Solve the following equation for x involving mixed recurring decimals and express the solution as vulgar fraction. 3x + 2.3666... = 5.4666...
$\frac{31}{30}$
The equation mixes recurring (repeating) decimals, so the reliable route is to convert each repeating decimal into a fraction, or — even faster here — to notice that the two decimals have the same repeating tail and simply subtract.
The equation is 3x + 2.3666… = 5.4666…. Isolating x:
- 3x = 5.4666… − 2.3666…
- Because both numbers repeat the digit 6 identically, the recurring parts cancel on subtraction, leaving 3x = 3.1 exactly.
- 3.1 as a fraction is 31/10, so 3x = 31/10.
- x = 31/10 ÷ 3 = 31/30.
We can confirm this with the formal fraction conversion. For a mixed recurring decimal, subtract the non-repeating figures from the whole figures and divide by 9s and 0s: 2.3666… = 2 + (36 − 3)/90 = 2 + 33/90 = 71/30, and 5.4666… = 5 + (46 − 4)/90 = 5 + 42/90 = 82/15. Then 3x = 82/15 − 71/30 = 164/30 − 71/30 = 93/30 = 31/10, giving the same x = 31/30 (which is about 1.0333…).
Hence the correct choice is the option showing 31/30. Any option displaying a value like 31/10 has stopped at 3x without the final division by 3, while options not equal to 31/30 come from mishandling the recurring-decimal-to-fraction conversion.
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Important Questions from Decimals
The value of \(\frac{1}{4} + \frac{{[{{(20.35)}^2} - {{(8.35)}^2}] \times 0.0175}}{{{{(1.05)}^2} + (1.05)(27.65)}}\) is:
The value of \(0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7 \) is:
The value of \(\frac{48.3\times[(4.95)^2+4.95\times13.25]}{[(12.55)^2-(5.65)^2]\times19.8} \) is:
Find the value of (1.6) 3 - (0.9) 3 - (0.7) 3.
What is the value of x, if \(5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}\) ?
