Base 5 number 3042 to decimal – UGC NET Question

In base 5, the number 3042 means 3×5^3 + 0×5^2 + 4×5^1 + 2×5^0. This equals 3×125 + 0×25 + 4×5 + 2×1. Calculating the sum gives 375 + 0 + 20 + 2 = 397. Therefore, 397 is the correct decimal equivalent of 3042₅.

Option A:
Option A, 397, matches the detailed place value computation. Each digit is multiplied by the appropriate power of 5 and then summed. Because the arithmetic leads exactly to 397, this option correctly represents the decimal value of 3042 in base 5.

Option B:
Option B, 389, is smaller than the computed decimal result. To obtain 389, we would need different digit coefficients or powers. Since the base-5 structure of 3042 fixes these values, 389 cannot be the correct conversion.

Option C:
Option C, 402, is slightly larger than 397, again conflicting with the precise place value sum. Any correct conversion must preserve equality when the number is expanded. Because 402 does not equal 397, this option is not correct.

Option D:
Option D, 423, differs by a larger margin from the computed value. It would correspond to a different base-5 numeral altogether. Thus, 423 cannot represent 3042₅ when converted properly to decimal.