Factoring Algebraic Fractions

Let's examine the problem:

$\frac{12ab}{?}=1$Now let's think logically, and remember the known fact that dividing any number by itself always yields the result 1,

Therefore, in order to get the result 1 from dividing two numbers, the only way is to divide the number by itself, meaning-

The missing expression in the denominator of the fraction on the left side is the complete expression that appears in the numerator of the same fraction:

$12ab$.

Therefore- the correct answer is answer D.

After examining the problem, proceed to write the expression on the right side as a fraction (using the fact that dividing a number by 1 does not change its value):

$\frac{16ab}{?}=2b \\ \downarrow\\ \frac{16ab}{?}=\frac{2b}{1}$

Remember the fraction reduction operation,

In order for the fraction on the left side to be deemed reducible, we want all the terms in its denominator to have a common factor. Additionally, we want to reduce the number 16 in order to obtain the number 2. Furthermore we want to reduce the term $a$ from the fraction's denominator given that in the expression on the right side it does not appear. Therefore we will choose the expression:

$8a$

Due to the fact that:

$16=8\cdot 2$

Let's verify that with this choice we indeed obtain the expression on the right side:

$\frac{16ab}{?}=\frac{2b}{1} \\ \downarrow\\ \frac{\not{16}\not{a}b}{\textcolor{red}{\not{8}\not{a}}}\stackrel{?}{= }\frac{2b}{1} \\ \downarrow\\ \boxed{\frac{2b}{1}\stackrel{!}{= }\frac{2b}{1} }$

Therefore this choice is indeed correct.

In other words - the correct answer is answer B.

Using the formula:

$\frac{x}{y}=\frac{z}{w}\xrightarrow{}x\cdot y=z\cdot y$

We first convert the 8 into a fraction, and multiply

$\frac{16ab}{?}=\frac{8}{1}$

$16ab\times1=8a$

$16ab=8a$

We then divide both sides by 8a:

$\frac{16ab}{8a}=\frac{8a}{8a}$

$2b$

Upon examining the problem, proceed to write down the expression on the right side as a fraction (using the fact that dividing a number by 1 doesn't change its value):

$\frac{19ab}{?}=a \\ \downarrow\\ \frac{19ab}{?}=\frac{a}{1}$
Remember the fraction reduction operation,

In order for the fraction on the left side to be deemed reducible, we want all the terms in its denominator to have a common factor. Additionally, we want to reduce the number 19 in order to obtain the number 1 as well as reducing the term $b$ from the fraction's numerator given that in the expression on the right side it doesn't appear. Therefore we'll choose the expression:

$19b$

Let's verify that this choice results in the expression on the right side:

$\frac{19ab}{?}=\frac{a}{1} \\ \downarrow\\ \frac{\not{19}a\not{b}}{\textcolor{red}{\not{19}\not{b}}}\stackrel{?}{= }\frac{a}{1} \\ \downarrow\\ \boxed{\frac{a}{1}\stackrel{!}{= }\frac{a}{1} }$

Therefore this choice is indeed correct.

In other words - the correct answer is answer D.

Upon examining the problem, proceed to write the expression on the right side as a fraction (using the fact that dividing a number by 1 does not change its value):

$\frac{27ab}{\text{?}}=3ab\\ \downarrow\\ \frac{27ab}{\text{?}}=\frac{3ab}{1}$

Remember the fraction reduction operation,

Note that both in the numerator of the expression on the right side and in the numerator of the expression on the left side the expression $ab$ is present. Therefore in the expression we are looking for there are no variables (since we are not interested in reducing them from the expression in the numerator on the left side),

Next, determine which number was chosen to be in the denominator of the expression on the left side in order that its reduction with the number 27 yields the number 3. The answer to this - the number 9,

Due to the fact that:

$27=9\cdot 3$

Let's verify that this choice indeed gives us the expression on the right side:

$\frac{27ab}{\text{?}}=\frac{3ab}{1} \\ \downarrow\\ \frac{\not{27}ab}{\textcolor{red}{\not{9}}}\stackrel{?}{= }\frac{3ab}{1} \\ \downarrow\\ \boxed{\frac{3ab}{1}\stackrel{!}{= }\frac{3ab}{1} }$

Therefore this choice is indeed correct.

In other words - the correct answer is answer A.