You know that 71=0.142857. Can you predict what the decimal expansions of 72,73,74,75,76 are, without actually doing the long division? If so, how? (Hint: Study the remainders while finding the value of 71 carefully.).

Given, 71=0.142857.Then,72=2×71=2×0.142857=0.28571473=3×71=3×0.142857=0.42857174=4×71=4×0.142857=0.57142875=5×71=5×0.142857=0.71428576=6×71=6×0.142857=0.857142 Here, the first digit in the decimal expansion continued in the repeated numbers afterwards.

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You know that {1}{7}=0.{142857}. Can you predict what the deci

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You know that $\frac{1}{7} = 0. \overline{142857}$ . Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how? (Hint: Study the remainders while finding the value of $\frac{1}{7}$ carefully.).

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You know that

\frac{1}{7} = 0. \overline{142857}

. Can you predict what the decimal expansions of

\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}

are, without actually doing the long division? If so, how? (Hint: Study the remainders while finding the value of

\frac{1}{7}

carefully.).

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Text solutionVerified

Given,

\frac{1}{7} = 0. \overline{142857}

Then,

\frac{2}{7} = 2 \times \frac{1}{7} = 2 \times 0. \overline{142857} = 0. \overline{285714}

\frac{3}{7} = 3 \times \frac{1}{7} = 3 \times 0. \overline{142857} = 0. \overline{428571}

\frac{4}{7} = 4 \times \frac{1}{7} = 4 \times 0. \overline{142857} = 0. \overline{571428}

\frac{5}{7} = 5 \times \frac{1}{7} = 5 \times 0. \overline{142857} = 0. \overline{714285}

\frac{6}{7} = 6 \times \frac{1}{7} = 6 \times 0. \overline{142857} = 0. \overline{857142}

Here, the first digit in the decimal expansion continued in the repeated numbers afterwards.

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Question Text	You know that $\frac{1}{7} = 0. \overline{142857}$ . Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how? (Hint: Study the remainders while finding the value of $\frac{1}{7}$ carefully.).
Answer Type	Text solution:1
Upvotes	150

You know that 71​=0.142857. Can you predict what the decimal expansions of 72​,73​,74​,75​,76​ are, without actually doing the long division? If so, how? (Hint: Study the remainders while finding the value of 71​ carefully.).

Text solutionVerified

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