Addition of two numbers with a twist

Design a system that gets two 4-bit numbers X₃X₂X₁X₀ , Y₃Y₂Y₁Y₀ in the range of 0-9 and the output of the system is the addition result between these two numbers (Z₄Z₃Z₂Z₁Z₀) while the MSB, Z₄is the tens and Z₃Z₂Z₁Z₀ are the units digits.

For example: X₃X₂X₁X₀= 8 (1000) , Y₃Y₂Y₁Y₀= 5 (0101) , Z₄Z₃Z₂Z₁Z₀ = 13 (1-0011)

X₃X₂X₁X₀= 9 (1001) , Y₃Y₂Y₁Y₀= 1 (0001) , Z₄Z₃Z₂Z₁Z₀ = 10 (1-0000)

X₃X₂X₁X₀= 4 (0100) , Y₃Y₂Y₁Y₀= 3 (0011) , Z₄Z₃Z₂Z₁Z₀ = 7 (0-0111)

In order to design the system you have two 4-bit adders, OR and AND gates as many as you like and constants (‘0’ and ‘1’).

Answer

We need to implement logic that detects when the result of the first addition between the two inputs is greater, or equal to ten. The Carry (Cout bit) will be ‘1’ in case that the result is more than 15 and S3 will be ‘1’ in case of a result greater than 8. Let’s write down some numbers

	Cout	S3	S2	S1	S0
10	0	1	0	1	0
11	0	1	0	1	1
12	0	1	1	0	0
13	0	1	1	0	1
14	0	1	1	1	0
15	0	1	1	1	1
16	1	0	0	0	0

From this table we can define the function F = Cout + S3(S2+S1) and it’s output will be connected directly to Z4.
If the result is greater than 9 then we need to convert it to the new representation. Let’s say that the result is 10 (1010), the new representation in such case is 10000 (16), so we need to use the second adder and to add 6 in order to get the correct Units digit representation.

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