Question

1/10
27÷\frac{3}{1}
1
\frac{27}{1}÷\frac{3}{1}
2
\frac{1}{27}×\frac{1}{3}
\frac{27}{1}×\frac{3}{1}

Explanation:

Step1: Recall division of fractions rule

To divide by a fraction, multiply by its reciprocal. The reciprocal of $\frac{3}{1}$ is $\frac{1}{3}$, but wait, actually, $27$ can be written as $\frac{27}{1}$. So $27\div\frac{3}{1}$ is $\frac{27}{1}\div\frac{3}{1}$. And dividing by a fraction $\frac{a}{b}$ is multiplying by $\frac{b}{a}$, so $\frac{27}{1}\div\frac{3}{1}=\frac{27}{1}\times\frac{1}{3}$? Wait no, wait the first option is $\frac{27}{1}\div\frac{3}{1}$. Wait, let's re - express $27$ as $\frac{27}{1}$. Then $27\div\frac{3}{1}=\frac{27}{1}\div\frac{3}{1}$. When dividing two fractions $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$. So here, $a = 27$, $b = 1$, $c = 3$, $d = 1$. So $\frac{27}{1}\div\frac{3}{1}=\frac{27}{1}\times\frac{1}{3}$? No, wait no, $\frac{c}{d}=\frac{3}{1}$, so reciprocal is $\frac{d}{c}=\frac{1}{3}$? Wait no, $\frac{3}{1}$ is 3, reciprocal of 3 is $\frac{1}{3}$, but the first option is $\frac{27}{1}\div\frac{3}{1}$, which is the same as $27\div3$, because $\frac{27}{1}\div\frac{3}{1}=\frac{27\div3}{1\div1}=9$. Wait, let's check each option:

Option 1: $\frac{27}{1}\div\frac{3}{1}$. Let's compute this. Using the rule $\frac{a}{b}\div\frac{c}{d}=\frac{a\times d}{b\times c}$, so $\frac{27\times1}{1\times3}=\frac{27}{3}=9$. And $27\div\frac{3}{1}=27\div3 = 9$.

Option 2: $\frac{1}{27}\times\frac{1}{3}=\frac{1}{81}
eq9$.

Option 3: $\frac{27}{1}\times\frac{3}{1}=81
eq9$.

Wait, but the original problem is $27\div\frac{3}{1}$, which is $27\div3 = 9$. And $\frac{27}{1}\div\frac{3}{1}=\frac{27}{3}=9$, which is equal to $27\div\frac{3}{1}$.

Wait, maybe I made a mistake earlier. Let's re - express $27$ as $\frac{27}{1}$. Then $27\div\frac{3}{1}=\frac{27}{1}\div\frac{3}{1}$. The rule for dividing fractions: $\frac{a}{b}\div\frac{c}{d}=\frac{a\times d}{b\times c}$. So here, $a = 27$, $b = 1$, $c = 3$, $d = 1$. So $\frac{27\times1}{1\times3}=\frac{27}{3}=9$. And $27\div\frac{3}{1}=27\div3 = 9$. So the first option $\frac{27}{1}\div\frac{3}{1}$ is equivalent to $27\div\frac{3}{1}$.

Step2: Verify each option

• Option 1: $\frac{27}{1}\div\frac{3}{1}=\frac{27}{3}=9$, and $27\div\frac{3}{1}=27\div3 = 9$. So this is equivalent.
• Option 2: $\frac{1}{27}\times\frac{1}{3}=\frac{1}{81}

eq9$.

• Option 3: $\frac{27}{1}\times\frac{3}{1}=81

eq9$.

Answer:

The correct option is the first one: $\boldsymbol{\frac{27}{1}\div\frac{3}{1}}$ (or the first boxed option with the expression $\frac{27}{1}\div\frac{3}{1}$)