GATE IT 2005 | Question: 47

Question

$(34.4)_{8} × \left ( 23.4 \right )_{8}$ evaluates to

• A. $(1053.6)_{8}$
• B. $(1053.2)_{8}$
• C. $(1024.2)_{8}$
• D. None of these

Comments

______

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A shortcut->

https://www.youtube.com/watch?v=Cvk7FyepPXQ

 

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Better to go with straight-forward approach which will take 1-2 extra minutes but will give a high probability of getting correct answer than to take shortcut to save 1-2 minutes but yeilding a wrong result as our brain is not trained to do calculations in base 8 and probability of doing a mistake is very high especially in exam. :P

Answer 1

Simply convert $(34.4)_8$ and $(23.4)_8$ to decimal.

$(34.4)_8 =  28.5$ in decimal and $(23.4)_8 = 19.5$ in decimal.

$28.5 \times 19.5 = 555.75$

Now convert $555.75$ back to octal which is $(1053.6)_8$

$(\overset{\leftarrow}{34}.\overset{\rightarrow}4)_8$ to decimal
$\quad = 3 \times 8^1 + 4 \times 8^0 + 4\times 8^{-1}$
$\quad = 24 + 4 + 0.5$
$\quad = (28.5)_{10}$

$(\overset{\leftarrow}{23}.\overset{\rightarrow}4)_8$ to decimal
$\quad = 2 \times 8^1 + 3 \times 8^0 + 4\times 8^{-1}$
$\quad = 16 + 3 + 0.5$
$\quad = (19.5)_{10}$

$(28.5)_{10} \times (19.5)_{10} = (555.75)_{10}$

Now, $$(555.75)_{10} = (?)_8$$

To convert the integer part

$$\begin{array}{r|lcc}
8 & 555 & \\\hline
8&69 &3\\\hline
8&8 &5\\\hline
8&1& 0\\\hline
8&0& 1 & {\uparrow}
\end{array}$$

We get $1053.$

To convert the decimal, keep multiplying by $8$ till decimal part becomes $0.$

$0.75 \times 8 \to \underbrace{6}_{\text{keep the integral part}}.\underbrace{00}_{0 \text{ decimal part}}$

$$\therefore (555.75)_{10} = (1053.6)_8$$

Correct Answer: $A$

Comments

clear cut approach bro

Answer 2

$(34.4)_8 = \underset{3}{011}\;\underset{4}{100}.\underset{0.4}1 = 16+8+4+0.5 = 28.5$

$(23.4)_8 = \underset{2}{010}\;\underset{3}{011}.\underset{0.4}1 = 16+2+1+0.5 = 19.5$

$28.5 ⨯ 19.5 = 555.75 = (1053.6)_8$

$(0.75 \times 8 = 0.6)$ and if we group together the remainders of dividing $555$ by $8,$ we get $1053.$

Answer 3

You can also multiply in octal base itself without converting to any other base.

Answer 4

Convert it into decimal then multiply and again bring back to octal ..i got option a

Answer 5

Group in 3 bits to convert to octal. 

Answer 6

Here is the Explanation