1. **State the problem:** Calculate the value of the expression $$n = (1 \times 3) + \left[1 \times 3 \times \frac{1}{4}\right] + \left(3 \times 3 \times \frac{1}{2}\right) + \left(2 \times 3 \times \frac{3}{4}\right) + (2 \times 4) + \left(3 \times 5 \times \frac{1}{4}\right)$$ 2. **Recall multiplication and fraction rules:** - Multiplying integers and fractions involves multiplying numerators and denominators. - Addition is performed after all multiplications are simplified. 3. **Calculate each term:** - First term: $1 \times 3 = 3$ - Second term: $1 \times 3 \times \frac{1}{4} = 3 \times \frac{1}{4} = \frac{3}{4}$ - Third term: $3 \times 3 \times \frac{1}{2} = 9 \times \frac{1}{2} = \frac{9}{2}$ - Fourth term: $2 \times 3 \times \frac{3}{4} = 6 \times \frac{3}{4} = \frac{18}{4}$ - Fifth term: $2 \times 4 = 8$ - Sixth term: $3 \times 5 \times \frac{1}{4} = 15 \times \frac{1}{4} = \frac{15}{4}$ 4. **Simplify fractions where possible:** - $\frac{18}{4} = \frac{\cancel{18}}{\cancel{4}} = \frac{9}{2}$ (dividing numerator and denominator by 2) 5. **Rewrite the expression with simplified terms:** $$n = 3 + \frac{3}{4} + \frac{9}{2} + \frac{9}{2} + 8 + \frac{15}{4}$$ 6. **Find a common denominator to add fractions:** - The denominators are 4 and 2; common denominator is 4. - Convert $\frac{9}{2} = \frac{18}{4}$ 7. **Rewrite all fractions with denominator 4:** $$n = 3 + \frac{3}{4} + \frac{18}{4} + \frac{18}{4} + 8 + \frac{15}{4}$$ 8. **Add all fractions:** $$\frac{3}{4} + \frac{18}{4} + \frac{18}{4} + \frac{15}{4} = \frac{3 + 18 + 18 + 15}{4} = \frac{54}{4}$$ 9. **Simplify $\frac{54}{4}$:** $$\frac{54}{4} = \frac{\cancel{54}}{\cancel{4}} = \frac{27}{2}$$ (dividing numerator and denominator by 2) 10. **Add integer terms:** $$3 + 8 = 11$$ 11. **Add integer sum and fraction sum:** $$n = 11 + \frac{27}{2} = \frac{22}{2} + \frac{27}{2} = \frac{49}{2}$$ 12. **Final answer:** $$n = \frac{49}{2} = 24.5$$