Solving For A * B When A/b = 5/4 A Math Guide

Hey guys! Today, we're diving into a cool math problem that involves ratios and finding the value of a product. We've got this equation: a/b = 5/4, and our mission is to figure out what a * b equals. It might seem tricky at first, but don’t worry, we’ll break it down step by step so it’s super clear. Let’s jump right in and solve this together!

Understanding the Problem

Before we dive into solving, let's make sure we fully grasp the problem. We're given a fraction, a/b = 5/4. This tells us the ratio between two numbers, a and b. The key here is that b cannot be zero (b ≠ 0) because division by zero is undefined in mathematics. Our goal is to find the value of the product a * b. This isn't as straightforward as just plugging in numbers; we need to manipulate the given ratio to find our answer. Think of it like a puzzle – we have the pieces, now we just need to fit them together correctly. To kick things off, let's explore what the ratio a/b = 5/4 actually means in simpler terms. It means that for every 5 units of a, there are 4 units of b. This relationship is crucial for unlocking the solution. So, stick with me, and let’s get started!

Breaking Down the Ratio

Alright, let’s really break down this ratio thing. When we see a/b = 5/4, it's like saying that a and b are in proportion to each other. Imagine you're mixing a drink – for every 5 parts of one ingredient (a), you need 4 parts of another (b). The fraction 5/4 is the constant of proportionality. This means we can express a and b in terms of a common variable. Let’s use k as our variable. We can say that a = 5k and b = 4k. This is a super handy trick because it allows us to work with actual values instead of just ratios. Now, why is this important? Well, remember, our mission is to find the value of a * b. By expressing a and b in terms of k, we can substitute these values into our target expression and see what happens. This is where the magic starts to happen! So, with a = 5k and b = 4k in our toolkit, let’s move on to the next step and see how we can use these values to crack the problem.

Expressing a and b with a Common Variable

So, as we discussed, expressing a and b with a common variable is a game-changer. We’ve established that a = 5k and b = 4k. This is super crucial because it transforms the problem from abstract ratios to concrete expressions. Think of k as a scaling factor. It helps us understand how a and b relate to each other in a tangible way. For instance, if k were 1, then a would be 5 and b would be 4. If k were 2, a would be 10 and b would be 8, and so on. The ratio 5/4 remains constant, but the actual values of a and b change. Now, here’s where the fun begins. We need to find a * b, and we now have expressions for both a and b in terms of k. This means we can substitute these expressions into the product a * b. Get ready, because this is where we’ll start to see the solution take shape. Let’s head over to the next section where we’ll actually perform this substitution and simplify the expression.

Calculating a * b

Alright, let's get down to the nitty-gritty and calculate a * b. We know that a = 5k and b = 4k. So, to find a * b, we simply multiply these two expressions together. This gives us:

a * b = (5k) * (4k)

Now, we just need to simplify this. Remember your basic algebra – when you multiply terms with the same variable, you multiply the coefficients and add the exponents. In this case, we multiply 5 and 4 to get 20, and we multiply k by k, which gives us k². So, our expression simplifies to:

a * b = 20k²

This is a significant step! We now have an expression for a * b in terms of k. But wait, we're not quite done yet. The problem doesn’t give us a specific value for k, and the answer choices are numerical values. This means we need to think a little more strategically. What can we infer from this expression? Well, 20k² tells us that the product a * b depends on the value of k. So, how do we figure out which of the given options is the correct one? Let's dive into some clever reasoning in the next section!

Substituting the Expressions

Okay, let's talk about substituting those expressions. We've landed at the equation a * b = 20k². This is a big step forward, but remember, the question gives us specific numerical answer choices. We need to figure out how 20k² relates to those options. Now, here's the thing: the value of k isn't explicitly given, but we know a/b = 5/4. The beauty of this equation is that it describes a ratio, not specific values. This means there are infinitely many pairs of a and b that satisfy this ratio. For example, if k = 1, then a = 5 and b = 4, and a * b = 20. But if k = 2, then a = 10 and b = 8, and a * b = 160. You see, the value of a * b changes depending on k. This is crucial! Since the problem doesn't give us a fixed value for k, and the possible answers are fixed numbers, it indicates that we need to analyze the general form of the solution rather than finding a specific numerical answer. We've got an expression for a * b, and now we need to think about what that expression tells us in the context of the given choices. Let’s explore this in more detail in the next part.

Simplifying the Product

Let’s simplify this product even further. We've established that a * b = 20k². Now, we need to think about what this expression really means. The k² term is key here. Since k is any non-zero number (because b cannot be zero), k² will always be a positive number (or zero, but if k were zero, then a and b would both be zero, which contradicts the ratio a/b = 5/4). So, we know that a * b will be 20 times a positive number. This is a pretty important piece of the puzzle! Now, let’s think about the answer choices we were given: a) 20, b) 9, c) 5/4, and d) 4/5. We need to see which of these options could possibly fit our expression 20k². Notice that 20 is one of the choices. What happens if k = 1? Well, then a * b = 20(1)² = 20. Aha! This tells us that 20 is a possible value for a * b. But are there any other possibilities? The other choices are 9, 5/4, and 4/5. These are all numbers less than 20. Can 20k² ever be less than 20? Remember, k² is always positive (or zero), so if k² is less than 1, then 20k² would indeed be less than 20. This is where it gets interesting. We need to carefully consider if any of these other options could be valid. Let's dig deeper in the next section.

Evaluating the Answer Choices

Okay, let's put on our detective hats and evaluate those answer choices! We've deduced that a * b = 20k², and we know that 20 is a possible answer when k = 1. But what about the other choices: 9, 5/4, and 4/5? Could any of these be valid? Let’s think critically. We know a * b is 20 times k². So, for a * b to equal 9, we would need 20k² = 9. If we solve for k², we get k² = 9/20. This is a perfectly valid value for k², so 9 is a potential answer. This is an important realization! For a * b to equal 5/4, we would need 20k² = 5/4. Solving for k², we get k² = (5/4) / 20 = 1/16. Again, this is a valid value for k², so 5/4 is also a potential answer. Lastly, for a * b to equal 4/5, we need 20k² = 4/5. Solving for k², we get k² = (4/5) / 20 = 1/25. This too is a valid value for k². So, here’s the twist: all the answer choices could potentially be correct, depending on the value of k! However, math problems like these usually have one most correct answer, and we need to look for a subtle clue. The fact that 20 is the simplest solution (when k = 1) and is a whole number might be a hint. But let’s dig a bit deeper and really scrutinize the question and our approach.

Analyzing the Options

Let's really analyze these options and make sure we haven’t missed anything. We’ve shown that all the choices—20, 9, 5/4, and 4/5—could be possible values for a * b, depending on the value of k. This is a bit unusual for a multiple-choice question, right? Usually, there's one clear-cut answer. So, what’s going on here? We need to step back and think about what the question is really asking. The question states,