Sum of $(87A)_{16}+(50B)_{16}$ is:

Question

Accepted Answer

The correct option is (1):
To find the sum of $(87A)*{16}$ and $(50B)*{16}$, we first perform hexadecimal addition. Students should recall that in hexadecimal, the digits A, B, C, D, E, F represent the decimal values 10 through 15, respectively. Starting from the rightmost digit, we add $A + B$, which is $10 + 11 = 21$ in decimal. Since the base is 16, we find the remainder and the carry: $21\_{10} = 1 \times 16 + 5$, so we write down '5' and carry '1'. Moving to the middle column, we add $7 + 0 + (\text{carry } 1) = 8$. Finally, for the leftmost digits, we add $8 + 5 = 13$, which is represented by 'D' in hexadecimal. The resultant hexadecimal sum is $(D85)*{16}$. Since Option 1 is in decimal, we must convert $(D85)*{16}$ to decimal: $(D \times 16^2) + (8 \times 16^1) + (5 \times 16^0) = (13 \times 256) + (8 \times 16) + (5 \times 1) = 3328 + 128 + 5 = 3461\_{10}$.

The incorrect option is (2):
Option (2), $(3472)*8$, is a representation in the octal number system. Converting the correct hexadecimal sum, $(D85)*{16}$, to octal results in $(6605)*8$, not $(3472)*8$. This indicates that even if the student intended to convert the answer to octal, this specific value is mathematically incorrect. Furthermore, converting $3472\_8$ to decimal yields $1850*{10}$, which is nearly half the value of the correct sum of $3461*{10}$. This option likely arises from a significant arithmetic or base conversion error during or after the addition.

The incorrect option is (3):
Option (3), $(9EA)*{16}$, is presented in the hexadecimal base but contains a major calculation error. The correct hexadecimal sum, which is equivalent to $3461*{10}$, is $(D85)*{16}$. The value $9EA*{16}$ translates to a decimal value of $2538\_{10}$, confirming it is not the correct result. Errors leading to $9EA\_{16}$ often involve a miscalculation in the most significant digits, such as $8+5$ incorrectly resulting in $9\_{16}$ (decimal 9) or $E\_{16}$ (decimal 14) combined with a misplaced carry from the lower-order additions.

The incorrect option is (4):
Option (4), $(63215)*8$, is an octal number that is far too large to be the sum of two three-digit hexadecimal numbers. To illustrate its magnitude, converting $(63215)*8$ to decimal gives $26253*{10}$, which is roughly seven times the magnitude of the correct sum, $3461*{10}$. This option suggests a fundamental misunderstanding of the number bases, potentially confusing octal with another base or making a severe misstep in carry propagation, leading to a vastly inflated and incorrect final value.