Math Problem: Finding The Smaller Number | 1988 ÖYS

Hey guys! Let's break down this interesting math problem from the 1988 ÖYS (Öğrenci Seçme ve Yerleştirme Sınavı - Student Selection and Placement Examination). We'll take it step by step to make sure everyone understands how to get to the solution. Math problems can be intimidating, but with a clear approach, we can tackle anything!

Understanding the Problem

Okay, so the question states: If the sum of two real numbers is 242, and when the larger number is divided by the smaller, the quotient is 4 and the remainder is 22, what is the smaller number? To effectively solve this, the initial step involves translating the word problem into mathematical equations. This involves identifying the knowns and unknowns, and expressing the given relationships using algebraic symbols. Let's break down the information we have:

• We have two real numbers. Let's call the larger number 'x' and the smaller number 'y'.
• Their sum is 242. This translates to the equation: x + y = 242.
• When the larger number (x) is divided by the smaller number (y), the quotient is 4 and the remainder is 22. This translates to the equation: x = 4y + 22. Remember the division rule: Dividend = (Divisor * Quotient) + Remainder.

Now we have two equations with two unknowns. This is a classic system of equations that we can solve using either substitution or elimination. We'll use the substitution method in this case because we already have x expressed in terms of y in the second equation. This initial setup is crucial. Without a correct understanding of the problem and its translation into mathematical form, any further calculation would be futile. Therefore, spending time at this stage to ensure clarity is an investment in successfully solving the problem. Keep reading, we're about to dive into the solution!

Solving the Equations

Now that we have our equations, let’s solve them to find the value of 'y', which represents the smaller number. Remember our equations are:

1. x + y = 242
2. x = 4y + 22

We'll use the substitution method. Since we already have 'x' isolated in the second equation, we can substitute the expression 4y + 22 for 'x' in the first equation. This gives us:

(4y + 22) + y = 242

Now we have a single equation with just one variable, 'y'. Let's simplify and solve for 'y'. First, combine like terms:

5y + 22 = 242

Next, subtract 22 from both sides of the equation to isolate the term with 'y':

5y = 242 - 22 5y = 220

Finally, divide both sides by 5 to solve for 'y':

y = 220 / 5 y = 44

So, we've found that the smaller number, 'y', is 44. That's one step closer to the final answer! But it’s always a good idea to double-check our work. Let's plug this value back into our original equations to see if it holds true. This validation step is vital in problem-solving, as it helps identify any potential errors made during the calculation process. By verifying the solution, we can be confident in our answer and proceed with the assurance that the problem has been solved correctly. Up next, we'll confirm our solution and wrap things up!

Verifying the Solution

We've found that the smaller number, y, is 44. To make sure this is correct, let's substitute this value back into our original equations and see if they hold true. This step is super important for preventing errors!

First, let's find the value of x using the first equation, x + y = 242:

x + 44 = 242

Subtract 44 from both sides:

x = 242 - 44 x = 198

So, the larger number, x, is 198. Now, let's check if these values satisfy our second equation, x = 4y + 22:

198 = (4 * 44) + 22 198 = 176 + 22 198 = 198

The equation holds true! This confirms that our values for x and y are correct. Therefore, the smaller number is indeed 44.

It's great to verify our solution. It not only boosts our confidence in the answer but also reinforces the understanding of the mathematical principles involved. This is particularly beneficial in exams where accuracy is paramount. Now that we've verified, let's conclude and highlight the key takeaways from this problem-solving exercise.

Conclusion and Key Takeaways

Alright, guys! We successfully solved this math problem from the 1988 ÖYS. The smaller number is 44 (Option D).

Here are the key takeaways from this problem:

1. Translate word problems into equations: This is the most crucial step. Identifying the unknowns and expressing the relationships mathematically allows us to use algebraic techniques to solve the problem. Pay close attention to keywords like "sum," "quotient," and "remainder" as they indicate specific mathematical operations.
2. System of equations: This problem involved a system of two equations with two unknowns. Remember the methods for solving these systems, such as substitution and elimination. Choosing the most efficient method can save you time and effort.
3. Verification is key: Always, always, always verify your solution by plugging the values back into the original equations. This helps catch any arithmetic errors and ensures the accuracy of your answer. It can be a lifesaver in exams!
4. Understanding division with remainders: The equation x = 4y + 22 represents the division process accurately. Make sure you understand the relationship between the dividend, divisor, quotient, and remainder. This is a fundamental concept in arithmetic.

By mastering these concepts and practicing regularly, you'll be well-equipped to tackle similar math problems with confidence. Keep practicing, and remember that every problem you solve makes you a better problem-solver!

Hope this breakdown was helpful! Let me know if you have any questions or want to try another problem. You've got this!