Solving Ratios: Find A, B, C | Ab/10 = Bc/35 = Ac/14
Hey guys! Today, we're diving into a fun math problem that involves ratios and a little bit of algebra. We're going to figure out how to find the values of a, b, and c when given the equations ab/10 = bc/35 = ac/14 and a + b + c = 126. Sounds like a puzzle, right? Let's break it down step by step and make it super easy to understand. So grab your pencils, and let's get started!
Understanding the Problem
Okay, let’s first understand exactly what we're dealing with. The core of our problem lies in these two key equations:
- ab/10 = bc/35 = ac/14: This tells us that the ratios of the products of these variables are equal. It's like saying we have three fractions that all simplify to the same value. This is a crucial piece of information because it links a, b, and c together.
- a + b + c = 126: This is a straightforward equation that tells us the sum of our three variables. It's the constraint we have to work within, making sure our final answers add up correctly. Think of it as the total pie, and we need to figure out how to slice it into a, b, and c.
To solve this, we need to find a way to relate these equations. We can’t just solve them independently; they are interconnected. The ratios give us a proportional relationship, and the sum gives us a total. Our goal is to use these relationships to isolate our variables and find their specific values. This is where the magic of algebra comes in, allowing us to manipulate these equations and uncover the hidden numbers. We'll use the proportional relationships to express some variables in terms of others, and then use the sum equation to tie everything together. It's like building a bridge, each step relying on the one before it. So, let’s roll up our sleeves and start constructing that bridge!
Setting Up Proportions
The key to cracking this problem is recognizing that equal ratios mean proportional relationships. So, let's take those ratios and set up some clear proportions. This is where we start turning the complex problem into manageable pieces. We'll use these proportions to express some of the variables in terms of the others, which will simplify our calculations later on.
From the given equation ab/10 = bc/35 = ac/14, we can derive two separate proportions:
- ab/10 = bc/35
- bc/35 = ac/14
These two proportions are like smaller puzzles within the bigger one. Each one gives us a direct relationship between two variables, allowing us to express one in terms of the other. This is a fundamental technique in algebra – simplifying complex relationships by breaking them down into smaller, more manageable parts. It's like taking a giant jigsaw puzzle and sorting the pieces by color and shape before trying to assemble the whole picture.
Now, let's dive into each of these proportions and see how we can simplify them further. We'll manipulate these equations using basic algebraic principles to isolate variables. This might involve multiplying or dividing both sides by the same value, or even canceling out common factors. The goal is to get each proportion into a form where we can easily see the relationship between the variables. Think of it as fine-tuning a radio signal to get a clear and strong connection. Once we have these clear relationships, we'll be much closer to solving for a, b, and c. So, let’s get to work on these proportions!
Solving the Proportions
Alright, let’s get our hands dirty and start simplifying these proportions. This is where we'll put our algebraic skills to the test and manipulate the equations to reveal the relationships between a, b, and c. We'll take each proportion one at a time, applying the rules of algebra to isolate variables and make the equations easier to work with. It’s like being a detective, following the clues to uncover the hidden values.
First Proportion: ab/10 = bc/35
To simplify this, we can start by cross-multiplying. This is a common technique when dealing with proportions and it helps eliminate the fractions. It's like turning a complex fraction problem into a simpler multiplication problem.
So, we get:
35 * ab = 10 * bc
35ab = 10bc
Now, we want to isolate the variables. We can divide both sides by 'b' (assuming b is not zero) to simplify further:
35a = 10c
Next, we can simplify this equation by dividing both sides by their greatest common divisor, which is 5:
7a = 2c
This gives us a crucial relationship: c = (7/2)a. We've successfully expressed 'c' in terms of 'a'! This is a major step forward because it means we can substitute this value of 'c' into other equations, reducing the number of unknowns.
Second Proportion: bc/35 = ac/14
Let's tackle the second proportion in the same way. Again, we start by cross-multiplying to get rid of the fractions:
14 * bc = 35 * ac
14bc = 35ac
Now, divide both sides by 'c' (assuming c is not zero):
14b = 35a
Simplify by dividing both sides by their greatest common divisor, which is 7:
2b = 5a
This gives us another valuable relationship: b = (5/2)a. Now we've expressed 'b' in terms of 'a' as well! This is fantastic news because we now have both 'b' and 'c' in terms of 'a'. This means we can substitute these values into the equation a + b + c = 126 and solve for 'a'. It’s like fitting the final pieces of a puzzle together, bringing us closer to the solution. So, let's move on to the next step and use these relationships to find the value of 'a'.
Solving for 'a'
Okay, guys, we've done some great work so far! We've managed to express both 'b' and 'c' in terms of 'a'. This is a huge step because it means we can now use the equation a + b + c = 126 to solve for 'a'. It's like we've narrowed down our search area and are now homing in on the exact location of our treasure.
We know:
- b = (5/2)a
- c = (7/2)a
And we also know:
- a + b + c = 126
Now, let's substitute the values of 'b' and 'c' into the equation:
a + (5/2)a + (7/2)a = 126
To make this equation easier to work with, let's get rid of the fractions. We can do this by multiplying the entire equation by 2:
2 * [a + (5/2)a + (7/2)a] = 2 * 126
2a + 5a + 7a = 252
Now, we can combine the terms on the left side:
14a = 252
To isolate 'a', we divide both sides by 14:
a = 252 / 14
a = 18
Fantastic! We've found the value of 'a'! This is a major breakthrough, as it unlocks the door to finding the values of 'b' and 'c' as well. We're now on the home stretch. We've built the foundation, and now we can start filling in the remaining pieces. So, let's use this value of 'a' to find 'b' and 'c'.
Finding 'b' and 'c'
Alright! Now that we've successfully found the value of 'a', which is 18, the next step is to use this information to determine the values of 'b' and 'c'. Remember those relationships we established earlier? They're going to be super useful now. This is where all our hard work pays off, and we see the solution coming together.
We know:
- b = (5/2)a
- c = (7/2)a
And we know:
- a = 18
Let's start by finding 'b'. We substitute the value of 'a' into the equation:
b = (5/2) * 18
b = (5 * 18) / 2
b = 90 / 2
b = 45
Great! We've found 'b' as well. Now, let's do the same for 'c':
c = (7/2) * 18
c = (7 * 18) / 2
c = 126 / 2
c = 63
Excellent! We've found the values of all three variables: a, b, and c. This is like reaching the summit of a mountain after a long climb – we've achieved our goal! We now have all the pieces of the puzzle, and we know exactly how they fit together. But before we celebrate too much, let’s take a moment to verify our solution and make sure everything checks out.
Verifying the Solution
Okay, we've found a = 18, b = 45, and c = 63. But before we shout from the rooftops, it's always a smart idea to double-check our work. This is like proofreading an essay or testing a new software – it's a crucial step to ensure accuracy. We need to make sure these values satisfy both of our original equations. Let's plug them in and see!
First Equation: a + b + c = 126
Let's substitute our values:
18 + 45 + 63 = 126
Adding them up:
126 = 126
Perfect! The first equation checks out. This gives us confidence that we're on the right track.
Second Equation: ab/10 = bc/35 = ac/14
This one is a bit more complex, as it involves three ratios. Let's calculate each ratio separately and see if they are equal.
- ab/10 = (18 * 45) / 10 = 810 / 10 = 81
- bc/35 = (45 * 63) / 35 = 2835 / 35 = 81
- ac/14 = (18 * 63) / 14 = 1134 / 14 = 81
Wow! All three ratios are equal to 81. This is the ultimate confirmation that our solution is correct. We've not only found the values of a, b, and c, but we've also verified that they perfectly fit the given conditions. It's like putting the final stamp of approval on a job well done.
Conclusion
Alright, guys! We did it! We successfully solved for a, b, and c given the equations ab/10 = bc/35 = ac/14 and a + b + c = 126. We found that a = 18, b = 45, and c = 63. This problem might have seemed daunting at first, but we broke it down step by step, using proportions and algebraic manipulation to reach our solution. Remember, the key is to understand the relationships between the variables and use them to simplify the equations. Keep practicing, and you'll become a master problem-solver in no time! Great job, everyone!