To convert the repeating decimal \(0.7\overline{7}\) into a fraction, we can use the following method:
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Let \(x = 0.7777...\)
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Since the decimal part repeats every one digit, we can multiply \(x\) by 10 to shift the decimal point one place to the right: \[ 10x = 7.7777... \]
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Now, we have two equations: \[ x = 0.7777... \quad (1) \] \[ 10x = 7.7777... \quad (2) \]
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Next, we can subtract equation (1) from equation (2): \[ 10x - x = 7.7777... - 0.7777... \] Simplifying this gives: \[ 9x = 7 \]
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Now, solve for \(x\): \[ x = \frac{7}{9} \]
Thus, the fraction equivalent of \(0.7\overline{7}\) is \(\frac{7}{9}\).
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