Integer Word Problems

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

Marcos has $840$ pesos in his bank account. If he makes a purchase in an app store for his cell phone that costs $1\,080$ pesos, what will his new balance be? Express it as an integer.

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

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

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

At a high-mountain weather station, the thermometer read $-12\ ^\circ\text{C}$ at six in the morning. At noon, because of the sunrise, the temperature rose $15\ ^\circ\text{C}$, but at nightfall a storm made it drop $8\ ^\circ\text{C}$.

Write an expression with integers that represents this situation and calculate the final temperature at the end of the day.

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

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

An underwater scientific station has its command center at level $0$ (sea level). It has residential floors going upwards (positive) and laboratories going downwards (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 $4$ meters high, at what depth (in meters) relative to sea level is the researcher at the end of their journey?

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

An exploration submarine is investigating a marine trench at a depth of $125$ meters below sea level. To avoid an obstacle, it ascends $40$ meters. At the same time, a bird flies over the area at a height of $15$ meters above sea level.

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

An archaeologist is studying a civilization that flourished in $450$ BC and disappeared in $120$ AD. During excavations in an ancient tower, he finds an inscription on the top level, $+18$ meters high, and a hidden chamber located at exactly the same distance from the ground but downwards (underground).

1. How many years did this civilization last?

2. If the archaeologist is in the hidden chamber and climbs up to the top level inscription, how many meters has he traveled in total? Express the levels as integers.

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

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

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

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

A logistics company has a cold storage chamber to preserve food. The initial temperature of the chamber is $4\ ^\circ\text{C}$. Due to a power failure, the temperature rises by $3\ ^\circ\text{C}$ every hour during 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 chamber be?

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

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

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

A futsal team has a goal difference of $-15$. In their last three matches, the results were:

• Match 1: Won $4$ to $1$.

• Match 2: Lost $0$ to $3$.

• Match 3: Drew $2$ to $2$.

What is the team's goal difference after these three matches? (Remember that goal difference is the difference between goals scored and goals conceded).

A weather exploration drone takes off from a base located on a mountain at $+1200$ meters above sea level. The temperature at the base is $+8$ $^\circ C$. The drone ascends $600$ meters, where the temperature drops by $5$ $^\circ C$ for every $200$ meters of ascent. It then descends rapidly to an underwater chasm located at $-150$ meters relative to sea level.

1. What is the temperature at the highest point of the flight?

2. What is the total altitude difference between the highest point reached and the bottom of the chasm?

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" has played $24$ matches in total. It is known that they have won three times as many matches as they have lost, and they have received $6$ time penalties. Write a single expression with integers to calculate the team's final score and solve it.

An industrial robot moves along a vertical axis. Its programming forces it to repeat the following sequence of movements: go up $12$ cm, go down $15$ cm and go up $7$ cm.

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

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

8th grade is organizing the end-of-year trip and is keeping a joint account of income and expenses by term. In the first term, they made $720\,000$ pesos from calendar sales. In the second term, they spent $900\,000$ pesos on booking the buses. In the third term, the expenses were double those of the second term (paying for the hostel). So that by the end of the school year the class is not in the red (final balance equal to $0$ pesos), how much will they need to earn from fourth-term activities? Write down the complete calculation using integers to represent income and expenses.