	2/3 (1-5)
	3/4 (6-8)
	1/2(1/3-2)
	1/5(3+1/2)
	2/5 (1+3/4)
	3/4 [2-(1+3)+5]
	1/2 [4+2(1+5)-6]
	2/3[2-1/3 (1-5)-2]
	3{1/2-4[2-1/3 (1-5)-5}
	3/4{2/5-3[4+5(1-2)-1/3)+6
	5/6{1/2-1/4 [3-1/2 (1+3)]-1/5
	2/3{3/4+2/9[1/2-3/5](2-1/5)+2]


Let's simplify each expression step by step:

1. **\(\frac{2}{3}(1 - 5)\)**  
   \[
   1 - 5 = -4
   \]
   \[
   \frac{2}{3} \times (-4) = \frac{-8}{3}
   \]
   Answer: \(\frac{-8}{3}\)

2. **\(\frac{3}{4}(6 - 8)\)**  
   \[
   6 - 8 = -2
   \]
   \[
   \frac{3}{4} \times (-2) = \frac{-6}{4} = \frac{-3}{2}
   \]
   Answer: \(\frac{-3}{2}\)

3. **\(\frac{1}{2}\left(\frac{1}{3} - 2\right)\)**  
   \[
   \frac{1}{3} - 2 = \frac{1}{3} - \frac{6}{3} = \frac{-5}{3}
   \]
   \[
   \frac{1}{2} \times \frac{-5}{3} = \frac{-5}{6}
   \]
   Answer: \(\frac{-5}{6}\)

4. **\(\frac{1}{5}(3 + \frac{1}{2})\)**  
   \[
   3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}
   \]
   \[
   \frac{1}{5} \times \frac{7}{2} = \frac{7}{10}
   \]
   Answer: \(\frac{7}{10}\)

5. **\(\frac{2}{5}(1 + \frac{3}{4})\)**  
   \[
   1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}
   \]
   \[
   \frac{2}{5} \times \frac{7}{4} = \frac{14}{20} = \frac{7}{10}
   \]
   Answer: \(\frac{7}{10}\)

6. **\(\frac{3}{4}\left[2 - (1 + 3) + 5\right]\)**  
   \[
   1 + 3 = 4
   \]
   \[
   2 - 4 + 5 = 3
   \]
   \[
   \frac{3}{4} \times 3 = \frac{9}{4}
   \]
   Answer: \(\frac{9}{4}\)

7. **\(\frac{1}{2}\left[4 + 2(1 + 5) - 6\right]\)**  
   \[
   1 + 5 = 6
   \]
   \[
   2 \times 6 = 12
   \]
   \[
   4 + 12 - 6 = 10
   \]
   \[
   \frac{1}{2} \times 10 = 5
   \]
   Answer: 5

8. **\(\frac{2}{3}\left[2 - \frac{1}{3}(1 - 5) - 2\right]\)**  
   \[
   1 - 5 = -4
   \]
   \[
   \frac{1}{3} \times (-4) = \frac{-4}{3}
   \]
   \[
   2 - \frac{-4}{3} - 2 = 0 + \frac{4}{3} = \frac{4}{3}
   \]
   \[
   \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}
   \]
   Answer: \(\frac{8}{9}\)

9. **\(3\left\{\frac{1}{2} - 4\left[2 - \frac{1}{3}(1 - 5) - 5\right]\right\}\)**  
   First, handle the inner expression:
   \[
   1 - 5 = -4
   \]
   \[
   \frac{1}{3} \times (-4) = \frac{-4}{3}
   \]
   \[
   2 - \frac{-4}{3} - 5 = 2 + \frac{4}{3} - 5 = \frac{6}{3} + \frac{4}{3} - \frac{15}{3} = \frac{-5}{3}
   \]
   Now,
   \[
   4 \times \frac{-5}{3} = \frac{-20}{3}
   \]
   \[
   \frac{1}{2} - \frac{-20}{3} = \frac{1}{2} + \frac{20}{3} = \frac{3}{6} + \frac{40}{6} = \frac{43}{6}
   \]
   Finally,
   \[
   3 \times \frac{43}{6} = \frac{129}{6} = \frac{43}{2}
   \]
   Answer: \(\frac{43}{2}\)

10. **\(\frac{3}{4}\left\{\frac{2}{5} - 3\left[4 + 5(1 - 2) - \frac{1}{3}\right] + 6\right\}\)**  
   This expression has several steps, so I'll break it down in parts. Let's start with the innermost expression:
   \[
   1 - 2 = -1
   \]
   \[
   5 \times (-1) = -5
   \]
   \[
   4 + (-5) = -1
   \]
   \[
   -1 - \frac{1}{3} = -1 - \frac{1}{3} = \frac{-3}{3} - \frac{1}{3} = \frac{-4}{3}
   \]
   Now, moving outward:
   \[
   3 \times \frac{-4}{3} = -4
   \]
   \[
   \frac{2}{5} - (-4) + 6 = \frac{2}{5} + 4 + 6 = \frac{2}{5} + 10
   \]
   Converting \(10\) to a fraction with denominator 5:
   \[
   \frac{2}{5} + \frac{50}{5} = \frac{52}{5}
   \]
   Finally,
   \[
   \frac{3}{4} \times \frac{52}{5} = \frac{156}{20} = \frac{39}{5}
   \]
   Answer: \(\frac{39}{5}\)

11. **\(\frac{5}{6}\left\{\frac{1}{2} - \frac{1}{4}\left[3 - \frac{1}{2}(1 + 3)\right] - \frac{1}{5}\right\}\)**  
   First, simplify the inner terms:
   \[
   1 + 3 = 4
   \]
   \[
   \frac{1}{2} \times 4 = 2
   \]
   \[
   3 - 2 = 1
   \]
   Now,
   \[
   \frac{1}{4} \times 1 = \frac{1}{4}
   \]
   \[
   \frac{1}{2} - \frac{1}{4} - \frac{1}{5} = \frac{5}{10} - \frac{2.5}{10} = <ratio

2/3 (1-5) 3/4 (6-8) 1/2(1/3-2) 1/5(3+1/2) 2/5 (1+3/4) 3/4 [2-(1+3)+5] 1/2 [4+2(1+5)-6] 2/3[2-1/3 (1-5)-2] 3{1/2-4[2-1/3 (1-5)-5} 3/4{2/5-3[4+5(1-2)-1/3)+6 5/6{1/2-1/4 [3-1/2 (1+3)]-1/5 2/3{3/4+2/9[1/2-3/5](2-1/5)+2]

Learn how to simplify complex fractions and algebraic expressions with step-by-step explanations. This problem covers key mathematical concepts such as the order of operations, multiplication, addition, and subtraction of fractions. Suitable for students in Grades 9-12.

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