All Operations in Signed Numbers: Complete the equation

Let's factor 24 into a multiplication exercise:

$?:-12\times8=-8\times3$

We'll simplify the 8 in both terms and get:

$?:-12=-3$

Let's multiply by negative 12:

$\frac{?}{-12}\times-12=-3\times-12$

We'll simplify between negative 12 and get:

$?=-3\times-12$

Let's note that we are multiplying two negative numbers, so the result will necessarily be a positive number:

$?=+36$

Let's write the exercise in the following form:

$\frac{-6\times6}{-3}:?=24$

We'll factor the 6 in the numerator into a multiplication exercise:

$\frac{-3\times2\times6}{-3}:?=24$

We'll cancel out the minus 3 found in the numerator and denominator of the fraction and get:

$12:?=24$

We'll multiply the question mark:

$12=24\times?$

We'll divide by 24:

$\frac{12}{24}=?$

We'll factor 24 into a multiplication exercise:

$\frac{12}{12\times2}=\text{?}$

We'll cancel out the 12 in the numerator and denominator of the fraction:

$\frac{1}{2}=\text{?}$

Maintaining the question mark let's proceed to write the exercise as follows:

$\frac{20}{?}\times-4=-80$

Divide by negative 4:

$\frac{20}{?}\times\frac{-4}{-4}=\frac{-80}{-4}$

Break down the 80 into a multiplication exercise:

$\frac{20}{?}\times\frac{-4}{-4}=\frac{20\times-4}{-4}$

Due to the fact that we are dividing two negative numbers the result must be a positive number:

$\frac{20}{?}\times\frac{4}{4}=\frac{20\times4}{4}$

Let's now reduce the 4 in both the numerator and denominator of the fraction and we should obtain the following:

$\frac{20}{?}\times1=20$

Multiply by the question mark:

$20=20\times\text{?}$

Divide by 20:

$\frac{20}{20}=?$

$1=\text{?}$

Let's write the exercise in the following way:

$\frac{-9\times-7}{?}=-3$

Note that in the numerator we are multiplying between two negative numbers, therefore the result must be a positive number:

$\frac{9\times7}{?}=-3$

Let's multiply by the question mark and get:

$9\times7=-3\times\text{?}$

Let's divide by negative 3 and get:

$\frac{9\times7}{-3}=\text{?}$

Let's factor the 9 into a multiplication exercise:

$\frac{3\times3\times7}{-3}=\text{?}$

Let's reduce between the 3 in the numerator and denominator, noting that we are dividing a positive number by a negative number, therefore the result must be a negative number:

$-3\times7=\text{?}$

$-21=\text{?}$

Let's add 38 to the multiplication exercise in the fraction in the following way:

$\frac{?\times38}{-4}:-9=-5$

Let's multiply by minus 9:

$\frac{?\times38}{-4}=-5\times-9$

Let's multiply by minus 4:

$?\times38=-5\times-9\times-4$

Let's solve the right side first.

Note that we are first multiplying between two negative numbers, so the result must be a positive number:

$-5\times-9=+45$

Now we are multiplying a positive number by a negative number, so the result must be a negative number:

$45\times-4=-180$

Now we got the exercise:

$38?=-180$

Let's divide by 38:

$?=-\frac{180}{38}$

Let's break down 180 and 38 into multiplication exercises:

$?=-\frac{2\times90}{2\times19}$

Let's reduce between the 2 in the numerator and denominator:

$?=-\frac{90}{19}$