Do You Remember How To Subtract Repeating Decimals?

Number Theory Level 1

0. 42 ‾ − 0. 35 ‾ = ? \large \displaystyle {0. \overline{42}-0.\overline{35}= \, ?}


Note: 0. a b ‾ = 0. a b a b a b a b … 0.\overline{ab}=0.abababab \ldots

0.7 0.7 0. 7 ˉ 0.\bar{7} 7 9 \dfrac{7}{9} 4 13 \dfrac{4}{13} 7 99 \dfrac{7}{99} None of the above choices

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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4 solutions

Arulx Z
May 20, 2015

x = 0. 42 ‾ 100 x = 42. 42 ‾ 100 x − x = 42. 42 ‾ . − 0. 42 ‾ 99 x = 42 x = 42 99 x=0.\overline { 42 } \\ 100x=42.\overline { 42 } \\ 100x-x=42.\overline { 42 } .-0.\overline { 42 } \\ 99x=42\\ x=\frac { 42 }{ 99 }

So 0. 42 ‾ = 42 99 0.\overline { 42 } =\frac { 42 }{ 99 } \\

In the same way, 0. 35 ‾ = 35 99 0.\overline { 35 } =\frac { 35 }{ 99 } \\

So

0. 42 ‾ − 0. 35 ‾ = 42 99 − 35 99 = 7 99 0.\overline { 42 } -0.\overline { 35 } =\frac { 42 }{ 99 } -\frac { 35 }{ 99 } \\ =\frac { 7 }{ 99 }

The answer is in fact 0. 07 ‾ 0.\overline{07} .

51 Helpful 3 Interesting 7 Brilliant 1 Confused

Moderator note:

Right, this is similar to a geometric series. Bonus question: Can you give a proof for the infamous statement: 0.9999 … = 1 0.9999\ldots = 1 ?

In response to Challenge Master's note

The number can be expressed as 0. 9 ‾ 0.\overline{9} . Let this = x

Therefore, 10x= 9. 9 ‾ 9.\overline{9}

Therefore, 9x=9

x=1.

Therefore, 0. 9 ‾ = 1 0.\overline{9}=1

Mehul Arora - 6 years ago

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You could also show that 1 3 = 0. 3 ‾ ⇔ 3 ∗ 1 3 = 3 ∗ 0. 3 ‾ ⇔ 1 = 0. 9 ‾ \frac{1}{3} = 0.\overline{3} \Leftrightarrow 3*\frac{1}{3} = 3*0.\overline{3} \Leftrightarrow 1 = 0.\overline{9}

Patrick Engelmann - 6 years ago

One more way to prove (Slightly longer) -

0.\overline { 9 } =\sum _{ n = 1 }^{ \infty }{ \frac { 9 }{ { 10 }^{ n } } }

which when simplified gives the answer as 1 1 .

Arulx Z - 6 years ago

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Ohh yeah. Good one.

Mehul Arora - 6 years ago

Consider 1/3 = 0.3333.. As equation 1 And 2/3 = 0.6666.. As equation 2 Adding eqn 1 and 2 we get LHS = 1/3 + 2/3 = 3/3 = 1 RHS = 0.333333...+0.666666 = 0.9999 But we are adding two equals so the whole are equal ... Therefore LHS=RHS So 1 = 0.9999999999..

Ashwin Kumar - 6 years ago

0 is an integer number

Saloni Singh - 6 years ago

Check you third row. missing a 0.

Gavin Johnstone - 5 years, 6 months ago

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Thanks for pointing it out!

Arulx Z - 5 years, 6 months ago
Jochem Jonges
Dec 24, 2017

0. 42 ‾ − 0. 35 ‾ = 0. 07 ‾ 0.\overline{42}-0.\overline{35} = 0.\overline{07}

x = 0. 07 ‾ x=0.\overline{07}

100 x − 7 = x 100x-7=x

x = 7 99 x=\frac{7}{99}

5 Helpful 3 Interesting 1 Brilliant 0 Confused
Soumen Bhowmik
May 20, 2015

The expression can be presented as: (42/99) - (35/99) = 7/99

5 Helpful 1 Interesting 1 Brilliant 0 Confused

Moderator note:

Your solution have no relevant working. Why it can be converted as such?

Poorly structured. The site usually includes 4 potential answers. The "none of the above" answer appeared to be the last option on my laptop screen. Had I seen the remaining possible answers, this could have been answered quite easily. I suspect that is why there is a low correct response rate (56%) for a fairly straightforward question.

JQ user - 4 years ago

x=0.4242....

.'. 100x=42.4242.....

.'.(100-1)x=42

.'.x=42/99

Ansh Bhatt - 6 years ago

You need to covert the fraction into rational then it is easy to substract

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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