Johnston Bowen
07/23/2024 · Primary School
The value of \( 2 . \overline{6}-0 . \overline{9} \) is \( \begin{array}{ll}\text { (a) } 4 & \text { (b) } \frac{1}{4}\end{array} \)
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Tutor-Verified Answer
Step-by-step Solution
To solve the expression \( 2 . \overline{6} - 0 . \overline{9} \), we first need to convert the repeating decimals into fractions.
1. **Convert \( 2 . \overline{6} \) to a fraction:**
Let \( x = 2 . \overline{6} \).
This can be expressed as:
\[
x = 2.66666\ldots
\]
To eliminate the repeating part, we can multiply by 10:
\[
10x = 26.66666\ldots
\]
Now, we subtract the original \( x \) from this equation:
\[
10x - x = 26.66666\ldots - 2.66666\ldots
\]
\[
9x = 24
\]
\[
x = \frac{24}{9} = \frac{8}{3}
\]
Thus, \( 2 . \overline{6} = \frac{8}{3} \).
2. **Convert \( 0 . \overline{9} \) to a fraction:**
Let \( y = 0 . \overline{9} \).
This can be expressed as:
\[
y = 0.99999\ldots
\]
To eliminate the repeating part, we can multiply by 10:
\[
10y = 9.99999\ldots
\]
Now, we subtract the original \( y \) from this equation:
\[
10y - y = 9.99999\ldots - 0.99999\ldots
\]
\[
9y = 9
\]
\[
y = 1
\]
Thus, \( 0 . \overline{9} = 1 \).
3. **Now we can compute \( 2 . \overline{6} - 0 . \overline{9} \):**
\[
2 . \overline{6} - 0 . \overline{9} = \frac{8}{3} - 1
\]
To perform the subtraction, we convert \( 1 \) to a fraction with a denominator of 3:
\[
1 = \frac{3}{3}
\]
Now we can subtract:
\[
\frac{8}{3} - \frac{3}{3} = \frac{8 - 3}{3} = \frac{5}{3}
\]
Thus, the value of \( 2 . \overline{6} - 0 . \overline{9} \) is \( \frac{5}{3} \).
Since neither of the provided options (4 or \( \frac{1}{4} \)) matches \( \frac{5}{3} \), it seems there may be an error in the options given. However, the correct answer is:
\[
\frac{5}{3}
\]
Quick Answer
The value is \( \frac{5}{3} \).
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