Write the missing digits in each number sentence.

Here are the missing digits for each number sentence:

: $\frac{\square}{\square} \text{ of } 54 = \frac{1}{\square} \text{ of } 27$ Step 1: Let the value of the expression be $X$. We have $X = \frac{1}{c} \times 27 = \frac{27}{c}$ and $X = \frac{a}{b} \times 54 = \frac{54a}{b}$. Step 2: Equate the two expressions: $\frac{54a}{b} = \frac{27}{c}$. Divide both sides by 27 to simplify: $\frac{2a}{b} = \frac{1}{c}$. Step 3: To find single-digit values for $a, b, c$, let $a=1$. Then $\frac{2}{b} = \frac{1}{c}$, which implies $b=2c$. If $c=1$, then $b=2$. So, $\frac{1}{2}$ of $54 = 27$ and $\frac{1}{1}$ of $27 = 27$. The missing digits are $\boxed{\frac{1}{2}}$ and $\boxed{1}$.

: $\frac{1}{4} \text{ of } 28 = \frac{1}{\square} \text{ of } \square$ Step 1: Calculate the value of the left side: $\frac{1}{4} \times 28 = 7$. Step 2: We need to find $x$ and $y$ such that $7 = \frac{1}{x} \times y$. The simplest unit fraction representation for 7 is $\frac{1}{1}$ of $7$. The missing digits are $\boxed{1}$ and $\boxed{7}$.

: $\frac{\square}{\square} \text{ of } 30 = \frac{1}{\square} \text{ of } 20$ Step 1: Let the value of the expression be $X$. We have $X = \frac{1}{c} \times 20 = \frac{20}{c}$ and $X = \frac{a}{b} \times 30 = \frac{30a}{b}$. Step 2: Equate the two expressions: $\frac{30a}{b} = \frac{20}{c}$. Divide both sides by 10 to simplify: $\frac{3a}{b} = \frac{2}{c}$. This implies $3ac = 2b$. Step 3: To find single-digit values for $a, b, c$, let $a=1$. Then $3c = 2b$. Since $2b$ is even, $3c$ must be even, so $c$ must be even. If $c=2$, then $3 \times 2 = 2b \Rightarrow 6 = 2b \Rightarrow b=3$. So, $\frac{1}{3}$ of $30 = 10$ and $\frac{1}{2}$ of $20 = 10$. The missing digits are $\boxed{\frac{1}{3}}$ and $\boxed{2}$.

: $\frac{1}{\square} \text{ of } 90 = \frac{\square}{\square} \text{ of } 45$ Step 1: Let the value of the expression be $X$. We have $X = \frac{1}{b} \times 90 = \frac{90}{b}$ and $X = \frac{a}{c} \times 45 = \frac{45a}{c}$. Step 2: Equate the two expressions: $\frac{90}{b} = \frac{45a}{c}$. Divide both sides by 45 to simplify: $\frac{2}{b} = \frac{a}{c}$. This implies $2c = ab$. Step 3: To find single-digit values for $a, b, c$, let $a=1$. Then $2c=b$. If $c=1$, then $b=2$. So, $\frac{1}{2}$ of $90 = 45$ and $\frac{1}{1}$ of $45 = 45$. The missing digits are $\boxed{2}$ and $\boxed{\frac{1}{1}}$.

: $\frac{1}{\square} \text{ of } 36 = \frac{\square}{\square} \text{ of } \square$ Step 1: Let the value of the expression be $X$. Let the missing denominator on the left be $d_1$. So $X = \frac{1}{d_1} \times 36 = \frac{36}{d_1}$. Step 2: To find a simple integer value for $X$, choose $d_1$ as a factor of 36. Let $d_1=4$. Then $X = \frac{36}{4} = 9$. Step 3: We need to find $n_2, d_2, N_3$ such that $9 = \frac{n_2}{d_2} \times N_3$. The simplest unit fraction representation for 9 is $\frac{1}{1}$ of $9$. So $n_2=1, d_2=1, N_3=9$. The missing digits are $\boxed{4}$, $\boxed{\frac{1}{1}}$ and $\boxed{9}$.

: $\frac{1}{3} \text{ of } 24 = \frac{1}{8} \text{ of } \square$ Step 1: Calculate the value of the left side: $\frac{1}{3} \times 24 = 8$. Step 2: We need to find $N$ such that $8 = \frac{1}{8} \times N$. Step 3: Multiply both sides by 8: $8 \times 8 = N \Rightarrow N = 64$. The missing digit is $\boxed{64}$.