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#### paulmdrdo

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- May 13, 2013

- 386

0.17777777777 convert into a ratio.

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- #1

- May 13, 2013

- 386

0.17777777777 convert into a ratio.

Hi,

This is [tex]0.1 + 0.077777=\frac{1}{10}+\frac{7}{100}+\frac{7}{1000}+...[/tex] where you have a GP to sum.

Or [tex] \text{Let } x=0.0777..[/tex] so that [tex]10x=0.777..[/tex].

Subtracting gives [tex]9x=0.7[/tex] and so [tex]x=\frac{7}{90}[/tex]. Now just add [tex]\frac{1}{10}+\frac{7}{90}[/tex] and simplify.

I should also say that we can write a decimal as a fraction but we can't write it as a ratio.

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- May 13, 2013

- 386

what do you mean by "GP"?Hi,

This is [tex]0.1 + 0.077777=\frac{1}{10}+\frac{7}{100}+\frac{7}{1000}+...[/tex] where you have a GP to sum.

Or [tex] \text{Let } x=0.0777..[/tex] so that [tex]10x=0.777..[/tex].

Subtracting gives [tex]9x=0.7[/tex] and so [tex]x=\frac{7}{90}[/tex]. Now just add [tex]\frac{1}{10}+\frac{7}{90}[/tex] and simplify.

I should also say that we can write a decimal as a fraction but we can't write it as a ratio.

Sorry, I have to stop using abbreviations.what do you mean by "GP"?

A GP is a geometric progression: [tex]a, ar, ar^2, ar^3...[/tex].

If you haven't met this then the second method I posted is fine.

Hello, paulmdrdo!

[tex]\text{Convert }\,0.1777\text{...}\,\text{ to a fraction.}[/tex]

[tex]\begin{array}{ccc}\text{We have:} & x &=& 0.1777\cdots \\ \\ \text{Multiply by 100:} & 100x &=& 17.777\cdots \\ \text{Multiply by 10:} & 10x &=& \;\;1.777\cdots \\ \text{Subtract:} & 90x &=& 16\qquad\quad\; \end{array}[/tex]

Therefore: .[tex]x \;=\;\frac{16}{90} \;=\;\frac{8}{45}[/tex]

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- #6

- May 13, 2013

- 386

for example i have 3.5474747474... how would you convert this one?

If you use 1000 and 10 you will get

1000x=3547.474747...

10x=35.474747...

So 990x=3512 and x=3512/990=1756/495.

I'm adopting Soroban's approach as I prefer it to what I did earlier.

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- #8

- May 13, 2013

- 386

"a difference of two in the powers of ten" -- what do you mean by this? sorry, english is not my mother tongue. bear with me.

If you use 1000 and 10 you will get

1000x=3547.474747...

10x=35.474747...

So 990x=3512 and x=3512/990=1756/495.

I'm adopting Soroban's approach as I prefer it to what I did earlier.

Last edited:

No problem."a difference of two in the powers of ten" -- what do you me by this? sorry, english is not my mother tongue. bear with me.

We have 10^3 and 10^1.

The difference between 3 and 1 is 3-1=2

You want to multiply by a power of 10 which enables you to only have the repeating digits shown, and then multiply by a higher power of ten to have

for example i have 3.5474747474... how would you convert this one?

So in this case, since the 47 repeats, you want the first to read "something.4747474747..." and the second to read "something-else.4747474747..."

What powers of 10 will enable this?

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- #11

1.) \(\displaystyle x=0.1\overline{7}\)

\(\displaystyle 10x=1.\overline{7}=1+\frac{7}{9}=\frac{16}{9}\)

\(\displaystyle x=\frac{16}{90}=\frac{8}{45}\)

2.) \(\displaystyle x=3.5\overline{47}\)

\(\displaystyle 10x=35.\overline{47}=35+\frac{47}{99}=\frac{3512}{99}\)

\(\displaystyle x=\frac{3512}{990}=\frac{1756}{495}\)