Calculate the following expression: $\frac{2}{5} + \frac{1}{3} - \frac{7}{10} + \frac{3}{4} + \frac{1}{6} - \frac{11}{30} =$

To calculate the expression:

$\frac{2}{5} + \frac{1}{3} - \frac{7}{10} + \frac{3}{4} + \frac{1}{6} - \frac{11}{30}$

we will follow these steps:

Step 1: Find a Common Denominator

First, identify the least common denominator (LCD) for all the fractions. The denominators are: 5, 3, 10, 4, 6, and 30.

• The prime factors:
  
  
  
  
  
  • 5: $5$
  
  
  
  • 3: $3$
  
  
  
  • 10: $2 \times 5$
  
  
  
  • 4: $2^2$
  
  
  
  • 6: $2 \times 3$
  
  
  
  • 30: $2 \times 3 \times 5$
  
  
  
  
  
  

The LCD is the product of the highest powers of all primes present:

$2^2 \times 3 \times 5 = 60$

Step 2: Convert Each Fraction to Have the LCD

Convert each fraction to an equivalent fraction with a denominator of 60:

$\frac{2}{5} = \frac{2 \times 12}{5 \times 12} = \frac{24}{60}$

$\frac{1}{3} = \frac{1 \times 20}{3 \times 20} = \frac{20}{60}$