6,8*5/13*2,7*3 3/25:5 2/5 : 1/175
Let's carefully evaluate the given expression step by step:
Expression:
\[ 6.8 \times 5 / 13 \times 2.7 \times 3 \, \frac{3}{25} : 5 \, \frac{2}{5} : \frac{1}{175} \]
Step 1: Convert mixed numbers to improper fractions for clarity
- \( 3 \frac{3}{25} = \frac{3 \times 25 + 3}{25} = \frac{75 + 3}{25} = \frac{78}{25} \)
- \( 2 \frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5} \)
Step 2: Rewrite the expression
\[ 6.8 \times 5 \div 13 \times 2.7 \times 3 \times \frac{78}{25} \div \frac{12}{5} \div \frac{1}{175} \]
Note: The colons (:) denote division.
Rearranged with explicit division signs:
\[ (6.8 \times 5) \div 13 \times 2.7 \times 3 \times \frac{78}{25} \div \frac{12}{5} \div \frac{1}{175} \]
Step 3: Simplify step by step
a) Calculate \(6.8 \times 5\):
\[ 6.8 \times 5 = 34 \]
b) Now, divide by 13:
\[ 34 \div 13 = \frac{34}{13} \]
c) Multiply by 2.7:
\[ \frac{34}{13} \times 2.7 = \frac{34 \times 2.7}{13} \]
Calculate numerator:
\[ 34 \times 2.7 = (34 \times 2) + (34 \times 0.7) = 68 + 23.8 = 91.8 \]
So:
\[ \frac{91.8}{13} \]
d) Multiply by 3:
\[ \frac{91.8}{13} \times 3 = \frac{91.8 \times 3}{13} = \frac{275.4}{13} \]
e) Multiply by \(\frac{78}{25}\):
\[ \frac{275.4}{13} \times \frac{78}{25} = \frac{275.4 \times 78}{13 \times 25} \]
Calculate numerator:
\[ 275.4 \times 78 \]
Let's compute this:
- \( 275.4 \times 78 = (275.4 \times 80) - (275.4 \times 2) \)
\[ 275.4 \times 80 = 22,032 \] \[ 275.4 \times 2 = 550.8 \]
Subtract:
\[ 22,032 - 550.8 = 21,481.2 \]
Denominator:
\[ 13 \times 25 = 325 \]
So, current numerator/denominator:
\[ \frac{21,481.2}{325} \]
f) Divide by \(\frac{12}{5}\):
Dividing by a fraction is multiplying by its reciprocal:
\[ \frac{21,481.2}{325} \times \frac{5}{12} = \frac{21,481.2 \times 5}{325 \times 12} \]
Calculate numerator:
\[ 21,481.2 \times 5 = 107,406 \]
Calculate denominator:
\[ 325 \times 12 = 3,900 \]
Now, we have:
\[ \frac{107,406}{3,900} \]
g) Divide by \(\frac{1}{175}\):
Again, dividing by a fraction is multiplying by its reciprocal:
\[ \frac{107,406}{3,900} \times 175 = \frac{107,406 \times 175}{3,900} \]
Calculate numerator:
\[ 107,406 \times 175 \]
Break down:
- \(107,406 \times 100 = 10,740,600\)
- \(107,406 \times 75 = 107,406 \times (50 + 25) = 5,370,300 + 2,685,150 = 8,055,450\)
Sum:
\[ 10,740,600 + 8,055,450 = 18,796,050 \]
Now, numerator = 18,796,050
Division:
\[ \frac{18,796,050}{3,900} \]
Simplify:
Divide numerator and denominator by 150:
- numerator: \( 18,796,050 \div 150 = 125,307 \)
- denominator: \( 3,900 \div 150 = 26 \)
Thus:
\[ \frac{125,307}{26} \]
Finally, divide numerator by denominator:
\[ 125,307 \div 26 \approx 4,824.9 \]
Final Answer:
\[ \boxed{approximate \, value \approx 4,824.9} \]
So, the value of the entire expression is approximately 4,824.9.
