GATE CSE 1999 | Question: 2.16

Question

The number of full and half-adders required to add $16$-bit numbers is

• A. $8$ half-adders, $8$ full-adders

• B. $1$ half-adder, $15$ full-adders

• C. $16$ half-adders, $0$ full-adders

• D. $4$ half-adders, $12$ full-adders

Comments

@Mohnish option is c is incorrect because HA doesn’t take input carry which we need in this case else the answer of addition come out to be wrong. 

So either use 16 FA or use 1HA+15FA.

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Before getting to the solution the question is about implementing the 16 bit number using both full and half adders as combinations that is in the same implementation we will use both full and half adders

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is this equation  generalization or it can work only for few variations?

Answer 1

Answer is B.

For LSB addition we do not need a full adder.

For addition of subsequent bits we need full adders since carry from previous addition has to be fed into the addition operation.

Comments

@jatin khachane 1 while subtraction borrow is generated only in one case when 1 is being subtracted from 0. 

subtraction = A ex or B (it will give 1 as A =0 and B=1)

Borrow = ~A.B (It will give one for the same input.)

So in LSB of result we get 1 which is desired(10-1) and for all the next bits we have to deal with 3 bits in worst case so full adder.

Hence 15 full adder + 1 half adder.

 

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In subtraction we add $\bar{B}$  to A by giving 1 to M that make $B{}'$ (2s complement of B)

Hence $C_{0}$ also 1..At LSB we have to add 3 Bits and from LSB onward we may have to add 3 bits

check 5 - 1

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Circuit is built in such a way that it takes sum with 1s complement first and then add 1 to it. So possibility of 3 1s at LSB is not possible. 

Answer 2

Answer : Option B

 

Comments

Thanks for this detailed explanation.

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Thank you buddy :)

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One small doubt is it acting as ripple carry adder?

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@Shamim Ahmed

yes it is ripple carry. You can check carry look ahead adder for better version of this.

Answer 3

N Full adder= N-1 Full adder  + 1 half adder

                      N-1 (2 half adder + 1 OR gate ) + 1 half adder 

                      2N-2+1 (half adder ) + N-1(OR gate )

                      2N-1 half adder + N-1 OR gate 

Answer 4

if we have to add two n-bit numbers by using half and full adders 

To add LSBs we need a Half adder as only 2 bits are to be added as there is no carry 

For rest of bits full adder is needed as carry from previous bits is added to next two bits 

so we need $n-1$ Full adders & $1 $Half Adder 

here n=16

15 Full adders

1 half adder