Fractions and mixed numerals

(a) Convert the following mixed numbers to improper fractions. i. 3 5/8 ii. 7 7/6
(b) Convert the following improper fraction to a mixed number. i. 13/4 ii. 78/5
(c) Simplify these fractions to their lowest terms. i. 36/42 ii. 27/45

(d) evaluate the following expressions with fractions and mixed numerals:
i. 1/2 - 3/5 + 4/7=
ii. 1/2 ÷ 3/5 × 4/7=
iii. (4/7 + 1/2) ÷ 5 3/7=
iv. 2/5 × 2 1/3 + 3/2=
v. 20 - (4 × 2/7 ÷ 2/21)

Correct answer:

a₁ =  3 ⁵⁄₈ = 29/8
a₂ =  8 ¹⁄₆ = 49/6
b₁ =  3 ¹⁄₄ = 13/4
b₂ =  15 ³⁄₅ = 78/5
c₁ =  6/7
c₂ =  3/5
d₁ =  33/70
d₂ =  10/21
d₃ =  15/76
d₄ =  2 ¹³⁄₃₀ = 73/30
d₅ =  8

Step-by-step explanation:

$a_1=3\frac{5}{8}=3+\frac{5}{8}=\frac{3\cdot 8+5}{8}=\frac{24+5}{8}=\frac{29}{8}=3\frac{5}{8}=3.625$

$a_2=7\frac{7}{6}=7+\frac{7}{6}=\frac{7\cdot 6+7}{6}=\frac{42+7}{6}=\frac{49}{6}=8\frac{1}{6}=8.1667$

$b_1=\frac{13}{4}=3\frac{1}{4}=3.25$

$b_2=\frac{78}{5}=15\frac{3}{5}=15.6$

$c_1=\frac{36}{42}=\frac{6}{7}=0.8571$

$c_2=\frac{27}{45}=\frac{3}{5}=0.6$

$d_1=\frac{1}{2}-\frac{3}{5}+\frac{4}{7}=\frac{35}{70}-\frac{42}{70}+\frac{40}{70}=\frac{35-42+40}{70}=\frac{33}{70}=0.4714$

$d_2=\frac{1}{2}/\frac{3}{5}\cdot \frac{4}{7}=\frac{10}{21}=0.4762$

$d_3=(\frac{4}{7}+\frac{1}{2})/5\frac{3}{7}=\frac{15}{76}=0.1974$

$d_4=\frac{2}{5}\cdot 2\frac{1}{3}+\frac{3}{2}=\frac{73}{30}=2\frac{13}{30}=2.4333$

$d_5=20-(4\cdot \frac{2}{7}/\frac{2}{21})=8$

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