Problem Solving with Integers

At a ski resort in the Rocky Mountains, the thermometer showed $-5^\circ\text{C}$ at eight in the morning. By midday, the temperature had risen by $9^\circ\text{C}$. What temperature did the thermometer show at noon?

Marcus has $\$14$ in his bank account. If he makes a purchase in an app store for his cell phone that costs $\$18$, what will his new balance be? Express it as an integer.

A diver is doing a dive at a depth of $12$ feet below sea level. If they decide to descend another $15$ feet to observe some coral, at what depth will they be now? Express it as an integer.

An elevator in a shopping mall is on floor $2$. First, it goes down $4$ floors to go to the parking garage, and then it goes up $3$ floors to go to the movie theater. On which floor is the movie theater?

Lucy has a balance of $\$120\ \$$ in her bank account. During the morning, the bank charges her an internet bill of $\$45\ \$$, and she makes a purchase at a clothing store for $\$82\ \$$. In the afternoon, her grandfather sends her a $\$20\ \$$ Venmo transfer.

Find the final balance of Lucy's account and explain whether she is "in the red" (negative balance) or not.

An underwater research station has its command center on level $0$ (at sea level). It has residential floors going up (positive) and laboratory floors going down (negative). A researcher leaves their cabin on floor $+3$, goes down $8$ floors to collect some samples, then goes up $5$ floors to analyze them, and finally goes down another $4$ floors to store them in the security vault.

1. On which floor is the security vault located?

2. If each floor is $4$ feet high, at what depth (in feet) relative to sea level is the researcher at the end of their trip?

Calculate the vertical distance separating a helicopter flying at a height of $250$ feet above sea level from a submarine navigating at a depth of $45$ feet.

An underwater drone is exploring an ocean trench at a depth of $125$ feet below sea level. To avoid an obstacle, it ascends $40$ feet. At the same time, a bird flies over the area at a height of $15$ feet above sea level.

1. a) Represent the position of the drone and the bird with integers.
2. b) Calculate the vertical distance separating the bird from the drone in its final position.

An industrial robot moves along a vertical axis. Its program requires it to repeat the following cycle of movements: go up $12$ in, go down $15$ in, and go up $7$ in.

1. If the robot repeats this cycle $8$ consecutive times, what is its final position relative to the starting point?

2. What is the total distance (in absolute value) that the robot has traveled after completing the $8$ cycles?

A Roman emperor was born in the year $27$ BC and died in the year $14$ AD. At what age did this emperor die? (Assume he had already had his birthday on the date of his death).

A famous philosopher from ancient Greece was born in the year $384$ BC and died in the year $322$ BC. A contemporary historian claims that he lived for more than $65$ years.

Using the subtraction of integers (considering BC dates as negative numbers), determine how many years he lived and explain whether the historian is correct.

An archaeologist is studying a civilization that flourished in the year $450$ BC and disappeared in the year $120$ AD. During their excavations in an ancient tower, they find an inscription on the upper level, at a height of $+18$ feet, and a hidden chamber located at exactly the same distance below ground level.

1. How many years did this civilization last?

2. If the archaeologist is in the hidden chamber and climbs up to the inscription on the upper level, how many feet have they traveled in total? Express the levels as integers.

A logistics company has a cold room for frozen food which is at a temperature of $-18^\circ\text{C}$. Due to a power failure, the temperature starts to rise by $2^\circ\text{C}$ every hour. What temperature will the room show after $5$ hours?

A logistics company has a cold room to preserve food. The initial temperature of the room is $4\ ^\circ\text{C}$. Due to a power failure, the temperature rises by $3\ ^\circ\text{C}$ every hour for the first $5$ hours. Once the fault is repaired, a rapid cooling system manages to lower the temperature at a rate of $5\ ^\circ\text{C}$ per hour.

If the cooling system runs for $4$ hours, what will the final temperature of the room be?

In a high-precision cold storage room, the temperature drops by $3$ $^\circ C$ every $15$ minutes until it reaches the storage temperature of $-24$ $^\circ C$. If the internal temperature was $+12$ $^\circ C$ when the room was turned on:

1. How many hours must pass for the room to reach $-24$ $^\circ C$?

2. If, after reaching that temperature, there is a power outage and the temperature rises by $2$ $^\circ C$ every hour, what will the temperature be after $7$ hours of outage?

In a board game, each correct answer adds $10$ points and each mistake deducts $15$ points. If Sarah has $45$ points and gets the next three questions wrong, what score will she have at the end?

In a video game, each level completed awards $500$ points, but every time the character loses a life, $150$ points are deducted, and each magic item collected adds $75$ points.

If a player has completed $4$ levels, lost $3$ lives and collected $12$ magic items, what is their total score? Write the calculation using combined operations with integers.

An indoor soccer team has a goal difference of $-15$. In the last three games, the results have been:

• Game 1: They won $4$ to $1$.

• Game 2: They lost $0$ to $3$.

• Game 3: They drew $2$ to $2$.

What is the team's goal difference after these three games? (Remember that the goal difference is the difference between goals for and goals against).

A content creator on a streaming platform starts the month with a balance of $\$-125$ due to purchasing a new microphone. During the first week, they receive donations worth $\$450$. However, the platform charges a fee of $\$2$ for each of the $15$ donations received. In addition, they have to pay for $3$ software subscriptions at $\$12$ each. Calculate the creator's final balance at the end of the week and explain, using the concept of absolute value, how much total money has "moved" (both income and expenses) in their account.

In an eSports tournament, a team's score is calculated as follows: $+100$ points for each win, $-40$ points for each loss, and $-15$ points for each time penalty. Team "Cipher" played $25$ matches in total. It is known that they won three times as many matches as they lost, and they received $6$ time penalties. Write a single expression with integers to calculate the team's final score and evaluate it.

A textile company analyzes its quarterly financial results. In the first quarter, it made a profit of $\$12000$. In the second, it made a loss of $\$15000$. In the third, the losses were twice as much as in the second. For the company to break even at the end of the year (final balance equal to $\$0$), what must the profit for the fourth quarter be? Write the complete calculation using integers to represent profits and losses.