Problems with Integers

Salma has $\$34$ in her money box. She decides to buy a book that costs $\$15$ and a backpack that costs $\$25$. As she does not have enough money, her parents tell her she can owe them the remainder. What integer represents Salma's final balance with respect to her parents?

In a video game tournament, "The Gamers" team has a total of $-150$ points accumulated for committing fouls. If in the final round they get a bonus of $225$ points, how many points do they have on their scoreboard now?

At a ski resort in Arteaga, Coahuila, the thermometer read a temperature of -6^° C at 8:00 in the morning. By 12:00 noon, the temperature had risen by 9^° C, and by 20:00 in the evening it had fallen by 7^° C compared to noon. What temperature did the thermometer read at the end of the day?

A food delivery driver enters the elevator of a large shopping center on the third basement floor, which is shown as floor $-3$. First, they go up $7$ floors to deliver an order and, immediately afterward, go down $2$ floors to pick up another order. Which floor is the delivery driver on after these movements?

A diver is submerged in the ocean at a depth of $18$ meters below sea level ($-18\ m$). If he ascends $7$ meters to observe a coral reef and then descends another $4$ meters, at what depth is he finally? Express the result as an integer.

The highest peak of a volcanic island is located $1\ 200$ meters above sea level. On the same island, there is an underwater cave whose base is $85$ meters below sea level. What is the difference in height between the top of the peak and the base of the cave?

The summit of Mount Everest is at an altitude of $+8,848$ meters above sea level, while the deepest point of the Mariana Trench is at $-11,034$ meters. What is the total vertical distance between the highest point and the lowest point on the planet?

A historian is studying the life of an Ancient Greek sculptor. According to records, the sculptor was born in $495$ BC and died in $429$ BC. How many years did this sculptor live?

Pythagoras, the famous Greek mathematician and philosopher, was born in $570$ BC and died in $495$ BC. At what age did Pythagoras die? Use integers (associating years BC with negative numbers) to carry out the calculation.

The Greek philosopher Hypatia of Alexandria died in the year $415$ AD at the age of $45$. Her father, Theon of Alexandria, was a famous mathematician who was born $80$ years before she died.

1. a) In which year was Hypatia born? (Express it as an integer).
2. b) In which year was her father Theon born?
3. c) How old was Theon when his daughter was born?

An Amazon delivery driver starts their workday on floor $3$ of a large logistics center. First, they go down $5$ floors to pick up a package in the basement, then go up $7$ floors to deliver it to an office and, finally, go down another $3$ floors to rest in the cafeteria. Which floor is the cafeteria on? Express the result as an integer.

A diver exploring a coral reef is $14$ meters below sea level. If he decides to descend another $18$ meters to observe some archaeological remains and then climbs $9$ meters to get closer to the surface, at what depth is he now? Represent the situation using integers.

A scientist is carrying out a chemical experiment. They place a compound in a special freezer that lowers the temperature of the sample by $4^\circ C$ every hour. If the sample was initially at a room temperature of $18^\circ C$, what temperature will the sample be after $6$ hours? Express the result using a single calculation combining integers.

In a board game, you gain $15$ points for each correct answer, but you lose $8$ points for each wrong answer and $4$ points for each question left unanswered. If out of a total of $20$ questions you got $12$ correct, $5$ wrong, and left the rest unanswered, what is your final score?

In a scientific laboratory, an experiment is being conducted with a gaseous substance. The initial temperature of the sample is $12$ ºC below zero. The process follows these stages:

1. During the first $5$ minutes, the temperature drops by $3$ ºC each minute.

2. Then, it is placed in a heating chamber where the temperature rises by $48$ ºC in total.

3. Finally, the chamber is turned off and the temperature decreases at a constant rate of $4$ ºC each minute until it reaches the freezing point of the sample, which is $-25$ ºC.

How long did the final cooling stage (step 3) last? What was the maximum temperature reached during the whole experiment?

In a video game tournament, the scoring system works as follows:

• Each win adds $100$ points.

• Each loss subtracts $60$ points.

• Each "special achievement" multiplies the score accumulated at that time by $2$ (if positive).

• Each penalty divides the accumulated score by $2$ (if positive).

If a player starts with $0$ points and achieves: $3$ consecutive wins, one "special achievement", $2$ losses, one penalty, and finally, $1$ win:

1. a) Write a single combined expression that calculates their final score.
2. b) What is the final score?

A young entrepreneur decides to open an online shop selling custom T-shirts. During their first month, they record the following transactions in their bank account:

• Receives an initial investment of $24\,000$ pesos.

• Pays platform rent and advertising: $9\,000$ pesos.

• Buys materials worth $12\,000$ pesos.

• Sells $45$ T-shirts at $300$ pesos each.

• $8$ defective T-shirts are returned, for which they have to refund the full amount to the customers.

• At the end of the month, the bank charges a fee of $240$ pesos.

Calculate the final balance for the month and explain whether they made a profit or a loss relative to their initial investment.

An underwater cave explorer begins their dive from a platform located $4$ meters above sea level.

• First, they descend at a constant speed of $2$ meters per minute for $15$ minutes.

• Then, they encounter an updraft that lifts them $12$ meters.

• Finally, they decide to descend to the bottom of the cave, which is at a depth three times their current position.

What depth is the bottom of the cave at? Express the result as an integer relative to sea level.

A delivery driver uses an elevator in a large logistics center that has floors above and below ground level (floor $0$). They take the following route:

1. Goes up from floor $0$ to floor $+12$.

2. Goes down $15$ floors to collect a package.

3. Goes up $7$ floors to deliver it.

4. Goes down twice as many floors as they have just gone up.

5. Finally, goes up to halfway between the floor they are on and floor $+8$.

Which floor is the delivery driver on at the end of their shift? How many floors have they traveled in total (counting both up and down)?

A forest surveillance drone takes off from a base located at an altitude of $850$ meters. During its patrol, it makes the following movements:

1. Climbs $120$ meters to avoid a mountain.

2. Descends $300$ meters to analyze an area of low vegetation.

3. Moves horizontally and then descends another $150$ meters.

4. Due to a technical fault, it falls to the bottom of a ravine at an altitude of $-20$ meters (below sea level).

Calculate how many meters the drone descended in its final fall (step 4). If the drone were able to climb directly from the ravine to the initial base, how many meters would it have to ascend?

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