Question

Transforme essas dízimas em frações;
a) 10,151515... =
b) 2,666.... =
c) 1,2777... =
d) 2,245555.... =
e) 1,4333... =

Answer 1

a) 335/33

n = 10,151515 (equação 1)

1000000 × n = 10151515,151515 (equação 2)

1000000 × n = 10151515,151515

     1 × n = 10,151515

999999 × n = 10151505

n = $\frac{10151505}{999999} =\frac{10151505:30303}{999999:30303} = \frac{335}{33}$

b) 8/3

n = 2,666 (equação 1)

1000 × n = 2666,666 (equação 2)

1000 × n = 2666,666

  1 × n = 2,666

999 × n = 2664

$\frac{2664}{999} =\frac{2664:33}{999:333} = \frac{8}{3}$

c) 23/18

n = 1,2777 (equação 1)

1000 × n = 1277,7777 (equação 2)

1000 × n = 1277,777 7

  1 × n = 1,2777

999 × n = 1276,5

$\frac{1276,5}{999} \frac{1765.10}{999.10} \frac{17650}{9990}$

$\frac{17650}{999} = \frac{17650:555}{9990:555} =\frac{23}{18}$

d) 2021/900

n = 2,245555 (equação 1)

10000 × n = 22455,555555 (equação 2)

10000 × n = 22455,555555

   1 × n = 2,245555

9999 × n = 22453,31

$\frac{22453,31}{9999} =\frac{22453,31.100}{9999.100} = \frac{2245331}{999900}$

n =$\frac{2245331}{99990} =\frac{2245331:1111}{999900:1111} = \frac{2021}{900}$

e) 43/30

n = 1,43333 (equação 1)

10000 × n = 14333,33333 (equação 2)

10000 × n = 14333,33333

   1 × n = 1,43333

9999 × n = 14331,9

$\frac{14331,9}{9999} =\frac{14331,9.10}{9999.10} \frac{1433190}{99990}$

n = $\frac{1433190}{99990} =\frac{1433190:3333}{9999:3333}= \frac{43}{30}$

Answer 2

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$\tt{a)}\\\sf{x=10,151515...\cdot100}\\\sf{100x=1015,151515....}\\-\underline{\begin{cases}\sf{100x=1015,\diagdown\!\!\!\1\!\!15\diagdown\!\!\!\1\!\!15\diagdown\!\!\!\1\!\!15...}\\\sf{x=10,\diagdown\!\!\!\1\!\!15\diagdown\!\!\!\1\!\!15\diagdown\!\!\!\1\!\!15...}\end{cases}}\\\sf{99x=1005}\\\sf{x=\dfrac{1005\div3}{99\div3}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{x=\dfrac{335}{33}}}}}}$

$\tt{b)}~\sf{w=2,666...\cdot10}\\\sf{10w=26,666...}\\-\underline{\begin{cases}\sf{10w=26,\diagdown\!\!\!\1\!\!6\diagdown\!\!\!\1\!\!6\diagdown\!\!\!\1\!\!6...}\\\sf{w=2,\diagdown\!\!\!\1\!\!6\diagdown\!\!\!\1\!\!6\diagdown\!\!\!\1\!\!6...}\end{cases}}\\\sf{9w=24}\\\sf{w=\dfrac{24\div3}{9\div3}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{w=\dfrac{8}{3}}}}}}$

$\tt{c)}\\\sf{k=1,2777...\cdot100}\\\sf{100k=127,777...\cdot10}\\\sf{1000k=1277,777...}\\-\underline{\begin{cases}\sf{1000k=1277,\diagdown\!\!\!\1\!\!7\diagdown\!\!\!\1\!\!7\diagdown\!\!\!\1\!\!7...}\\\sf{100k=127,\diagdown\!\!\!\1\!\!7\diagdown\!\!\!\1\!\!7\diagdown\!\!\!\1\!\!7...}\end{cases}}\\\sf{900k=1150}\\\sf{k=\dfrac{1150\div50}{900\div50}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{k=\dfrac{23}{18}}}}}}$

$\tt{d)}~\sf{\partial=2,245555...\cdot1000}\\\sf{1000\partial=2245,555...\cdot10}\\\sf{10~000\partial=22455,555....}\\-\underline{\begin{cases}\sf{10~000\partial=22455,\diagdown\!\!\!\1\!\!5\diagdown\!\!\!\1\!\!5\diagdown\!\!\!\1\!\!5...}\\\sf{1000\partial=2245,\diagdown\!\!\!\1\!\!5\diagdown\!\!\!\1\!\!5\diagdown\!\!\!\1\!\!5...}\end{cases}}\\\sf{9000\partial=20210}\\\sf{\partial=\dfrac{20210\div10}{9000\div10}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{\partial=\dfrac{2021}{900}}}}}}$

$\tt{e)}~\sf{\Sigma=1,4333...\cdot10}\\\sf{10\Sigma=14,333...\cdot10}\\\sf{100\Sigma=143.333...}\\-\underline{\begin{cases}\sf{100\Sigma=143,\diagdown\!\!\!\1\!\!3\diagdown\!\!\!\1\!\!3\diagdown\!\!\!\1\!\!3...}\\\sf{10\Sigma=14,\diagdown\!\!\!\1\!\!3\diagdown\!\!\!\1\!\!3\diagdown\!\!\!\1\!\!3...}\end{cases}}\\\sf{90\Sigma=129}\\\sf{\Sigma=\dfrac{129\div3}{90\div3}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{\Sigma=\dfrac{43}{30}}}}}}$