Finding Values Of A: When A+b=4 And A=240/a + 240/b

Hey guys! Let's dive into this interesting math problem where we need to figure out the possible values of A, given that a + b = 4 and A is defined as 240/a + 240/b, with both 'a' and 'b' being non-zero. This might sound a bit complex at first, but don't worry, we'll break it down step by step. Our goal here is to provide a clear and comprehensive explanation so that anyone, even those who aren't math whizzes, can understand the solution. So, grab your thinking caps, and let's get started!

Understanding the Problem

At the heart of this problem lies a relationship between two variables, a and b, which sum up to 4. We're also introduced to a new variable, A, which depends on the reciprocals of a and b. Specifically, A = 240/a + 240/b. The key challenge is to determine what values A can take, given the constraint that a and b are non-zero. This constraint is crucial because division by zero is undefined in mathematics. So, we need to keep this in mind as we explore possible solutions. We are essentially looking for a range or specific set of values that A can hold under these conditions. The fact that a and b are non-zero also hints that we might be dealing with inequalities or a range of values rather than a single, definitive answer. To solve this, we’ll need to manipulate the equations and possibly use some algebraic techniques to express A in terms of a single variable.

Setting up the Equations

Okay, let's get the basics down. We have two crucial pieces of information here. First, we know that:

a + b = 4

This is a simple linear equation that tells us how a and b are related. If we know the value of one, we can easily find the other. The second piece of information is the equation for A:

A = 240/a + 240/b

This equation defines A in terms of a and b. Our main goal now is to somehow combine these two equations to express A in a way that we can easily determine its possible values. A common strategy in these scenarios is to try and eliminate one of the variables (either a or b) so that we can work with a single equation. Since we have a + b = 4, we can easily express b in terms of a (or vice versa) and substitute it into the equation for A. This will allow us to have an equation where A is expressed solely in terms of a, which will make it easier to analyze.

Simplifying the Equation for A

Now comes the fun part – simplification! From our first equation, a + b = 4, we can express b in terms of a like this:

b = 4 - a

This is a simple rearrangement, but it's a powerful move. Now we can substitute this expression for b into the equation for A:

A = 240/a + 240/(4 - a)

This looks a bit messy, but we're making progress. We've eliminated b, and now A is expressed only in terms of a. To make this equation more manageable, let's find a common denominator and combine the fractions. This will help us see the relationship between A and a more clearly. Finding a common denominator is a standard algebraic technique when dealing with fractions, and it often simplifies complex expressions. Once we combine the fractions, we'll have a single fraction for A, which should make it easier to analyze and find possible values.

To combine the fractions, we need a common denominator, which in this case is a(4 - a). So, we rewrite the equation as:

A = [240(4 - a) + 240a] / [a(4 - a)]

Now, let's simplify the numerator:

A = (960 - 240a + 240a) / [a(4 - a)]

Notice that the -240a and +240a terms cancel out, which simplifies things nicely:

A = 960 / [a(4 - a)]

This is a much cleaner expression for A! We've successfully expressed A as a single fraction in terms of a. This form allows us to see directly how the value of A changes as a changes. The denominator, a(4 - a), is a quadratic expression, which means we can analyze its behavior to understand the possible values of A. This simplified equation is the key to unlocking the solution.

Analyzing the Denominator

The simplified equation we have is:

A = 960 / [a(4 - a)]

The value of A is heavily influenced by the denominator, which is the expression a(4 - a). To understand how A behaves, we need to analyze this denominator. Let's rewrite the denominator to make it clearer:

Denominator = a(4 - a) = 4a - a²

This is a quadratic expression, and it represents a parabola when graphed. To find the maximum or minimum value of this parabola, we can complete the square or find the vertex. Completing the square is a technique that rewrites a quadratic expression in a form that reveals its vertex. The vertex represents the point where the parabola reaches its maximum or minimum value. Knowing the vertex will help us understand the range of values the denominator can take, and consequently, the range of values for A.

Let's complete the square for the denominator:

4a - a² = -(a² - 4a)

To complete the square, we need to add and subtract (4/2)² = 4 inside the parentheses:

-(a² - 4a + 4 - 4) = -( (a - 2)² - 4 )

Distributing the negative sign, we get:

4 - (a - 2)²

Now we can see that the denominator is in the form 4 minus a squared term. This tells us a lot about its behavior. The term (a - 2)² is always non-negative (it's a square), so subtracting it from 4 means the maximum value of the denominator occurs when (a - 2)² is zero. This happens when a = 2. At a = 2, the denominator is:

4 - (2 - 2)² = 4

So, the maximum value of the denominator is 4. This is a crucial piece of information. It tells us that the denominator can never be greater than 4. Also, since a cannot be 0 or 4 (because b would be 4 or 0 respectively, and we can't divide by zero), the denominator will always be positive within the allowed range of a.

Determining Possible Values of A

We've made excellent progress! We know that:

A = 960 / [a(4 - a)]

And we've determined that the maximum value of the denominator a(4 - a) is 4. This happens when a = 2. Now we can use this information to find the minimum value of A. Since A is 960 divided by the denominator, A will be smallest when the denominator is largest. So, when the denominator is 4, we have:

A_min = 960 / 4 = 240

This is the minimum possible value of A. But what about the maximum value? To figure that out, we need to think about what happens to the denominator as a approaches its limits. Remember, a cannot be 0 or 4, because that would make either a or b zero, and we can't divide by zero. As a gets closer to 0 or 4, the denominator a(4 - a) gets closer to 0. This means that the fraction 960 / [a(4 - a)] gets larger and larger. In other words, as the denominator approaches 0, A approaches infinity.

So, A can take on any value greater than or equal to 240. There's no upper limit to the value of A, it can become infinitely large as a approaches 0 or 4. The fact that A approaches infinity as a nears 0 or 4 is a key insight. It demonstrates that A is not bounded above, meaning it can take on extremely large values. This understanding completes our analysis of the possible values of A.

Conclusion

So, guys, we've successfully navigated this math problem! We started with the equations a + b = 4 and A = 240/a + 240/b, and after some algebraic manipulation and analysis, we've found that the possible values of A are all numbers greater than or equal to 240. That's a pretty cool result! We used techniques like substitution, combining fractions, completing the square, and analyzing the behavior of a quadratic expression. These are all valuable tools in the mathematician's toolkit. Understanding how to apply these techniques can help you tackle a wide range of problems. Remember, math isn't just about memorizing formulas, it's about understanding relationships and using logic to solve puzzles. Hopefully, this explanation has made the process clear and accessible. Keep practicing, and you'll become a math whiz in no time!