Try the following conversions. (i) (514) base 8 = ? base 10 (ii) (220) base 8 = ? base 2 (iii) (76F) base 16 = ? base 10 (iv) (4D9) base 16 = ? base 10 (v) (11001010) base 2 = ? base 10 (vi) (1010111) base 2 = ? base 10
Question
Try the following conversions.
(i) $(514)_{8}=\left(\right. ? {) }_{10}$
(ii) $(220)_{8}=(\text { ? })_{2}$
(iii) $(76 \mathrm{~F})_{16}=(\text { ? })_{10}$
(iv) $(4 \mathrm{D} 9)_{16}=(?)_{10}$
(v) $(11001010)_{2}=(?)_{10}$
(vi) $(1010111)_{2}=(\text { ? })_{10}$
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Answer
Conversion (i) $ (514)_8 = (?)_{10} $
To convert from octal to decimal:
$ 514_8 = 5 \times 8^2 + 1 \times 8^1 + 4 \times 8^0 $
$ 5 \times 64 + 1 \times 8 + 4 \times 1 $
$ 320 + 8 + 4 = 332 $
So, $(514)_8 = (332)_{10}$
Conversion (ii) $ (220)_8 = (?)_{2} $
To convert from octal to binary:
Replace each octal digit with its 3-bit binary equivalent:
2 → 010
2 → 010
0 → 000
Combine the binary values: 0 100 100 000
So, $(220)_8 = (100100000)_2$
Conversion (iii) $ (76F)_{16} = (?)_{10} $
To convert from hexadecimal to decimal:
$ 76F_{16} = 7 \times 16^2 + 6 \times 16^1 + 15 \times 16^0 $
$ 7 \times 256 + 6 \times 16 + 15 $
$ 1792 + 96 + 15 = 1903 $
So, $(76F)_{16} = (1903)_{10}$
Conversion (iv) $ (4D9)_{16} = (?)_{10} $
To convert from hexadecimal to decimal:
$ 4D9_{16} = 4 \times 16^2 + 13 \times 16^1 + 9 \times 16^0 $
$ 4 \times 256 + 13 \times 16 + 9 $
$ 1024 + 208 + 9 = 1241 $
So, $(4D9)_{16} = (1241)_{10}$
Conversion (v) $ (11001010)_{2} = (?)_{10} $
To convert from binary to decimal:
Compute the positional value:
$ 1 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 $
$ 1 \times 128 + 1 \times 64 + 0 \times 32 + 0 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 0 \times 1 $
$ 128 + 64 + 8 + 2 = 202 $
So, $(11001010)_2 = (202)_{10}$
Conversion (vi) $ (1010111)_{2} = (?)_{10} $
To convert from binary to decimal:
Compute the positional value:
$ 1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 $
$ 1 \times 64 + 0 \times 32 + 1 \times 16 + 0 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times 1 $
$ 64 + 16 + 4 + 2 + 1 = 87 $
So, $(1010111)_2 = (87)_{10}$
Summary:
$(514)_8 = (332)_{10}$
$(4D9)_{16} = (1241)_{10}$
$(220)_8 = (100100000)_2$
$(11001010)_2 = (202)_{10}$
$(76F)_{16} = (1903)_{10}$
$(1010111)_2 = (87)_{10}$
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