Solving Ratio Problems Find Value Of (4a^2 + 5b^2) / (3a + 4b) Given A B = 7 8
In the realm of mathematics, ratio and proportion problems often present intriguing challenges that require a blend of algebraic manipulation and conceptual understanding. One such problem involves finding the value of a complex expression given a specific ratio between two variables. In this article, we will delve into a problem where we are given the ratio a : b = 7 : 8 and tasked with determining the value of the expression (4a² + 5b²) / (3a + 4b). This exploration will not only provide a step-by-step solution but also highlight the underlying principles of ratios and how they can be effectively applied to solve algebraic expressions.
At its core, a ratio is a way of comparing two quantities. It expresses how much of one thing there is compared to another. In our case, the ratio a : b = 7 : 8 signifies that for every 7 units of a, there are 8 units of b. This relationship is crucial for solving the problem, as it allows us to express one variable in terms of the other. Understanding proportions is equally important. A proportion is an equation that states that two ratios are equal. This concept helps in scaling quantities while maintaining their relative sizes. For instance, if we double the value of a, we must also proportionally adjust the value of b to maintain the given ratio. The beauty of ratios and proportions lies in their ability to simplify complex problems by providing a framework for relative comparison.
In the context of our problem, the ratio a : b = 7 : 8 implies a direct proportionality between a and b. This means that a and b can be represented as multiples of a common factor. This representation is a key step in simplifying the given expression and arriving at a numerical solution. By grasping the fundamental principles of ratios and proportions, we can confidently tackle problems that might initially appear daunting. The ability to manipulate ratios, set up proportions, and express variables in terms of each other forms the bedrock of solving a wide range of mathematical problems, making this concept invaluable in both academic and practical settings.
Before diving into the solution, let's restate the problem clearly. We are given that the ratio of a to b is 7 to 8, written as a : b = 7 : 8. Our objective is to find the value of the expression (4a² + 5b²) / (3a + 4b). This expression involves both linear and quadratic terms of a and b, which adds a layer of complexity. The initial approach to this problem involves recognizing the fundamental relationship between a and b dictated by the given ratio. Since a : b = 7 : 8, we can express a and b in terms of a common variable. This is a crucial step because it allows us to reduce the number of variables in the expression and simplify the calculations.
One common technique is to introduce a variable, say k, such that a = 7k and b = 8k. This substitution maintains the ratio 7 : 8 while allowing us to express both a and b in terms of a single variable. By substituting these expressions into the given expression, we can transform the problem from one involving two variables to one involving a single variable. This significantly simplifies the algebraic manipulations required to find the solution. The success of this approach hinges on the understanding that the ratio provides a fixed relationship between a and b, and expressing them in terms of a common variable preserves this relationship while making the problem more tractable. This technique is widely applicable in ratio and proportion problems, making it an essential tool in mathematical problem-solving.
Now, let's embark on the step-by-step solution to the problem. As we established earlier, given the ratio a : b = 7 : 8, we introduce a variable k such that a = 7k and b = 8k. This substitution is the cornerstone of our solution, allowing us to work with a single variable instead of two.
The next step is to substitute these expressions for a and b into the expression we want to evaluate: (4a² + 5b²) / (3a + 4b). Replacing a with 7k and b with 8k, we get:
(4(7k)² + 5(8k)²) / (3(7k) + 4(8k)).
Now, we simplify the expression by performing the operations within the parentheses. Squaring 7k and 8k, we have:
(4(49k²) + 5(64k²)) / (21k + 32k)
Next, we multiply the constants:
(196k² + 320k²) / (21k + 32k)
Now, we add the terms in the numerator and the denominator:
(516k²) / (53k)
At this point, we can simplify the expression further by dividing both the numerator and the denominator by k. This is a crucial step as it eliminates the variable k from the expression, leading us closer to a numerical solution:
516k / 53
Finally, we simplify the fraction by dividing 516 by 53:
516 / 53 = (4129) / 53*. Unfortunately, 516 is not evenly divisible by 53. I apologize for that error. Factoring 516 yields 2 * 2 * 3 * 43, which does not share factors with 53, so the result cannot be reduced further. Therefore, the expression simplifies to 516 / 53.
Let's break down the detailed calculation and simplification to ensure clarity and accuracy. We started with the expression:
(4(7k)² + 5(8k)²) / (3(7k) + 4(8k))
First, we square the terms inside the parentheses:
(4(49k²) + 5(64k²)) / (21k + 32k)
Next, we multiply the constants:
(196k² + 320k²) / (21k + 32k)
Now, we add the terms in the numerator and the denominator:
(516k²) / (53k)
This step combines the like terms, making the expression more concise. The numerator combines the k² terms, and the denominator combines the k terms. The next crucial step is to simplify the expression by canceling out the common factor k from both the numerator and the denominator. This simplification is valid as long as k is not zero. In the context of the given ratio, k cannot be zero because that would imply that both a and b are zero, which contradicts the ratio a : b = 7 : 8. Therefore, we can safely divide both the numerator and the denominator by k:
(516k² / k) / (53k / k) = 516k / 53
This simplification significantly reduces the complexity of the expression, leaving us with a linear term in the numerator and a constant term in the denominator. Finally, we observe that we can divide 516 by 53. The result of this division gives us the numerical value of the expression:
516 / 53 = 9.7358
However, notice that k should be canceled out completely. Let us verify the result. (516k²) / (53k) = (516/53) * (k²/k) = (516/53) * k We made an error by not canceling k from numerator and denominator. Rewriting it the correct way will get: (516k²) / (53k) = 516k²/53k = (516/53)k Therefore, (516k)/53 is incorrect. The final answer should have k involved. If we cancel k from the numerator and the denominator, we get: 516k²/53k = 516k/53 Since k is an arbitrary constant, we can say that the value cannot be determined.
In conclusion, by meticulously following the steps of substituting a = 7k and b = 8k into the given expression, simplifying the resulting terms, and canceling out the common factors, we arrive at the simplified expression 516k / 53. However, here’s where we encounter a critical observation: the value of the expression still depends on the variable k. Since k is an arbitrary constant that defines the scale of the ratio between a and b, its presence in the final expression means that the value of the expression (4a² + 5b²) / (3a + 4b) is not uniquely determined by the given ratio a : b = 7 : 8.
This realization leads us to the final answer. Because the value of the expression depends on k, and k can take on any non-zero value, the expression does not have a unique numerical value. Therefore, the correct answer is (D) Cannot be determined. This problem underscores the importance of careful simplification and the recognition of the limitations imposed by variables that do not cancel out during the simplification process. It also reinforces the understanding that ratios define a relationship between quantities but do not necessarily specify their absolute values.
