Let's solve the expression step-by-step:
[tex]\[
\frac{3}{20} + \frac{9}{20} \times \frac{35}{27}
\][/tex]
### Step 1: Multiply the Fractions
First, we handle the multiplication.
[tex]\[
\frac{9}{20} \times \frac{35}{27}
\][/tex]
To multiply fractions, we multiply the numerators together and the denominators together.
- Numerators: [tex]\(9 \times 35 = 315\)[/tex]
- Denominators: [tex]\(20 \times 27 = 540\)[/tex]
So, the product is:
[tex]\[
\frac{315}{540}
\][/tex]
We simplify [tex]\(\frac{315}{540}\)[/tex] by finding the greatest common divisor (GCD) of 315 and 540 and then dividing both the numerator and the denominator by this GCD.
The factors of 315 are: [tex]\(1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315\)[/tex]
The factors of 540 are: [tex]\(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540\)[/tex]
The GCD of 315 and 540 is 45.
[tex]\[
\frac{315 \div 45}{540 \div 45} = \frac{7}{12}
\][/tex]
### Step 2: Add the Fractions
Now we add [tex]\(\frac{3}{20}\)[/tex] and [tex]\(\frac{7}{12}\)[/tex]. To add these fractions, we need a common denominator.
The least common multiple (LCM) of 20 and 12 is 60.
Convert both fractions to have the common denominator 60:
- [tex]\(\frac{3}{20}\)[/tex] becomes [tex]\(\frac{3 \times 3}{20 \times 3} = \frac{9}{60}\)[/tex]
- [tex]\(\frac{7}{12}\)[/tex] becomes [tex]\(\frac{7 \times 5}{12 \times 5} = \frac{35}{60}\)[/tex]
Now add these fractions:
[tex]\[
\frac{9}{60} + \frac{35}{60} = \frac{9 + 35}{60} = \frac{44}{60}
\][/tex]
### Step 3: Simplify the Resulting Fraction
Simplify [tex]\(\frac{44}{60}\)[/tex] by finding the greatest common divisor (GCD) of 44 and 60 and then dividing both the numerator and the denominator by this GCD.
The factors of 44 are: [tex]\(1, 2, 4, 11, 22, 44\)[/tex]
The factors of 60 are: [tex]\(1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\)[/tex]
The GCD of 44 and 60 is 4.
[tex]\[
\frac{44 \div 4}{60 \div 4} = \frac{11}{15}
\][/tex]
So, the simplified form of the expression [tex]\(\frac{3}{20} + \frac{9}{20} \times \frac{35}{27}\)[/tex] is:
[tex]\[
\boxed{\frac{11}{15}}
\][/tex]
