Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 May 17th, 2010, 07:36 PM #1 Newbie   Joined: May 2010 Posts: 7 Thanks: 0 HELP PLEASE Hey All! Can someone please help me in doing this question? a) find the repetend for the fraction 1/103 b) find the five largest fractions amongst n/103 for n = 1,2,3.....,102 whose repetends have the cyclic order as the repetend for 1/103. c)find three fractions amongst n/103 for n = 1,2,3,....,102 whose repetends do not have the same cyclic order. d) represent the three different repetends in part c as A,B,C. Determine which of these repetends each of the fractions n/103 has for n = 1,2,3,....,10 and 93,94,95...,102. what pattern do you notice?
 May 18th, 2010, 03:25 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,469 Thanks: 2038 (a) 0097087378640776699029126213592233 (b) 102/103, 100/103, 95/102, 94/102, 93/102 (c) 1/103, 2/103, 4/103 (d) A,B,A,C,C,B,C,A,A,A and A,A,A,C,B,C,C,A,B,A (order reversed)
May 18th, 2010, 06:47 PM   #3
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Hello, Mariner!

Quote:
 a) Find the repetend for the fraction 1/103

Carefully divide 1 by 103.

We find a decimal with a 34-digit repeating cycle.

[color=beige]. . [/color]$\frac{1}{103} \;=\; 0.\,\overline{009\,708\,737\,862\,077\,669\,902\,9 12\,621\,359\,223\,3}\:\cdots$

Quote:
 b) Find the five largest fractions among n/103 for n = 1,2,3.....,102 [color=beige]. . [/color]whose repetends have the cyclic order as the repetend for 1/103.

Read along the cycle and pick out the five largest 6-digit (or 9-digit) decimals.

[color=beige]. . [/color]$0.901912\;\;\;0.0.912621\;\;\;0.922330\;\;\;0.0.97 0873\;\;\;0.990291$

Multiply each by 103:

[color=beige]. . [/color]$\begin{array}{cccccc}0.902912\,\times\,103=&92.999936=&\approx=&93 \\ \\ \\ 0.912621\,\times\,103=&93.999963=&\approx=&94 \\ \\ \\ 0.922330\,\times\,103=&94.999990=&\approx=&95 \\ \\ \\ 0.970873\,\times\,103=&99.999919=&\approx=&100 \\ \\ \\ 0.990291\,\times\,103=&101.999973=&\approx=$

$\text{Therefore: }\;\begin{Bmatrix}\frac{93}{103}=&0.\overline{902912\,\cdots} \\ \\ \\ \frac{94}{103}=&0.\overline{912621\,\cdots} \\ \\ \\ \frac{95}{103}=&0.\overline{922330\,\cdots} \\ \\ \\ \frac{100}{103}=&0.\overline{970873\,\cdots} \\ \\ \\ \frac{102}{103}=&0.\overline{990291\,\cdots} \end{Bmatrix}=$

Quote:
 c) Find three fractions among n/103 for n = 1,2,3,....,102 [color=beige]. . [/color]whose repetends do not have the same cyclic order.

$\frac{2}{103} \;=\;0.\overline{019\,417\,475\,728\,155\,339\,805 \,825\,242\,718\,446\,6} \,\cdots$

$\frac{4}{103}\;=\;0.\overline{038\,834\,951\,456\, 310\,679\,611\,650\,485\,436\,893\,2}\,\cdots$

$\frac{5}{103}\;=\;0.\overline{048\,543\,689\,320\, 388\,349\,514\,563\,106\,796\,116\,5}\,\cdots$

Quote:
 d) Represent the three different repetends in part c as A,B,C. Determine which of these repetends each of the fractions n/103 has for n = 1,2,3,....,10 and 93,94,95...,102. What pattern do you notice?

Not sure what they're asking . . .

 May 19th, 2010, 10:23 PM #4 Newbie   Joined: May 2010 Posts: 7 Thanks: 0 Re: HELP PLEASE Thank you very much!
 May 19th, 2010, 10:24 PM #5 Newbie   Joined: May 2010 Posts: 7 Thanks: 0 Re: Please Help! Thanks skipjack!

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### cyclic order repetend

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