Question #e3a65

1 Answer
Feb 16, 2018

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\qquad \qquad \qquad \qquad \qquad \qquad \quad \quad 3.777777777... = 17/45 \quad.

Explanation:

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"One neat way to do this is the following:"

"Let:"

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = .3777777777... \qquad \qquad \qquad \qquad \qquad \ "(Equation 1)"

"Multipy both sides by" \ 10, "and then move the decimal point one place to the right:"

\qquad \qquad \qquad \qquad \qquad \qquad \quad \ 10 x = 10 \cdot (.3777777777... )

\qquad \qquad \qquad \qquad \qquad \qquad \quad \ 10 x = 3.777777777... \qquad \qquad \qquad \qquad \qquad \ \ "(Equation 2)"

"Subtract (Equation 1) from (Equation 2):"

\ 10 x - x = 3.777777777... - .3777777777... \quad \ "(Eqn 2) - (Eqn 1)"

\qquad \qquad \ \ \ 9 x = 3.777777777... - .3777777777...

\qquad \qquad \qquad \quad \ = \qquad \quad " " \ 3.777777777...
\qquad \qquad \qquad \qquad \qquad \quad \ \ \ - \ \ \ \ .3777777777...

\qquad \qquad \qquad \quad \ = \qquad " " \ \ \ 3.400000000...

\qquad \qquad \qquad \quad \ = \qquad \quad " " \ 3.4

"So we conclude:"

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad 9 x = 3.4

"Multiply both sides by 10:"

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 10 (9 x) = 10 \cdot (3.4)

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ 90 x = 34

"This is easy to solve (!!):"

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = 34/90.

"Simplify to lowest terms:"

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = 17/45.

"Remembering what" \ \ x \ \ "is, by (Equation 1), we have:"

\qquad \qquad \qquad \qquad \qquad \qquad \quad \quad 3.777777777... = 17/45.

"This is our answer."

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"Summarizing:"

\qquad \qquad \qquad \qquad \qquad \qquad \quad \quad 3.777777777... = 17/45 \quad.