Math worksheet for evaluating expressions with rational numbers, designed for educational practice.

Here are the step-by-step solutions for each problem on the worksheet.

1) $\left(\frac{2}{5} \times \frac{-1}{2}\right) \div 2 - \frac{-11}{9} \div \frac{-3}{2}$
* First part: $\frac{2}{5} \times \frac{-1}{2} = \frac{-2}{10} = \frac{-1}{5}$. Then divide by 2: $\frac{-1}{5} \div 2 = \frac{-1}{10}$.
* Second part: $\frac{-11}{9} \div \frac{-3}{2} = \frac{-11}{9} \times \frac{2}{-3} = \frac{-22}{-27} = \frac{22}{27}$.
* Subtract: $\frac{-1}{10} - \frac{22}{27}$. Common denominator is 270.
* $\frac{-27}{270} - \frac{220}{270} = \frac{-247}{270}$.

2) $\left(\frac{-16}{9} + \frac{1}{5}\right)^2 - \left(\frac{11}{8} - \frac{-1}{10}\right)$
* Inside first bracket: $\frac{-16}{9} + \frac{1}{5} = \frac{-80}{45} + \frac{9}{45} = \frac{-71}{45}$.
* Square it: $\left(\frac{-71}{45}\right)^2 = \frac{5041}{2025}$.
* Inside second bracket: $\frac{11}{8} + \frac{1}{10} = \frac{55}{40} + \frac{4}{40} = \frac{59}{40}$.
* Subtract: $\frac{5041}{2025} - \frac{59}{40}$. Common denominator is 16,200.
* $\frac{40328}{16200} - \frac{23895}{16200} = \frac{16433}{16200}$.

3) $\frac{-8}{5} \div \frac{-4}{3} \times 1 \div \frac{1}{7} - \frac{4}{3}$
* Division/Multiplication left to right:
* $\frac{-8}{5} \div \frac{-4}{3} = \frac{-8}{5} \times \frac{3}{-4} = \frac{24}{20} = \frac{6}{5}$.
* $\frac{6}{5} \times 1 = \frac{6}{5}$.
* $\frac{6}{5} \div \frac{1}{7} = \frac{6}{5} \times 7 = \frac{42}{5}$.
* Subtract: $\frac{42}{5} - \frac{4}{3}$. Common denominator is 15.
* $\frac{126}{15} - \frac{20}{15} = \frac{106}{15}$.

4) $\frac{-11}{10} - 2 + \frac{9}{10} - \frac{6}{5} - \frac{1}{2}$
* Combine fractions with denominator 10: $\frac{-11}{10} + \frac{9}{10} = \frac{-2}{10} = \frac{-1}{5}$.
* Now we have: $\frac{-1}{5} - 2 - \frac{6}{5} - \frac{1}{2}$.
* Combine fifths: $\frac{-1}{5} - \frac{6}{5} = \frac{-7}{5}$.
* Now: $\frac{-7}{5} - 2 - \frac{1}{2}$. Common denominator is 10.
* $\frac{-14}{10} - \frac{20}{10} - \frac{5}{10} = \frac{-39}{10}$.

5) $\frac{-6}{5}\left(\frac{-9}{7} + 2 \div \frac{1}{2} - \frac{-4}{7}\right)$
* Inside parenthesis first (division): $2 \div \frac{1}{2} = 4$.
* Expression becomes: $\frac{-9}{7} + 4 + \frac{4}{7}$.
* Combine sevenths: $\frac{-9}{7} + \frac{4}{7} = \frac{-5}{7}$.
* Add 4: $4 - \frac{5}{7} = \frac{28}{7} - \frac{5}{7} = \frac{23}{7}$.
* Multiply by outside term: $\frac{-6}{5} \times \frac{23}{7} = \frac{-138}{35}$.

6) $\frac{-3}{2} \times \left(\frac{-1}{2} - 9\right)\left(\frac{-7}{4} - \frac{7}{8}\right)$
* First bracket: $\frac{-1}{2} - \frac{18}{2} = \frac{-19}{2}$.
* Second bracket: $\frac{-14}{8} - \frac{7}{8} = \frac{-21}{8}$.
* Multiply all three: $\frac{-3}{2} \times \frac{-19}{2} \times \frac{-21}{8}$.
* Numerator: $(-3)(-19)(-21) = -1197$.
* Denominator: $2 \times 2 \times 8 = 32$.
* Result: $\frac{-1197}{32}$.

7) $\frac{-6}{7} \times -10 \times \left(\frac{13}{10} + \frac{7}{5}\right) \div \frac{-8}{5}$
* Inside parenthesis: $\frac{13}{10} + \frac{14}{10} = \frac{27}{10}$.
* Rewrite expression: $\frac{-6}{7} \times \frac{-10}{1} \times \frac{27}{10} \times \frac{5}{-8}$ (flip last fraction).
* Simplify: The 10s cancel out. We have $\frac{-6}{7} \times -27 \times \frac{5}{-8}$.
* Numerator: $(-6)(-27)(5) = 810$. Wait, let's check signs. Two negatives make positive, times negative is negative. So numerator is $-810$.
* Denominator: $7 \times 8 = 56$.
* Fraction: $\frac{-810}{56}$. Divide top and bottom by 2: $\frac{-405}{28}$.

8) $\left(\frac{-5}{3}\right)^2 \times \left(\frac{-2}{3} - 2\right) \div \frac{1}{2}$
* Square: $\left(\frac{-5}{3}\right)^2 = \frac{25}{9}$.
* Bracket: $\frac{-2}{3} - \frac{6}{3} = \frac{-8}{3}$.
* Divide by $\frac{1}{2}$ is multiply by 2.
* Expression: $\frac{25}{9} \times \frac{-8}{3} \times 2$.
* Numerator: $25 \times -8 \times 2 = -400$.
* Denominator: $9 \times 3 = 27$.
* Result: $\frac{-400}{27}$.

9) $\left(\frac{-4}{3}\right)^2 \times \left(\frac{-2}{3} - \frac{5}{3}\right) \div \frac{-3}{2}$
* Square: $\frac{16}{9}$.
* Bracket: $\frac{-7}{3}$.
* Divide by $\frac{-3}{2}$ is multiply by $\frac{-2}{3}$.
* Expression: $\frac{16}{9} \times \frac{-7}{3} \times \frac{-2}{3}$.
* Numerator: $16 \times (-7) \times (-2) = 224$.
* Denominator: $9 \times 3 \times 3 = 81$.
* Result: $\frac{224}{81}$.

10) $\left(\frac{5}{7} - \left(\frac{10}{9} - \frac{9}{7}\right)\right) \div (1 - -1)$
* Inner bracket: $\frac{10}{9} - \frac{9}{7}$. Common denom 63. $\frac{70}{63} - \frac{81}{63} = \frac{-11}{63}$.
* Outer bracket: $\frac{5}{7} - (\frac{-11}{63}) = \frac{5}{7} + \frac{11}{63}$.
* Convert $\frac{5}{7}$ to $\frac{45}{63}$. Sum: $\frac{45+11}{63} = \frac{56}{63}$. This simplifies to $\frac{8}{9}$.
* Divisor: $1 - (-1) = 2$.
* Final: $\frac{8}{9} \div 2 = \frac{4}{9}$.

11) $\left(\frac{-3}{2} \times \frac{-11}{7} \times -1\right) \div \left(\frac{-7}{4} - 2\right)$
* Numerator part: $\frac{-3}{2} \times \frac{-11}{7} = \frac{33}{14}$. Times $-1$ is $\frac{-33}{14}$.
* Denominator part: $\frac{-7}{4} - \frac{8}{4} = \frac{-15}{4}$.
* Divide: $\frac{-33}{14} \div \frac{-15}{4} = \frac{-33}{14} \times \frac{4}{-15}$.
* Negatives cancel. Simplify: $33/15$ divides by 3 ($11/5$). $4/14$ divides by 2 ($2/7$).
* Result: $\frac{11}{7} \times \frac{2}{5} = \frac{22}{35}$.

12) $\left(\frac{-1}{4} - \frac{3}{2} \times \frac{2}{3} - 2\right) \div \frac{7}{4}$
* Multiplication first inside bracket: $\frac{3}{2} \times \frac{2}{3} = 1$.
* Bracket becomes: $\frac{-1}{4} - 1 - 2 = \frac{-1}{4} - 3$.
* $\frac{-1}{4} - \frac{12}{4} = \frac{-13}{4}$.
* Divide: $\frac{-13}{4} \div \frac{7}{4} = \frac{-13}{4} \times \frac{4}{7}$.
* The 4s cancel. Result: $\frac{-13}{7}$.

13) $\left(-6 \times \frac{-4}{3} \times \frac{-13}{9}\right) \div \left(\frac{2}{5} - \frac{4}{3}\right)$
* Numerator part: $-6 \times \frac{-4}{3} = \frac{24}{3} = 8$.
* $8 \times \frac{-13}{9} = \frac{-104}{9}$.
* Denominator part: $\frac{2}{5} - \frac{4}{3}$. Common denom 15. $\frac{6}{15} - \frac{20}{15} = \frac{-14}{15}$.
* Divide: $\frac{-104}{9} \div \frac{-14}{15} = \frac{-104}{9} \times \frac{15}{-14}$.
* Negatives cancel. Simplify: $15/9$ becomes $5/3$. $104/14$ becomes $52/7$.
* Result: $\frac{52}{7} \times \frac{5}{3} = \frac{260}{21}$.

14) $\frac{-11}{9} \div \left(\frac{-3}{2}\left(\frac{-5}{3} - \frac{7}{4}\right) \times 2\right)$
* Inner bracket: $\frac{-5}{3} - \frac{7}{4}$. Common denom 12. $\frac{-20}{12} - \frac{21}{12} = \frac{-41}{12}$.
* Middle part: $\frac{-3}{2} \times \frac{-41}{12}$. Simplify $3/12$ to $1/4$. Result $\frac{41}{8}$.
* Multiply by 2: $\frac{41}{8} \times 2 = \frac{41}{4}$.
* Final division: $\frac{-11}{9} \div \frac{41}{4} = \frac{-11}{9} \times \frac{4}{41}$.
* Result: $\frac{-44}{369}$.

15) $-1 + \frac{-5}{7} + 2 + \frac{1}{5} \times \frac{3}{10}$
* Multiplication first: $\frac{1}{5} \times \frac{3}{10} = \frac{3}{50}$.
* Integers: $-1 + 2 = 1$.
* Expression: $1 - \frac{5}{7} + \frac{3}{50}$.
* $1 - \frac{5}{7} = \frac{2}{7}$.
* Add $\frac{3}{50}$: $\frac{2}{7} + \frac{3}{50}$. Common denom 350.
* $\frac{100}{350} + \frac{21}{350} = \frac{121}{350}$.

16) $\frac{5}{6} \times 2 + \frac{3}{7} - \frac{-8}{7} \div \frac{5}{4}$
* Multiplication: $\frac{5}{6} \times 2 = \frac{10}{6} = \frac{5}{3}$.
* Division: $\frac{-8}{7} \div \frac{5}{4} = \frac{-8}{7} \times \frac{4}{5} = \frac{-32}{35}$.
* Expression: $\frac{5}{3} + \frac{3}{7} - (\frac{-32}{35}) = \frac{5}{3} + \frac{3}{7} + \frac{32}{35}$.
* Combine sevenths and thirty-fifths: $\frac{3}{7} = \frac{15}{35}$. So $\frac{15}{35} + \frac{32}{35} = \frac{47}{35}$.
* Add $\frac{5}{3}$: $\frac{5}{3} + \frac{47}{35}$. Common denom 105.
* $\frac{175}{105} + \frac{141}{105} = \frac{316}{105}$.

Final Answer:
1) -247/270
2) 16433/16200
3) 106/15
4) -39/10
5) -138/35
6) -1197/32
7) -405/28
8) -400/27
9) 224/81
10) 4/9
11) 22/35
12) -13/7
13) 260/21
14) -44/369
15) 121/350
16) 316/105