Sure! Let's solve this problem step-by-step.
We first need to express both repeating decimals as fractions.
### Step 1: Convert [tex]\( 0.\overline{54} \)[/tex] to a fraction
Let [tex]\( x = 0.\overline{54} \)[/tex].
Since [tex]\( 0.\overline{54} \)[/tex] means [tex]\( 0.54545454 \ldots \)[/tex], we can write:
[tex]\[ 100x = 54.545454 \ldots \][/tex]
Now, subtract the original [tex]\( x \)[/tex] from this:
[tex]\[ 100x - x = 54.545454 \ldots - 0.545454 \ldots \][/tex]
[tex]\[ 99x = 54 \][/tex]
[tex]\[ x = \frac{54}{99} \][/tex]
We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 54 and 99 is 9:
[tex]\[ x = \frac{54 \div 9}{99 \div 9} = \frac{6}{11} \][/tex]
So, [tex]\( 0.\overline{54} = \frac{6}{11} \)[/tex].
### Step 2: Convert [tex]\( 0.\overline{5} \)[/tex] to a fraction
Let [tex]\( y = 0.\overline{5} \)[/tex].
Since [tex]\( 0.\overline{5} \)[/tex] means [tex]\( 0.555555 \ldots \)[/tex], we can write:
[tex]\[ 10y = 5.555555 \ldots \][/tex]
Now, subtract the original [tex]\( y \)[/tex] from this:
[tex]\[ 10y - y = 5.555555 \ldots - 0.555555 \ldots \][/tex]
[tex]\[ 9y = 5 \][/tex]
[tex]\[ y = \frac{5}{9} \][/tex]
So, [tex]\( 0.\overline{5} = \frac{5}{9} \)[/tex].
### Step 3: Multiply the fractions
Now we need to find the product of [tex]\( \frac{6}{11} \)[/tex] and [tex]\( \frac{5}{9} \)[/tex]:
[tex]\[ \frac{6}{11} \times \frac{5}{9} = \frac{6 \times 5}{11 \times 9} = \frac{30}{99} \][/tex]
### Step 4: Simplify the product
We simplify [tex]\( \frac{30}{99} \)[/tex] by dividing both the numerator and the denominator by their GCD. The GCD of 30 and 99 is 3:
[tex]\[ \frac{30}{99} = \frac{30 \div 3}{99 \div 3} = \frac{10}{33} \][/tex]
So, the value of [tex]\( 0.\overline{54} \times 0.\overline{5} \)[/tex] is [tex]\( \frac{10}{33} \)[/tex].
Hence, the answer is:
[tex]\[ \boxed{\frac{10}{33}} \][/tex]
