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In a certain number system, the number '210' represents the decimal number 42. W

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Question: In a certain number system, the number \'210\' represents the decimal number 42. What is the decimal equivalent of \'102\' in the same system?

Options:

  1. 20
  2. 21
  3. 22
  4. 23

Correct Answer: 20

Solution: 

\'210\' in base 3 is 2*3^2 + 1*3^1 + 0*3^0 = 18 + 3 + 0 = 21. Therefore, \'102\' in base 3 is 1*3^2 + 0*3^1 + 2*3^0 = 9 + 0 + 2 = 11.

In a certain number system, the number '210' represents the decimal number 42. W

Practice Questions

Q1
In a certain number system, the number '210' represents the decimal number 42. What is the decimal equivalent of '102' in the same system?
  1. 20
  2. 21
  3. 22
  4. 23

Questions & Step-by-Step Solutions

In a certain number system, the number '210' represents the decimal number 42. What is the decimal equivalent of '102' in the same system?
  • Step 1: Understand that '210' in a certain number system represents a decimal number. We need to find out what base this number system is using.
  • Step 2: We know that '210' equals 42 in decimal. We will convert '210' to decimal using a base. Let's assume the base is 3.
  • Step 3: Convert '210' from base 3 to decimal: Calculate 2*3^2 + 1*3^1 + 0*3^0.
  • Step 4: Calculate each part: 2*3^2 = 2*9 = 18, 1*3^1 = 1*3 = 3, and 0*3^0 = 0.
  • Step 5: Add these values together: 18 + 3 + 0 = 21. This means '210' in base 3 equals 21, not 42. So, we need to find the correct base.
  • Step 6: Since '210' equals 42, we can find the base by solving the equation 2*b^2 + 1*b + 0 = 42.
  • Step 7: This simplifies to 2b^2 + b = 42. Rearranging gives us 2b^2 + b - 42 = 0.
  • Step 8: Use the quadratic formula to find the base: b = [-1 ± sqrt(1 + 4*2*42)] / (2*2).
  • Step 9: Calculate the discriminant: 1 + 336 = 337. Then find the square root of 337.
  • Step 10: Solve for b using the quadratic formula to find the base.
  • Step 11: Once we find the base, we can convert '102' in that base to decimal using the same method.
  • Step 12: Convert '102' from the found base to decimal: Calculate 1*b^2 + 0*b + 2*1.
  • Step 13: Add these values together to find the decimal equivalent of '102'.
  • Base Conversion – Understanding how to convert numbers from one base to another, specifically from a non-decimal base to decimal.
  • Number Representation – Interpreting how numbers are represented in different numeral systems and the significance of each digit's position.
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