What Is 1 4 Divided By 2 In Fraction Form

What is 1 4/5 Divided by 2 in Fraction Form? A Comprehensive Guide

This article will thoroughly explain how to divide the mixed number 1 4/5 by 2, ultimately expressing the answer in fraction form. We'll cover the steps involved, the underlying mathematical principles, and even address some frequently asked questions. Understanding this seemingly simple calculation lays a crucial foundation for more complex fraction problems encountered in algebra and beyond. By the end, you’ll not only know the answer but also confidently tackle similar problems.

Understanding Mixed Numbers and Improper Fractions

Before diving into the division, let's refresh our understanding of mixed numbers and improper fractions. A mixed number combines a whole number and a fraction (e.g., 1 4/5). An improper fraction, on the other hand, has a numerator larger than or equal to its denominator (e.g., 9/5). These two forms are interchangeable.

To convert a mixed number to an improper fraction, we follow these steps:

Multiply the whole number by the denominator: 1 * 5 = 5
Add the result to the numerator: 5 + 4 = 9
Keep the same denominator: 5

Therefore, 1 4/5 is equal to 9/5. This conversion is crucial for simplifying our division problem.

Dividing Fractions: The Reciprocal Method

Dividing fractions involves a clever trick: we don't actually divide; instead, we multiply by the reciprocal. The reciprocal of a fraction is simply the fraction flipped upside down. For example, the reciprocal of 2/3 is 3/2, and the reciprocal of 5 is 1/5 (since 5 can be written as 5/1).

Our problem is 1 4/5 ÷ 2. We've already converted 1 4/5 to 9/5. Now, we rewrite the division problem as a multiplication problem using the reciprocal of 2 (which is 1/2):

9/5 ÷ 2 becomes 9/5 * 1/2

Performing the Multiplication

Multiplying fractions is straightforward: we multiply the numerators together and the denominators together:

(9 * 1) / (5 * 2) = 9/10

Therefore, 1 4/5 divided by 2 is equal to 9/10.

Illustrative Example: Sharing Pizza

Let's visualize this with a real-world example. Imagine you have one whole pizza and four-fifths of another pizza. You want to share this total amount of pizza equally between two people. How much pizza does each person get?

First, we determine the total amount of pizza: 1 4/5 pizzas. Then, we divide this amount by 2 (the number of people). As we've calculated, each person receives 9/10 of a pizza. This visually reinforces the result of our fraction division.

A Deeper Dive into the Mathematics

The process of dividing by a fraction is inherently linked to the concept of reciprocals and the multiplicative inverse. Every non-zero number has a multiplicative inverse, a number that when multiplied by the original number results in 1. For fractions, this inverse is simply the reciprocal.

Consider the division problem a/b ÷ c/d. We can rewrite this as a fraction:

(a/b) / (c/d)

To simplify this complex fraction, we multiply both the numerator and denominator by the reciprocal of the denominator:

[(a/b) * (d/c)] / [(c/d) * (d/c)]

This simplifies to:

(ad/bc) / 1 = ad/bc

This demonstrates why the "flip and multiply" method works: it's a shortcut to arrive at the same result.

Expanding on the Concept: Dividing by Mixed Numbers

Let's extend our understanding to scenarios where we divide by another mixed number. For example, let’s solve 1 4/5 ÷ 2 1/2.

First, convert both mixed numbers into improper fractions:

1 4/5 = 9/5 2 1/2 = 5/2

Now, rewrite the division as multiplication using the reciprocal of 5/2 (which is 2/5):

9/5 ÷ 5/2 = 9/5 * 2/5

Multiply the numerators and denominators:

(9 * 2) / (5 * 5) = 18/25

Therefore, 1 4/5 divided by 2 1/2 is 18/25. This illustrates the consistent application of the reciprocal method for various division problems with fractions and mixed numbers.

Frequently Asked Questions (FAQs)

Q1: Can I divide the whole number part and the fractional part separately?

A1: No, you cannot. You must first convert the mixed number to an improper fraction before performing the division. Treating the whole and fractional parts independently will yield an incorrect answer.

Q2: What if I'm dividing by a fraction smaller than 1?

A2: The method remains the same. When you multiply by the reciprocal of a fraction less than 1 (a fraction where the numerator is smaller than the denominator), the result will be larger than the original number because you're essentially multiplying by a number greater than 1.

Q3: How do I simplify my answer after dividing?

A3: After performing the multiplication, check if the resulting fraction can be simplified. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD. For example, 18/25 cannot be simplified further because the GCD of 18 and 25 is 1.

Conclusion

Dividing mixed numbers, like 1 4/5 divided by 2, requires a methodical approach. Converting the mixed number into an improper fraction (9/5 in this case) is the crucial first step. Then, we utilize the reciprocal method, changing the division problem into a multiplication problem by using the reciprocal of the divisor. This provides a clear and efficient path to arriving at the correct fractional answer, which is 9/10. By understanding these fundamental principles and practicing the steps, you can confidently tackle any fraction division problems. Remember, mastery of fractions is a cornerstone of mathematical proficiency, opening doors to more advanced concepts in the future.