To solve the problem [tex]\(\frac{5}{11} \times\left(\frac{3}{4}-\frac{1}{5}\right)\)[/tex], we need to follow a series of steps to simplify the expression. Let's break it down step-by-step:
### Step 1: Simplify the Expression Inside the Parentheses
First, we need to evaluate the expression inside the parentheses: [tex]\(\frac{3}{4} - \frac{1}{5}\)[/tex].
To subtract these fractions, we need a common denominator. The denominators are 4 and 5. The least common multiple (LCM) of 4 and 5 is 20. We will convert both fractions to have a common denominator of 20:
[tex]\[
\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}
\][/tex]
[tex]\[
\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}
\][/tex]
Now we can subtract these fractions:
[tex]\[
\frac{15}{20} - \frac{4}{20} = \frac{15 - 4}{20} = \frac{11}{20}
\][/tex]
### Step 2: Multiply the Simplified Expression by [tex]\(\frac{5}{11}\)[/tex]
Now, we need to multiply [tex]\(\frac{5}{11}\)[/tex] by [tex]\(\frac{11}{20}\)[/tex]:
[tex]\[
\frac{5}{11} \times \frac{11}{20}
\][/tex]
When multiplying fractions, we multiply the numerators together and the denominators together:
[tex]\[
\text{Numerators: } 5 \times 11 = 55
\][/tex]
[tex]\[
\text{Denominators: } 11 \times 20 = 220
\][/tex]
So the product is:
[tex]\[
\frac{55}{220}
\][/tex]
### Step 3: Simplify the Resulting Fraction
Finally, we simplify the fraction [tex]\(\frac{55}{220}\)[/tex]. To do this, we need to find the greatest common divisor (GCD) of 55 and 220.
The GCD of 55 and 220 is 55. We divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{55 \div 55}{220 \div 55} = \frac{1}{4}
\][/tex]
### Final Answer
So, after simplifying, the proper fraction is:
[tex]\[
\boxed{\frac{1}{4}}
\][/tex]
