2/5. Therefore, the correct answer is C. 2/5.

Dividing fractions might seem challenging initially, but by following the "Keep, Change, Flip" method and understanding the concept of reciprocals, you can simplify the process significantly. Remember to always reduce your final answer to its simplest form for clarity and accuracy. Mastering fraction division is a crucial step in building a strong foundation in mathematics, and with practice, you'll become more confident in tackling such problems. Don’t hesitate to review these steps and try additional examples to solidify your understanding. Happy fraction dividing!

For further learning and practice on fractions, you can visit Khan Academy's Fraction Resources.