Sum $(89D)_{16}+(259)_{10}$ is:

Question

Accepted Answer

The correct option is (3):
To accurately find the sum of two numbers expressed in different numerical bases, the fundamental rule is to convert both operands into a common base, with the decimal (base 10) system being the easiest for arithmetic. The hexadecimal number $(89D)*{16}$ must first be converted to decimal. This is calculated as $8 \times 16^2 + 9 \times 16^1 + 13 \times 1$ (since $D$ is equivalent to 13 in decimal). The result of this conversion is $2048 + 144 + 13$, which equals $(2205)*{10}$. The addition is then performed in the decimal system: $2205 + 259$. The final, correct sum is exactly $(2464)\_{10}$, which corresponds perfectly to Option 3.

The incorrect option is (1):
Option 1, $(9EA)*{16}$, is incorrect because its decimal value is not $2464$. To verify this, we convert $(9EA)*{16}$ to decimal as $9 \times 16^2 + 14 \times 16^1 + 10 \times 1$, knowing that $E$ is 14 and $A$ is 10. Performing the arithmetic yields $2304 + 224 + 10$, resulting in $(2538)\_{10}$. While this value is numerically close, it demonstrates a calculation error, either in the initial conversion or the final addition step, proving it is not the correct sum.

The incorrect option is (2):
Option 2, $(3472)*8$, is presented in octal (base 8). Converting this back to the decimal system allows us to check its magnitude. The decimal equivalent is calculated as $3 \times 8^3 + 4 \times 8^2 + 7 \times 8^1 + 2 \times 8^0$. This expands to $3 \times 512 + 4 \times 64 + 7 \times 8 + 2 \times 1$, which equals $1536 + 256 + 56 + 2$. The total decimal value is $(1850)*{10}$. Since $1850$ is not equal to the correct sum of $2464$, this octal representation is incorrect.

The incorrect option is (4):
Option 4, $(63215)*8$, is an octal number that is excessively large compared to the required sum. Even a rough magnitude estimate shows this number is in the tens of thousands. Its complete decimal conversion is $6 \times 8^4 + 3 \times 8^3 + 2 \times 8^2 + 1 \times 8^1 + 5 \times 8^0$. This evaluates to $6 \times 4096 + 3 \times 512 + 2 \times 64 + 8 + 5$, which totals $(26253)*{10}$. This result is an order of magnitude too high, confirming that the representation in Option 4 is not the correct sum.