Science:Math Exam Resources/Courses/MATH101/April 2016/Question 01 (d)

MATH101 April 2016

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Question 01 (d)
Which fraction equals the decimal $0.{\overline {321}}=0.321321321\dots$? ${\begin{aligned}{{\textbf {T}}{:}}&\,1/3&{{\textbf {X}}{:}}\,102/333\\{{\textbf {V}}{:}}&\,322/999&{{\textbf {Y}}{:}}\,8/27\\{{\textbf {W}}{:}}&\,321/1000&{{\textbf {Z}}{:}}\,107/333\end{aligned}}$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Write it in the Geometric Series form and compute the series.

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Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Note: A Geometric Series has the form $a+ar+ar^{2}+ar^{3}+...,$ where $a$ is the first term, and $r$ is called the common ratio. We can compute this series using the formula $a+ar+ar^{2}+ar^{3}+...={\frac {\color {OrangeRed}a}{1-{\color {OliveGreen}r}}}$ We first write $0.321321321...$ in the Geometric Series form. Indeed, ${\begin{aligned}0.321321321...&=0.321+0.000321+0.000000321+...\\&={\frac {321}{10^{3}}}+{\frac {321}{10^{6}}}+{\frac {321}{10^{9}}}+....\end{aligned}}$ We can see that the first term is ${\color {OrangeRed}a={\frac {321}{10^{3}}}}$ and the common ration is ${\color {OliveGreen}r}={\frac {\text{the Second term}}{\text{the First term}}}={\frac {\frac {321}{10^{6}}}{\frac {321}{10^{3}}}}{\color {OliveGreen}={\frac {1}{10^{3}}}.}$. Therefore, ${\begin{aligned}0.321321321...&={\frac {321}{10^{3}}}+{\frac {321}{10^{6}}}+{\frac {321}{10^{9}}}+....\\&={\frac {\color {OrangeRed}{\frac {321}{10^{3}}}}{1-{\color {OliveGreen}{\frac {1}{10^{3}}}}}}={\frac {\frac {321}{10^{3}}}{\frac {999}{10^{3}}}}\\&={\frac {321}{999}}={\color {blue}{\frac {107}{333}}}\end{aligned}}$ So, the answer is Z.

Solution 2
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $x=0.{\overline {321}}=0.321321321\cdots$ Then, $1000x=321.321321321\cdots$, which implies $1000x-x=321.321321321\cdots -0.321321321\cdots =321$. Therefore $999x=321$ and hence $x={\frac {321}{999}}={\frac {107}{333}}$. The answer is Z.