Compare Fractions: 7/12 Vs 19/36 - Which Is Bigger?

by Omar Yusuf 52 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of fractions and tackling a common question: Which symbol correctly compares the fractions 7/12 and 19/36? This is a fundamental concept in mathematics, and mastering it will help you in various areas, from basic arithmetic to more complex algebraic equations. So, let's break it down step-by-step and make sure you've got a solid understanding.

Understanding Fractions: A Quick Refresher

Before we jump into the comparison, let's quickly refresh our understanding of fractions. A fraction represents a part of a whole. It consists of two main parts:

  • Numerator: The number on the top, which indicates how many parts we have.
  • Denominator: The number on the bottom, which indicates the total number of equal parts the whole is divided into.

For example, in the fraction 7/12, 7 is the numerator, and 12 is the denominator. This means we have 7 parts out of a total of 12 equal parts.

The Challenge: Comparing 7/12 and 19/36

Now, let's get to the heart of the problem. We need to determine whether 7/12 is less than (<), greater than (>), or equal to (=) 19/36. To do this effectively, we need a common ground for comparison. And that common ground, my friends, is a common denominator.

Finding a Common Denominator: The Key to Comparison

The key to accurately comparing fractions lies in having a common denominator. This means we need to rewrite the fractions so that they both have the same denominator. Why? Because when fractions have the same denominator, we can directly compare their numerators to see which represents a larger portion of the whole.

So, how do we find this magical common denominator? There are a couple of ways, but the most common is to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators divide into evenly.

In our case, the denominators are 12 and 36. Let's list out the multiples of each:

  • Multiples of 12: 12, 24, 36, 48, 60...
  • Multiples of 36: 36, 72, 108...

Aha! We see that 36 is the smallest number that appears in both lists. Therefore, the LCM of 12 and 36 is 36. This means our common denominator will be 36.

Converting Fractions to a Common Denominator: Making the Transformation

Now that we have our common denominator, we need to convert both fractions to have a denominator of 36. Let's start with 7/12.

To convert 7/12 to an equivalent fraction with a denominator of 36, we need to figure out what number we can multiply 12 by to get 36. The answer is 3 (because 12 x 3 = 36). But here's the crucial part: we must multiply both the numerator and the denominator by the same number to keep the fraction equivalent. Think of it like scaling a recipe – if you double the ingredients, you need to double everything to maintain the proportions.

So, we multiply both the numerator (7) and the denominator (12) of 7/12 by 3:

(7 x 3) / (12 x 3) = 21/36

Great! Now we have 7/12 expressed as 21/36.

Next, let's look at 19/36. Notice that it already has a denominator of 36! This makes our job easy. We don't need to change it.

The Grand Comparison: Numerators Tell the Tale

Alright, we've done the prep work. Now comes the exciting part: the comparison! We have:

  • 7/12 converted to 21/36
  • 19/36 (already in the correct form)

Now that both fractions have the same denominator, we can directly compare their numerators. We're asking: Is 21 greater than, less than, or equal to 19?

The answer, of course, is that 21 is greater than 19.

The Verdict: Choosing the Right Symbol

Since 21/36 is greater than 19/36, that means the original fraction 7/12 is also greater than 19/36. So, the correct symbol to use is > (greater than).

Therefore, the correct comparison is:

7/12 > 19/36

Why This Matters: The Importance of Fraction Comparison

You might be thinking,