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
What is the result when 0.129129129… is converted to a fraction?
Which of the following statement(s) is/are correct?
I. (3/11) > 0.3
II. (7/8) > 0.86
The value of \(1.\overline{3}+0.\overline{69}-0.5\overline{23}\) is equal to:
The value of \(0.\bar 4 + 0.5 \bar 9 - 0.4 \overline{23}\) is equal to:
Find the sum of \(\frac{0.01}{0.1}+\frac{0.1}{0.01}\)
