\( + \frac { 2 } { 5 } + \frac { 1 } { 3 } \quad \frac { 105 + 60 + 5 e } { 150 } ( \frac { 215 } { 750 } ) = 5 ( \frac { 43 } { 36 } ) \)

To solve the equation \[ + \frac{2}{5} + \frac{1}{3} \quad \frac{105 + 60 + 5e}{150} \left( \frac{215}{750} \right) = 5 \left( \frac{43}{36} \right), \] we will first simplify each part step by step. 1. **Simplifying the left side:** We start with the fractions \( \frac{2}{5} \) and \( \frac{1}{3} \). To add these fractions, we need a common denominator. The least common multiple of 5 and 3 is 15. \[ \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \] \[ \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \] Now we can add them: \[ \frac{6}{15} + \frac{5}{15} = \frac{11}{15} \] 2. **Simplifying the expression \( \frac{105 + 60 + 5e}{150} \):** First, we calculate \( 105 + 60 + 5e \): \[ 105 + 60 = 165 \quad \Rightarrow \quad 165 + 5e = 165 + 5e \] Now we can write: \[ \frac{165 + 5e}{150} \] 3. **Calculating \( \frac{215}{750} \):** We can simplify \( \frac{215}{750} \) by finding the greatest common divisor (GCD) of 215 and 750. The GCD is 5. \[ \frac{215 \div 5}{750 \div 5} = \frac{43}{150} \] 4. **Combining the two parts:** Now we multiply \( \frac{165 + 5e}{150} \) by \( \frac{43}{150} \): \[ \frac{165 + 5e}{150} \cdot \frac{43}{150} = \frac{(165 + 5e) \cdot 43}{150 \cdot 150} = \frac{43(165 + 5e)}{22500} \] 5. **Setting the left side equal to the right side:** The right side of the equation is: \[ 5 \left( \frac{43}{36} \right) = \frac{215}{36} \] Now we have: \[ \frac{11}{15} + \frac{43(165 + 5e)}{22500} = \frac{215}{36} \] 6. **Finding a common denominator for the left side:** The common denominator for \( 15 \) and \( 22500 \) is \( 22500 \): \[ \frac{11}{15} = \frac{11 \cdot 1500}{15 \cdot 1500} = \frac{16500}{22500} \] Now we can rewrite the left side: \[ \frac{16500 + 43(165 + 5e)}{22500} = \frac{215}{36} \] 7. **Cross-multiplying to eliminate the fractions:** \[ 16500 + 43(165 + 5e) = \frac{215}{36} \cdot 22500 \] Calculating the right side: \[ \frac{215 \cdot 22500}{36} = 215 \cdot 625 = 134375 \] So we have: \[ 16500 + 43(165 + 5e) = 134375 \] 8. **Solving for \( e \):** Rearranging gives: \[ 43(165 + 5e) = 134375 - 16500 \] \[ 43(165 + 5e) = 117875 \] \[ 165 + 5e = \frac{117875}{43} = 2745 \] \[ 5e = 2745 - 165 = 2580 \] \