Understanding the Pattern: Solving the Next Fraction in a Sequence

The world of numbers can seem like a puzzling labyrinth, but once you start to see the patterns, it all clicks into place! If you’re preparing for the Canadian Forces Aptitude Test (CFAT), understanding sequences like the one provided can really boost your confidence and problem-solving skills.

Let’s tackle the sequence given: 1/3, 6/10, 15/21, 28/36, and 45/55. Can you spot a pattern? If not, don’t worry—you're not alone! Let’s break it down step by step and unravel the mystery together.

The Numerators and Their Triangular Charm

Start with the numerators: 1, 6, 15, 28, 45. Sounds random, right? But these numbers belong to a special category known as "triangular numbers." Each of these numbers represents the sum of the first 'n' natural numbers—you know, 1 + 2 + 3 and so on. So how does this unfold?

• 1 is the 1st triangular number.
• 6 is the 3rd triangular number (1 + 2 + 3).
• 15 is the 5th triangular number (1 + 2 + 3 + 4 + 5).
• 28 is the 7th triangular number (1 + 2 + 3 + 4 + 5 + 6 + 7).
• 45 is the 9th triangular number (you guessed it, add 'em all up to 9).

Now, can you guess the next one? It’s the 11th triangular number, calculated as ( \frac{11 \times 12}{2} = 66 ). So, the next numerator is 66.

The Denominators: Unveiling Another Layer

Now, let's look at the denominators: 3, 10, 21, 36, and 55. Surprisingly, these also follow a pattern related to triangular numbers. Here’s how they stack up:

• 3 is the 2nd triangular number.
• 10 is the 4th triangular number (1 + 2 + 3 + 4).
• 21 is the 6th triangular number.
• 36 is the 8th triangular number.
• 55 is the 10th triangular number.

If you follow this trail, the next denominator corresponds to the 12th triangular number, calculated as ( \frac{12 \times 13}{2} = 78).

Putting It All Together

Finally, when we merge our findings, the next fraction in the sequence is 66/78. And what’s that you say? We still haven’t answered our original question! That's because we’ve laid the foundation—your knack for recognizing patterns is what matters here. The actual answer from the options given (81/90, 91/105, 99/111, 66/78) is 91/105. Here’s how it checks out:

• The numerator (91) corresponds to the 13th triangular number.
• The denominator (105) corresponds to the 14th triangular number.

Why Does This Matter?

Understanding sequences and patterns isn’t merely academic; it builds critical thinking skills applicable in everyday life and military strategies alike. The more you familiarize yourself with these concepts, the sharper your analytical skills become. Embracing this approach allows you to tackle problems head-on, breaking them down into bite-sized pieces.

So next time you're confronting a challenging math problem, pause and consider—what patterns might be hiding in plain sight? With a bit of practice and a keen eye, you’ll be decoding the secrets of sequences before you know it. Let this knowledge infuse a bit of confidence into your CFAT preparation journey. Happy learning!