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

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

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.

Steps: 13

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'.