Find Digit X: Base 10 To Base 2 Conversion

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Find Digit X: Base 10 to Base 2 Conversion

Hey guys! Today, we're diving into a cool math problem where we need to find a mystery digit, x, in the decimal system. We'll be doing some base conversions between base 10 (our regular number system) and base 2 (binary, the language of computers!). So, let's jump right in and crack these codes!

a) 3x (10) = 100101 (2)

Let's break down the first part of our problem: 3x in base 10 equals 100101 in base 2. This means we need to figure out what digit x makes this equation true. First, we need to convert the binary number 100101 (2) into its decimal (base 10) equivalent. Remember how binary works? Each digit represents a power of 2, starting from the rightmost digit as 2^0, then 2^1, 2^2, and so on.

So, 100101 (2) can be expanded as follows:

(1 * 2^5) + (0 * 2^4) + (0 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0)

Which simplifies to:

(1 * 32) + (0 * 16) + (0 * 8) + (1 * 4) + (0 * 2) + (1 * 1)

And that equals:

32 + 0 + 0 + 4 + 0 + 1 = 37

Alright, so we now know that 100101 (2) is equal to 37 in base 10. Our equation now looks like this: 3x (10) = 37 (10). The 3x here isn't multiplication; it's a two-digit number where 3 is the tens digit and x is the ones digit. Therefore, we can rewrite 3x as 30 + x. Now our equation becomes:

30 + x = 37

To solve for x, we simply subtract 30 from both sides of the equation:

x = 37 - 30

x = 7

Therefore, the digit x in the first equation is 7. This means that 37 in base 10 is equal to 100101 in base 2. We have successfully converted from binary to decimal and solved for our unknown digit. Isn't that neat? Understanding these base conversions is super useful in computer science and helps you appreciate how different number systems represent the same values. Always remember the place values when converting from binary to decimal; it's the key to getting the correct answer! And remember to double-check your calculations. A small mistake can throw off your entire result. Accuracy is your best friend in math!

b) x7 (10) = 10001 (2)

Now let's tackle the second part of the problem: x7 in base 10 equals 10001 in base 2. Just like before, we need to convert the binary number 10001 (2) into its decimal equivalent. We use the same principle of expanding the binary number based on powers of 2.

10001 (2) can be expanded as:

(1 * 2^4) + (0 * 2^3) + (0 * 2^2) + (0 * 2^1) + (1 * 2^0)

Which simplifies to:

(1 * 16) + (0 * 8) + (0 * 4) + (0 * 2) + (1 * 1)

And that equals:

16 + 0 + 0 + 0 + 1 = 17

So, 10001 (2) is equal to 17 in base 10. Now our equation looks like this: x7 (10) = 17 (10). Again, x7 represents a two-digit number in base 10, where x is the tens digit and 7 is the ones digit. So we can rewrite x7 as 10x + 7

Our equation becomes:

10x + 7 = 17

To solve for x, first subtract 7 from both sides of the equation:

10x = 17 - 7

10x = 10

Now, divide both sides by 10:

x = 10 / 10

x = 1

Therefore, the digit x in the second equation is 1. This means that 17 in base 10 is equal to 10001 in base 2. So, guys, we did it! We found the value of x by converting 10001(2) to decimal which became 17(10). Thus, the answer is x = 1. Converting from one number system to another might sound complex, but with practice, it becomes second nature.

Key Takeaways and Tips

  • Mastering Base Conversion: The core of these problems lies in understanding how to convert numbers between different bases, particularly between binary (base 2) and decimal (base 10). Always remember that each digit in a binary number represents a power of 2. Starting from the rightmost digit, these powers are 2^0, 2^1, 2^2, and so on. When converting from binary to decimal, multiply each binary digit by its corresponding power of 2 and then sum up the results. This will give you the decimal equivalent.
  • Understanding Place Value: In both base 10 and base 2, the position of a digit determines its value. For instance, in the number 3x, 3 is in the tens place and x is in the ones place. Similarly, in a binary number like 100101, each digit's position corresponds to a power of 2. Being clear on place value is crucial for accurate conversions and solving equations.
  • Setting Up Equations: Once you've converted the binary number to its decimal equivalent, the next step is to set up an equation that represents the problem. If you have a number like x7 in base 10, remember that this can be written as 10x + 7. This representation allows you to form an algebraic equation that you can then solve for x.
  • Solving for the Unknown: After setting up the equation, use basic algebraic principles to solve for the unknown digit, x. This typically involves isolating x on one side of the equation by performing operations like addition, subtraction, multiplication, or division on both sides. Double-check your work to ensure you haven't made any arithmetic errors.
  • Checking Your Answer: Once you've found a value for x, it's a good idea to check your answer by plugging it back into the original equation. Make sure that the equation holds true with the value of x that you've found. This will help you catch any mistakes and ensure that your answer is correct.
  • Practice Makes Perfect: Like any skill, mastering base conversions and solving these types of problems requires practice. The more you work with different numbers and conversions, the more comfortable you'll become with the process. Try working through additional examples and seeking out practice problems online or in textbooks.
  • Attention to Detail: Pay close attention to detail when converting numbers and setting up equations. A small mistake can lead to a wrong answer. Be careful when multiplying, adding, subtracting, and dividing, and double-check your work to minimize errors.
  • Understanding Binary and Decimal: A strong understanding of both binary and decimal number systems is essential for solving these types of problems. Know how each system works and how to convert between them. This will give you a solid foundation for tackling more complex problems in the future.

Conclusion

So there you have it! We successfully found the values of x in both equations by converting binary numbers to decimal and using basic algebra. Remember, practice makes perfect, so keep honing those skills. Understanding number systems and how to convert between them is not only useful for math problems but also super important in the world of computers and technology. Keep exploring and happy calculating, guys!