（2,48+ 3 5/9 *2 1/8 :(6.1-3.7）

Let's break down the expression step by step. The expression you provided is:

\[ 2 + 48 + \left( 3 \frac{5}{9} \times 2 \frac{1}{8} \right) : (6.1 - 3.7) \]

Step 1: Simplify \(6.1 - 3.7\)

\[ 6.1 - 3.7 = 2.4 \]

Step 2: Work out the fractional multiplication \(3 \frac{5}{9} \times 2 \frac{1}{8}\)

First, convert the mixed numbers to improper fractions:

• \(3 \frac{5}{9} = \frac{3 \times 9 + 5}{9} = \frac{27 + 5}{9} = \frac{32}{9}\)
• \(2 \frac{1}{8} = \frac{2 \times 8 + 1}{8} = \frac{16 + 1}{8} = \frac{17}{8}\)

Now, multiply the two fractions together:

\[ \frac{32}{9} \times \frac{17}{8} = \frac{32 \times 17}{9 \times 8} = \frac{544}{72} \]

We can simplify \( \frac{544}{72} \) by finding the greatest common divisor (GCD) of 544 and 72, which is 8:

\[ \frac{544 \div 8}{72 \div 8} = \frac{68}{9} \]

Step 3: Divide by \(2.4\)

Now we need to divide \( \frac{68}{9} \) by \(2.4\):

Convert \(2.4\) to a fraction:

\[ 2.4 = \frac{24}{10} = \frac{12}{5} \]

Now, perform the division:

\[ \frac{68}{9} \div \frac{12}{5} = \frac{68}{9} \times \frac{5}{12} = \frac{68 \times 5}{9 \times 12} = \frac{340}{108} \]

We can simplify \( \frac{340}{108} \):

The GCD of 340 and 108 is 4:

\[ \frac{340 \div 4}{108 \div 4} = \frac{85}{27} \]

Step 4: Combine Everything

Now substitute back into the original expression:

\[ 2 + 48 + \frac{85}{27} \]

First, add \(2 + 48\):

\[ 2 + 48 = 50 \]

Now express 50 as a fraction with a common denominator of 27 for easy addition:

\[ 50 = \frac{50 \times 27}{27} = \frac{1350}{27} \]

Now, add \(\frac{1350}{27} + \frac{85}{27}\):

\[ \frac{1350 + 85}{27} = \frac{1435}{27} \]

Final Result

So the final result of the expression \(2 + 48 + 3 \frac{5}{9} \times 2 \frac{1}{8} : (6.1 - 3.7)\) is:

\[ \frac{1435}{27} \]

If you want a mixed number or decimal, you can further calculate:

\[ \frac{1435}{27} \approx 53.52 \]

Hence, the final answer can be expressed in fractions or decimals as needed.