Math worksheet for evaluating algebraic expressions with fractions.

Let's evaluate each of the algebraic expressions step by step. We'll substitute the given values and simplify using fraction arithmetic.

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1) $ 4u + v $ at $ u = \frac{5}{6}, v = \frac{3}{4} $

$$
4u + v = 4 \cdot \frac{5}{6} + \frac{3}{4}
= \frac{20}{6} + \frac{3}{4}
= \frac{10}{3} + \frac{3}{4}
$$

Find a common denominator: LCM of 3 and 4 is 12.

$$
\frac{10}{3} = \frac{40}{12},\quad \frac{3}{4} = \frac{9}{12}
$$

$$
\frac{40}{12} + \frac{9}{12} = \frac{49}{12}
$$

✔ Answer: $ \boxed{\frac{49}{12}} $

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2) $ \frac{2a - 3b}{c} $ at $ a = \frac{1}{4}, b = \frac{8}{9}, c = \frac{5}{3} $

First compute numerator:

$$
2a = 2 \cdot \frac{1}{4} = \frac{2}{4} = \frac{1}{2}
$$
$$
3b = 3 \cdot \frac{8}{9} = \frac{24}{9} = \frac{8}{3}
$$
$$
2a - 3b = \frac{1}{2} - \frac{8}{3}
$$

Common denominator: 6

$$
\frac{1}{2} = \frac{3}{6},\quad \frac{8}{3} = \frac{16}{6}
\Rightarrow \frac{3}{6} - \frac{16}{6} = -\frac{13}{6}
$$

Now divide by $ c = \frac{5}{3} $:

$$
\frac{-\frac{13}{6}}{\frac{5}{3}} = -\frac{13}{6} \cdot \frac{3}{5} = -\frac{39}{30} = -\frac{13}{10}
$$

✔ Answer: $ \boxed{-\frac{13}{10}} $

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3) $ x - \frac{8w}{3} $ at $ x = \frac{1}{2}, w = -\frac{9}{8} $

$$
\frac{8w}{3} = \frac{8}{3} \cdot \left(-\frac{9}{8}\right) = -\frac{72}{24} = -3
$$

Wait, let's compute carefully:

$$
\frac{8}{3} \cdot \left(-\frac{9}{8}\right) = -\frac{72}{24} = -3
$$

Yes, because 8 cancels: $ \frac{8}{3} \cdot \left(-\frac{9}{8}\right) = -\frac{9}{3} = -3 $

Now:
$$
x - \frac{8w}{3} = \frac{1}{2} - (-3) = \frac{1}{2} + 3 = \frac{1}{2} + \frac{6}{2} = \frac{7}{2}
$$

✔ Answer: $ \boxed{\frac{7}{2}} $

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4) $ 6mn $ at $ m = \frac{7}{2}, n = \frac{4}{5} $

$$
6 \cdot \frac{7}{2} \cdot \frac{4}{5} = (6 \cdot \frac{7}{2}) \cdot \frac{4}{5}
$$

$ 6 \cdot \frac{7}{2} = \frac{42}{2} = 21 $

Then $ 21 \cdot \frac{4}{5} = \frac{84}{5} $

✔ Answer: $ \boxed{\frac{84}{5}} $

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5) $ pq^2 + r $ at $ p = \frac{6}{7}, q = \frac{2}{3}, r = \frac{5}{9} $

First compute $ q^2 $:

$$
q^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}
$$

Now $ pq^2 = \frac{6}{7} \cdot \frac{4}{9} = \frac{24}{63} = \frac{8}{21} $

Now add $ r $:

$$
\frac{8}{21} + \frac{5}{9}
$$

LCM of 21 and 9 is 63.

$$
\frac{8}{21} = \frac{24}{63},\quad \frac{5}{9} = \frac{35}{63}
\Rightarrow \frac{24 + 35}{63} = \frac{59}{63}
$$

✔ Answer: $ \boxed{\frac{59}{63}} $

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6) $ d - e $ at $ d = \frac{1}{3}, e = -\frac{1}{9} $

$$
d - e = \frac{1}{3} - (-\frac{1}{9}) = \frac{1}{3} + \frac{1}{9}
$$

LCM of 3 and 9 is 9:

$$
\frac{1}{3} = \frac{3}{9},\quad \frac{3}{9} + \frac{1}{9} = \frac{4}{9}
$$

✔ Answer: $ \boxed{\frac{4}{9}} $

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7) Evaluate $ \frac{y}{z} $ at:

#### i) $ y = \frac{7}{5}, z = \frac{1}{6} $

$$
\frac{y}{z} = \frac{7/5}{1/6} = \frac{7}{5} \cdot \frac{6}{1} = \frac{42}{5}
$$

✔ Answer: $ \boxed{\frac{42}{5}} $

#### ii) $ y = \frac{8}{7}, z = \frac{5}{4} $

$$
\frac{y}{z} = \frac{8/7}{5/4} = \frac{8}{7} \cdot \frac{4}{5} = \frac{32}{35}
$$

✔ Answer: $ \boxed{\frac{32}{35}} $

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✔ Final Answers Summary:

1) $ \boxed{\frac{49}{12}} $
2) $ \boxed{-\frac{13}{10}} $
3) $ \boxed{\frac{7}{2}} $
4) $ \boxed{\frac{84}{5}} $
5) $ \boxed{\frac{59}{63}} $
6) $ \boxed{\frac{4}{9}} $
7) i) $ \boxed{\frac{42}{5}} $, ii) $ \boxed{\frac{32}{35}} $

Let me know if you'd like these as mixed numbers or decimals!