Đáp án:
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Giải thích các bước giải:
a)
$1,2=1,2222...\\A=1,2222...\\\Rightarrow 10A=12,2222...\\\Rightarrow 10A-A=12,2222...-1,2222...\\\Rightarrow 9A=11\\\Rightarrow A=\dfrac{11}{9}\\2,3=2,3333...\\\Rightarrow A'=2,3333...\\\Rightarrow 10A'=23,3333...\\\Rightarrow 10A'-A'=23,3333...-2,3333...\\\Rightarrow 9A'=21\\\Rightarrow A'=\dfrac{21}{9}\\\Rightarrow 1,(2)+2,(3)\\=\dfrac{11}{9}+\dfrac{21}{9}\\=\dfrac{32}{9}$
b)
$2,2(12)=2,212121...\\\Rightarrow A=2,212121...\\\Rightarrow 100A=221,21212...\\\Rightarrow 100A-A=221,21212...-2,212121...\\\Rightarrow 99A=219\\\Rightarrow A=\dfrac{219}{99}\\4,(14)=4,141414...\\\Rightarrow A'=4,141414...\\\Rightarrow 100A'=414,141414...\\\Rightarrow 100A'-A'=414,141414...-4,141414...\\\Rightarrow 99A'=410\\\Rightarrow A'=\dfrac{410}{99}\\\Rightarrow 2,2(12)+4,(14)\\=\dfrac{219}{99}+\dfrac{410}{99}\\=\dfrac{629}{99}$
c)
$5,12(123)=5,12123123...\\\Rightarrow A=5,12123123...\\\Rightarrow 100\ 000A=512123,123123...\\\Rightarrow 100A=512,123123...\\\Rightarrow 100\ 000A-100A=512123,123123-512,123123\\\Rightarrow 99\ 900A=511\ 611\\\Rightarrow A=\dfrac{170\ 537}{33\ 300}\\2,11(101)=2,11101101...\\\Rightarrow A'=2,11101101...\\\Rightarrow 100\ 000A'=211101,101101...\\\Rightarrow100A'=211,101101...\\\Rightarrow 100\ 000A'-100A'=211\ 101,101101-211,101101\\\Rightarrow 99\ 900A'=210\ 890\\\Rightarrow A'=\dfrac{21089}{9\ 990}\\\Rightarrow 5,12(123)-2,11(101)\\=\dfrac{170\ 537}{33\ 300}-\dfrac{21\ 089}{9\ 990}\\=\dfrac{300\ 721}{99\ 900}$
d)
$2,(3).3,(2)+2,(3).5,(8)\\=2,(3).[3,(2)+5,(8)]\\=2,(3).9\\\Rightarrow 2,(3)=2,3333...\\\Rightarrow A=2,3333...\\\Rightarrow 10A=23,3333...\\\Rightarrow 10A-A=23,3333...-2,3333...\\\Rightarrow 9A=21\\\Rightarrow A=\dfrac{7}{3}\\\Rightarrow 2,(3).9\\=\dfrac73\cdot9\\=21$
