Solve Proportions: Find A + B + C + D

by Natalie Brooks 38 views

Hey guys! Ever stumbled upon a math problem that looks like a tangled web of fractions and variables? Well, today we're diving headfirst into one of those! We've got a proportional series that seems a bit daunting at first glance, but trust me, we'll break it down step-by-step and make it crystal clear. Our mission? To find the sum of a, b, c, and d in the given series. So, grab your thinking caps, and let's get started!

Decoding the Proportional Series

Proportional series can sometimes feel like deciphering a secret code, but with the right approach, we can crack it. Let's start by understanding what we're dealing with. We're given the series: 9/a = b/35 = 18/c = d/20. And to add a little extra twist, we also know that b - d = 9. Our ultimate goal is to find the value of a + b + c + d. Sounds like a fun puzzle, right?

Laying the Foundation: Understanding Proportions

Before we jump into solving, let's quickly revisit what proportions are all about. A proportion is simply a statement that two ratios are equal. In our case, we have a series of equal ratios. The key here is that if two ratios are equal, we can set them equal to each other and use cross-multiplication to solve for unknown variables. This is the fundamental principle we'll use to unravel our series.

Cracking the Code: Step-by-Step Solution

Now, let's get our hands dirty and start solving! Here’s how we can approach this problem:

  1. Isolate Pairs of Ratios: The beauty of a proportional series is that we can pick any two ratios and equate them. This gives us an equation we can work with. For example, we can equate 9/a = b/35 or 18/c = d/20. We'll strategically choose pairs that help us eliminate variables or introduce known relationships.
  2. Introduce a Constant (k): A clever trick to simplify things is to recognize that since all these ratios are equal, we can set them equal to a constant, let's call it 'k'. This means: 9/a = b/35 = 18/c = d/20 = k. This single constant now ties all our variables together, making it easier to express each variable in terms of k.
  3. Express Variables in Terms of k: Using our constant k, we can rewrite each fraction as an equation: 9/a = k implies a = 9/k, b/35 = k implies b = 35k, 18/c = k implies c = 18/k, and d/20 = k implies d = 20k. See how we've expressed a, b, c, and d, all in terms of this one little constant?
  4. Utilize the Given Condition (b - d = 9): Remember that extra tidbit of information we were given? b - d = 9. This is crucial! We can now substitute our expressions for b and d in terms of k into this equation: 35k - 20k = 9. This simplifies to 15k = 9, which means k = 9/15 = 3/5. We've just found the value of k!
  5. Calculate the Values of a, b, c, and d: Now that we know k, we can easily find the values of our variables. a = 9/k = 9/(3/5) = 15, b = 35k = 35*(3/5) = 21, c = 18/k = 18/(3/5) = 30, and d = 20k = 20*(3/5) = 12. We've successfully found the values of a, b, c, and d!

The Grand Finale: Finding a + b + c + d

We're almost there! Our final step is to add up the values we just calculated: a + b + c + d = 15 + 21 + 30 + 12. Adding these up, we get a + b + c + d = 78. And there you have it! The sum of a, b, c, and d is 78.

Why This Matters: Real-World Applications of Proportions

You might be wondering,