The value of \(0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7 \) is:
Understanding Recurring Decimals in Mathematics
Recurring decimals, also known as repeating decimals, are rational numbers that, when expressed in decimal form, have a sequence of digits that repeats infinitely. For example, \(1/3 = 0.333...\) is written as \(0.\overline 3\), and \(1/7 = 0.142857142857...\) is written as \(0.\overline {142857}\). A mixed recurring decimal has a non-repeating part and a repeating part, like \(0.4\overline 6\).
Converting Recurring Decimals to Fractions
To solve the given expression, the first step is to convert each recurring decimal into its equivalent fractional form (p/q). There are standard methods for this conversion.
- Converting \(0.4\overline 6\):
Let \(x = 0.4\overline 6 = 0.4666...\)
Multiply by 10 to shift the non-repeating part: \(10x = 4.666...\) (Equation 1)
Multiply by 100 to shift the repeating part one cycle past the decimal: \(100x = 46.666...\) (Equation 2)
Subtract Equation 1 from Equation 2:
\(100x - 10x = 46.666... - 4.666...\)
\(90x = 42\)
\(x = \frac{42}{90} = \frac{7}{15}\) - Converting \(0.7\overline {23}\):
Let \(y = 0.7\overline {23} = 0.7232323...\)
Multiply by 10 to shift the non-repeating part: \(10y = 7.232323...\) (Equation 3)
Multiply by 1000 (10 with power equal to total digits before and in repeating part) to shift the repeating part one cycle past the decimal: \(1000y = 723.2323...\) (Equation 4)
Subtract Equation 3 from Equation 4:
\(1000y - 10y = 723.2323... - 7.2323...\)
\(990y = 716\)
\(y = \frac{716}{990} = \frac{358}{495}\) - Converting \(0.3\overline 9\):
Let \(z = 0.3\overline 9 = 0.3999...\)
Multiply by 10: \(10z = 3.999...\) (Equation 5)
Multiply by 100: \(100z = 39.999...\) (Equation 6)
Subtract Equation 5 from Equation 6:
\(100z - 10z = 39.999... - 3.999...\)
\(90z = 36\)
\(z = \frac{36}{90} = \frac{2}{5}\)
Alternatively, \(0.\overline 9 = 1\), so \(0.3\overline 9 = 0.3 + 0.0\overline 9 = 0.3 + 0.1 = 0.4 = \frac{4}{10} = \frac{2}{5}\). - Converting \(0.\overline 7\):
Let \(w = 0.\overline 7 = 0.777...\)
Multiply by 10: \(10w = 7.777...\) (Equation 7)
Subtract the original equation from Equation 7:
\(10w - w = 7.777... - 0.777...\)
\(9w = 7\)
\(w = \frac{7}{9}\)
Evaluating the Mathematical Expression
Now, substitute the fractional values back into the original expression:
\(0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7 = \frac{7}{15} + \frac{358}{495} - \frac{2}{5} \times \frac{7}{9}\)
Following the order of operations (BODMAS/PEMDAS), perform the multiplication first:
\(\frac{2}{5} \times \frac{7}{9} = \frac{2 \times 7}{5 \times 9} = \frac{14}{45}\)
Now, the expression becomes:
\(\frac{7}{15} + \frac{358}{495} - \frac{14}{45}\)
To add and subtract these fractions, find a common denominator. The denominators are 15, 495, and 45. The least common multiple (LCM) of 15, 45, and 495 is 495.
| Fraction | Convert to Denominator 495 |
|---|---|
| \(\frac{7}{15}\) | \(\frac{7 \times (495 \div 15)}{15 \times (495 \div 15)} = \frac{7 \times 33}{15 \times 33} = \frac{231}{495}\) |
| \(\frac{358}{495}\) | \(\frac{358}{495}\) (Already has the denominator) |
| \(\frac{14}{45}\) | \(\frac{14 \times (495 \div 45)}{45 \times (495 \div 45)} = \frac{14 \times 11}{45 \times 11} = \frac{154}{495}\) |
Substitute the equivalent fractions back into the expression:
\(\frac{231}{495} + \frac{358}{495} - \frac{154}{495}\)
Now, perform the addition and subtraction:
\(\frac{231 + 358 - 154}{495} = \frac{589 - 154}{495} = \frac{435}{495}\)
Simplifying the Fraction and Converting Back to Decimal
The resulting fraction is \(\frac{435}{495}\). This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor (GCD). Both numbers are divisible by 5:
\(\frac{435 \div 5}{495 \div 5} = \frac{87}{99}\)
To convert the fraction \(\frac{87}{99}\) back to a decimal, divide 87 by 99. Fractions with a denominator of 9, 99, 999, etc., directly correspond to repeating decimals:
\(\frac{87}{99} = 0.\overline {87}\)
Conclusion
The value of the expression \(0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7 \) is \(0.\overline {87}\).
Revision Table: Recurring Decimal Conversion
| Recurring Decimal | Fraction Form | Method Hint |
|---|---|---|
| \(0.\overline d\) | \(\frac{d}{9}\) | Single repeating digit |
| \(0.\overline {dd}\) | \(\frac{dd}{99}\) | Two repeating digits |
| \(0.a\overline b\) | \(\frac{ab - a}{90}\) | One non-repeating, one repeating |
| \(0.ab\overline c\) | \(\frac{abc - ab}{900}\) | Two non-repeating, one repeating |
| \(0.a\overline {bc}\) | \(\frac{abc - a}{990}\) | One non-repeating, two repeating |
Additional Information: Operations with Fractions
When adding or subtracting fractions, it is essential to find a common denominator. The least common multiple (LCM) is usually the most efficient common denominator. Once fractions share a common denominator, add or subtract the numerators and keep the denominator the same.
For example:
\(\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{bc}{bd} = \frac{ad + bc}{bd}\)
\(\frac{a}{b} - \frac{c}{d} = \frac{ad}{bd} - \frac{bc}{bd} = \frac{ad - bc}{bd}\)
When multiplying fractions, multiply the numerators together and the denominators together:
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{ac}{bd}\)
Simplifying fractions to their lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD) is good practice.
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 \(\frac{48.3\times[(4.95)^2+4.95\times13.25]}{[(12.55)^2-(5.65)^2]\times19.8} \) is:
Find the value of (1.6) 3 - (0.9) 3 - (0.7) 3.
What is the value of x, if \(5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}\) ?
The value of \(11.\overline{4}\) + \(22.5\overline{67}\) – \(33.5\overline{9}\) is:
