LCM And HCF Problem: Find The Other Number
Let's break down this interesting math problem step by step, guys! We're dealing with the concepts of Least Common Multiple (LCM) and Highest Common Factor (HCF), and how they relate to two numbers. The goal is to find the second number when we know the relationship between the LCM and HCF, their sum, and one of the numbers. Buckle up, because we're about to dive into the solution!
Understanding the Problem
First, let's make sure we understand what the problem is asking. We're given that the LCM of two numbers is 20 times their HCF. We also know that the sum of the LCM and HCF is 2520. One of the numbers is 480, and we need to find the other number. Sounds like a puzzle, right? But don't worry, we'll solve it together!
Keywords: LCM (Least Common Multiple), HCF (Highest Common Factor), number theory, problem-solving, arithmetic.
To approach this problem effectively, we need to leverage the fundamental relationship between LCM, HCF, and the two numbers in question. Specifically, we know that the product of the LCM and HCF of two numbers is equal to the product of the numbers themselves. This relationship is crucial because it allows us to connect the given information (the relationship and sum of LCM and HCF, and one of the numbers) to what we need to find (the other number). Let's denote the two numbers as 'a' and 'b', where 'a' is given as 480, and 'b' is the number we're trying to find. If we let 'H' be the HCF and 'L' be the LCM, then we can express the core relationship as L * H = a * b. This equation is the key to unlocking the solution, as it bridges the gap between the known values and the unknown number we seek. Furthermore, understanding the properties of LCM and HCF can guide us in making educated guesses or approximations if the algebraic approach becomes cumbersome. For example, we know that the HCF must be a factor of both numbers, and the LCM must be a multiple of both numbers. These properties can provide additional context and help verify the reasonableness of our final answer. In summary, by carefully dissecting the problem statement and understanding the fundamental relationships between LCM, HCF, and the numbers involved, we can develop a strategic approach to find the elusive second number.
Setting Up the Equations
Let's use some algebra to represent the information given in the problem. Let the HCF be 'H' and the LCM be 'L'. According to the problem:
- L = 20H (The LCM is 20 times the HCF)
- L + H = 2520 (The sum of the LCM and HCF is 2520)
Now we can substitute the first equation into the second equation:
20H + H = 2520 21H = 2520 H = 2520 / 21 H = 120
So, the HCF (H) is 120. Now we can find the LCM (L):
L = 20 * H L = 20 * 120 L = 2400
Keywords: algebraic equations, substitution method, solving for variables, HCF calculation, LCM calculation.
Having established the HCF and LCM, we proceed to utilize the fundamental property that links these values to the two numbers in question. This property, expressed as L * H = a * b, where 'a' and 'b' are the two numbers, allows us to solve for the unknown number, 'b'. By substituting the known values of L, H, and 'a' (one of the numbers) into this equation, we create a direct pathway to find 'b'. This step is crucial as it leverages the interconnectedness of LCM, HCF, and the numbers themselves, demonstrating a deep understanding of number theory principles. Moreover, this approach highlights the power of algebraic manipulation in solving mathematical problems, as it transforms the initial problem statement into a solvable equation. It's not just about plugging in numbers; it's about recognizing the underlying relationships and using them to our advantage. As we move forward with the calculation, it's important to keep in mind the context of the problem. We're not just solving an abstract equation; we're finding a number that satisfies specific conditions related to LCM, HCF, and another known number. This context helps us interpret the result and ensure that it makes sense within the framework of the original problem statement. In essence, by carefully applying the algebraic equation and keeping the context in mind, we can confidently arrive at the solution for the unknown number, 'b'.
Finding the Other Number
We know that the product of the two numbers is equal to the product of their LCM and HCF.
So, a * b = L * H We know a = 480, L = 2400, and H = 120. Let's plug these values in:
480 * b = 2400 * 120 b = (2400 * 120) / 480 b = 288000 / 480 b = 600
Therefore, the other number is 600.
Keywords: product of numbers, LCM and HCF relationship, solving for unknown, arithmetic calculation, final answer.
The concluding step in this problem-solving journey involves interpreting the result we've obtained in the context of the original problem. After performing the calculations, we've arrived at the conclusion that the other number is 600. Now, it's crucial to take a moment and reflect on whether this answer makes sense and aligns with the information we were initially given. For instance, we can consider whether 600 and 480 could plausibly have an HCF of 120 and an LCM of 2400. This involves checking if 120 is indeed the greatest common factor of both numbers and if 2400 is the smallest multiple they both share. Additionally, we can think about the ratio between the two numbers and see if it corresponds with the ratio between their LCM and HCF. If everything checks out and the answer seems reasonable, we can confidently conclude that 600 is indeed the correct solution to the problem. This final step is not just about confirming the numerical accuracy of our calculations; it's about ensuring that we've truly understood the problem and arrived at a solution that is both mathematically sound and logically consistent with the given information. By taking the time to reflect on our answer and validate it against the context of the problem, we can strengthen our understanding of the underlying concepts and develop our problem-solving skills further.
Answer
The other number is 600. So the correct answer is:
d) 600
Conclusion
Great job, guys! We successfully solved the problem by understanding the relationship between LCM, HCF, and the two numbers. We set up equations, solved for the HCF and LCM, and then used that information to find the other number. Remember, practice makes perfect, so keep honing your math skills!
Keywords: problem-solving recap, LCM and HCF review, algebraic application, arithmetic practice, final solution.