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1.) Find the first ten digits after the decimal point in the decimal expansion of $\frac{10}{27}=0.abcdefghij\ldots$

 without a calculator.

(Express your answer as a ten-digit number.)

 

2.) The number $\frac{12}{13}$ can be expressed as a repeating decimal $0.\overline{abcdef}$. Find the repeating part $abcdef$ without a calculator.

 

3.) Convert $0.04\overline{55}$ to a fraction in simplest form.

 

4.) Convert $6032_8$ to decimal.

 

5.) Convert $999_{16}$ to decimal.

 

6.) Convert 35 from base 10 to base 2.

(You do not need to include the subscript 2 in this answer.)

 

7.) Convert $2231_4$ to base 2 without first converting to decimal.

(You do not need to include the subscript 2 in this answer.)

 

All help Appreciated(ASAP if possible)

 Jun 16, 2019
 #1
avatar+208 
-1

OK, IM guessing this is from AoPS from the dollar signs...... I don't really mind the asking and I would solve it but can you edit it so the latex is in latex???????? ok so this is my version of it...:

 

 

1.) Find the first ten digits after the decimal point in the decimal expansion of \(\frac{10}{27}=0.abcdefghij\ldots\)

 without a calculator.

(Express your answer as a ten-digit number.)

 

2.) The number \(\frac{12}{13}\) can be expressed as a repeating decimal \(0.\overline{abcdef}\). Find the repeating part \(abcdef\) without a calculator.

 

3.) Convert \(0.04\overline{55}\) to a fraction in simplest form.

 

4.) Convert \(6032_8\) to decimal.

 

5.) Convert \(999_{16}\) to decimal.

 

6.) Convert 35 from base 10 to base 2.

(You do not need to include the subscript 2 in this answer.)

 

7.) Convert \(2231_4\) to base 2 without first converting to decimal.

 Jun 17, 2019
edited by NoobGuest  Jun 17, 2019
 #2
avatar+208 
-1

1) one is kinda easy..... just start dividing and find a pattern. I found the pattern 0.370 repeating, and so your answer for 1 would be 0.3703703703

 

2)So I started manually dividing and after 6 digits, I stopped... I want to show it here but have no idea how... :( but your answer is 923076

 

3)to convert 0.045 repeating 5 to fraction form, just do this:

       1000x=45.55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555

             - x=0.045555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555

              _____________________________________________________________________________________________________

subtract and get 999x=45.1

                                 x=451/9990

                                 x=41/90

 

4) \(6032_8\) to decimal?! argh! ok, so start by doing (6*6^3)+(0*6^2)+(3*6^1)+(2*6^0)=1407.... and 1407 is 6032_8 in base 10.... in other words, decimal.

 

5)\(999__16\)ok I cant get the base thing to work in latex... so deal with this.

(9*16^2)+(9*16)+9=2457 in decimal..

 

6)35/2=17r1

  17/2=8r1

   8/2=4r0

   4/2=2r0

   2/2=1r0

   1/2=0r1

so i think 35 in base 2 is 100011

 

7)10101101...

NoobGuest  Jun 17, 2019
 #3
avatar+7711 
0

1) \(\dfrac{10}{27} = \dfrac{370}{999} = 0.\overline{370} = 0.370\;370\;370\;3...\)

.
 Jun 22, 2019
 #4
avatar+7711 
0

2) \(\dfrac{12}{13} = \dfrac{923076}{999999} = 0.\overline{923076}\)

.
 Jun 22, 2019
edited by MaxWong  Jun 22, 2019
 #5
avatar+7711 
0

3) 

\(\quad0.04\overline{55}\\ = 0.04 + 0.00\overline{5}\\ = \dfrac{4}{100} + \dfrac{0.\overline{5}}{100}\\ = \dfrac{4}{100} + \dfrac{5}{900}\\ = \dfrac{41}{900}\)

.
 Jun 22, 2019
 #6
avatar+7711 
0

4)

\(\quad 6032_8\\ = 6\cdot 8^3 + 3\cdot 8 + 2\\ = 3098\)

.
 Jun 22, 2019
edited by MaxWong  Jun 22, 2019
 #7
avatar+7711 
0

5)

\(\quad 999_{16}\\ = 9(16^2 + 16 + 1)\\ = 2457\)

.
 Jun 22, 2019
 #8
avatar+7711 
0

6)

\(35 = 2^5 + 2^1 + 2^0\\ 35 = 100011_2\)

.
 Jun 22, 2019
 #9
avatar+7711 
0

7)

\(\quad 2231_4\\ = 10101101_2\)

.
 Jun 22, 2019
edited by MaxWong  Jun 22, 2019

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