Integers & Negatives

$5-9=-4$

The expression becomes: $[(-3)\times (-4)+12]/(-2)-|-7|$

$(-3)\times (-4)=12$

The expression becomes: $[12+12]/(-2)-|-7|$

$12+12=24$

The expression becomes: $24/(-2)-|-7|$

$|-7|=7$

The expression becomes: $24/(-2)-7$

$24/(-2)=-12$

The expression becomes: $-12-7$

$-12-7=-19$

The final answer is $-19$.

Current depth: $-350+120=-230$ feet.

Current depth: $-230-210=-440$ feet.

Final depth: $-440-210=-650$ feet.

The final depth of the submarine is $650$ feet below sea level.

$b$ is a positive integer, so $b>0$.

$c$ is a negative integer, so $c<0$. This implies that $-c$ is a positive integer.

Therefore, $b-c$ can be rewritten as $b+(-c)$. Since both $b$ and $-c$ are positive, their sum $b+(-c)$ will always be positive.

Since $b-c$ is always positive, its absolute value is itself: $|b-c|=b-c$.

$(-c)$ is simply $-c$.

$a-|b-c|+(-c)$ becomes $a-(b-c)+(-c)$.

$a-b+c-c$

$=a-b$

The condition $|a|<|b|<|c|$ is given to establish specific relationships between the magnitudes of $a$, $b$, and $c$, but it does not alter the algebraic simplification of the expression itself. The final simplified expression is $a-b$.

The average of $N$ numbers is their sum divided by $N$. Given that the average of five integers is $-8$, their total sum is $5\times (-8)=-40$.

The four given integers are $12$, $-5$, $0$, and $-20$.

Sum = $12+(-5)+0+(-20)$

Sum = $12-5+0-20$

Sum = $7-20$

Sum = $-13$

Let the fifth integer be $x$. The sum of all five integers is the sum of the four known integers plus $x$.

So, $-13+x=-40$.

To isolate $x$, add $13$ to both sides of the equation:

$x=-40+13$

$x=-27$

The fifth integer is $-27$. All five integers ($12$, $-5$, $0$, $-20$, $-27$) are distinct, satisfying the condition given in the problem.