Is 72/96 Equivalent To 12/16

Introduction

In the world of mathematics, understanding the concept of equivalent fractions is essential. It allows us to simplify complex fractions, making them easier to work with. One common question that arises is whether the fractions 72/96 and 12/16 are equivalent. In this article, we will explore the concept of equivalent fractions, delve into the process of determining whether two fractions are equivalent, and ultimately answer the question: Is 72/96 equivalent to 12/16?

Detailed Explanation

Equivalent fractions represent the same portion of a whole, even though they may look different. For example, if you cut a pie into two equal pieces and take one piece, you have 1/2 of the pie. If you cut the same pie into four equal pieces and take two pieces, you still have the same amount of pie, but now it's represented as 2/4. Both 1/2 and 2/4 are equivalent fractions because they represent the same portion of the whole pie.

To determine if two fractions are equivalent, we need to simplify them to their lowest terms. This is done by dividing both the numerator (top number) and denominator (bottom number) by their greatest common factor (GCF). The GCF is the largest number that can divide both the numerator and denominator without leaving a remainder.

Step-by-Step or Concept Breakdown

Let's break down the process of determining if 72/96 and 12/16 are equivalent fractions:

1. Find the GCF of 72 and 96:

   • Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
   • Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
   • The GCF is 24.

2. Simplify 72/96 by dividing both the numerator and denominator by 24:

   • 72 ÷ 24 = 3
   • 96 ÷ 24 = 4
   • Simplified fraction: 3/4

3. Find the GCF of 12 and 16:

   • Factors of 12: 1, 2, 3, 4, 6, 12
   • Factors of 16: 1, 2, 4, 8, 16
   • The GCF is 4.

4. Simplify 12/16 by dividing both the numerator and denominator by 4:

   • 12 ÷ 4 = 3
   • 16 ÷ 4 = 4
   • Simplified fraction: 3/4

Real Examples

In real-life situations, equivalent fractions can be used to scale recipes, compare prices, or calculate discounts. For example, if a recipe calls for 3/4 cup of flour and you only have a 1/2 cup measuring tool, you can use the equivalent fraction 6/8 to measure the same amount of flour.

Scientific or Theoretical Perspective

From a mathematical standpoint, equivalent fractions are a fundamental concept in understanding the nature of rational numbers. Rational numbers are numbers that can be expressed as the ratio of two integers, such as 1/2, 3/4, or 7/8. Equivalent fractions help us understand that rational numbers can have multiple representations, all of which are equally valid.

Common Mistakes or Misunderstandings

One common mistake when working with equivalent fractions is failing to simplify the fractions to their lowest terms. This can lead to confusion and difficulty in comparing fractions. Another misunderstanding is the belief that equivalent fractions must have the same numerator and denominator, which is not the case.

FAQs

Q: Can equivalent fractions have different denominators? A: Yes, equivalent fractions can have different denominators. The key is that when simplified to their lowest terms, they represent the same portion of a whole.

Q: How do you find the GCF of two numbers? A: To find the GCF of two numbers, list out all the factors of both numbers and identify the largest factor they have in common.

Q: Are 4/6 and 6/9 equivalent fractions? A: No, 4/6 and 6/9 are not equivalent fractions. When simplified, 4/6 becomes 2/3, while 6/9 simplifies to 2/3 as well.

Q: Can equivalent fractions be used in calculations? A: Yes, equivalent fractions can be used in calculations. However, it's often easier to work with simplified fractions to avoid unnecessary complexity.

Conclusion

In conclusion, the fractions 72/96 and 12/16 are indeed equivalent, as they both simplify to 3/4 when reduced to their lowest terms. Understanding equivalent fractions is crucial for working with rational numbers and simplifying complex fractions. By following the step-by-step process outlined in this article, you can confidently determine whether any two fractions are equivalent and use this knowledge to solve a wide range of mathematical problems.