To calculate \((\frac{3}{4} + \frac{5}{12})(\frac{1}{7} + \frac{2}{3})\), we start by simplifying each expression within the parentheses.
-
Simplifying \(\frac{3}{4} + \frac{5}{12}\):
- Find a common denominator. The least common denominator (LCD) of 4 and 12 is 12.
- Convert \(\frac{3}{4}\) into twelfths: \[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]
- Now, add the fractions: \[ \frac{9}{12} + \frac{5}{12} = \frac{9 + 5}{12} = \frac{14}{12} \]
- Simplify \(\frac{14}{12}\): \[ \frac{14}{12} = \frac{7}{6} \]
-
Simplifying \(\frac{1}{7} + \frac{2}{3}\):
- Find a common denominator. The LCD of 7 and 3 is 21.
- Convert \(\frac{1}{7}\) into twenty-firsts: \[ \frac{1}{7} = \frac{1 \times 3}{7 \times 3} = \frac{3}{21} \]
- Convert \(\frac{2}{3}\) into twenty-firsts: \[ \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \]
- Now, add the fractions: \[ \frac{3}{21} + \frac{14}{21} = \frac{3 + 14}{21} = \frac{17}{21} \]
-
Multiplying \(\frac{7}{6}\) and \(\frac{17}{21}\): \[ \frac{7}{6} \times \frac{17}{21} = \frac{7 \times 17}{6 \times 21} \] Calculate \(7 \times 17 = 119\) and \(6 \times 21 = 126\): \[ = \frac{119}{126} \]
-
Lastly, simplify \(\frac{119}{126}\):
- The greatest common divisor (GCD) of 119 and 126 is 1 (since 119 is 7 times 17 and 126 is \(2 \times 3^2 \times 7\)).
- Therefore, \(\frac{119}{126}\) cannot be simplified further.
The final answer is: \[ \boxed{\frac{119}{126}} \]