Problems with Integers

Emma has $34\ \$$ in her piggy bank. 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 what she is missing. What integer represents Emma's final balance with respect to her parents?

Emma has $\$45$ in her bank account. During the weekend, she makes the following transactions: she pays a streaming platform subscription for $\$12$, receives a Zelle from her father for $\$20$ and buys a book that costs her $\$15$. If at the end of Sunday her bank charges her a maintenance fee of $\$3$, what is the final balance of her account?

A young entrepreneur decides to open an online personalized T-shirt shop. During his first month, he records the following movements in his bank account:

• He receives an initial investment of $\$1.200$.

• He pays the platform rent and advertising: $\$450$.

• He buys materials worth $\$600$.

• He sells $45$ T-shirts at $\$15$ each.

• $8$ faulty T-shirts are returned, for which he has to refund the full amount to the customers.

• At the end of the month, the bank charges him a fee of $\$12$.

Calculate the final balance for the month and explain whether he has made a profit or a loss compared to his initial investment.

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

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

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 accumulated score at that moment by $2$ (if it is positive).

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

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

1. a) Write a single expression with combined operations that calculates their final score.
2. b) What is the result of this score?

At a ski resort in the Rocky Mountains, the thermometer showed $-5^{\circ}C$ at seven in the morning. At twelve noon, the temperature had gone up by $8^{\circ}C$, but by midnight, it had gone down by $6^{\circ}C$ compared to the noon temperature. What temperature did the thermometer show at the end of the day?

A scientist is conducting a chemical experiment. He places 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 will the temperature of the sample be after $6$ hours? Express the result using a single expression with integers.

In a scientific laboratory, an experiment is being carried out with a gaseous substance. The initial temperature of the sample is $12$ ºF below zero. The process follows these phases:

1. During the first $5$ minutes, the temperature drops by $3$ ºF every minute.

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

3. Finally, the chamber is turned off and the temperature drops at a constant rate of $4$ ºF every minute until it reaches the freezing temperature of the sample, which is $-25$ ºF.

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

A food delivery driver enters the elevator of a large shopping mall on the third basement floor, which is indicated as floor $-3$. First, he goes up $7$ floors to deliver an order and, right after, he goes down $2$ floors to collect another order. On which floor is the delivery driver after these movements?

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

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

The highest peak of a volcanic island is at $1\ 200$ feet above sea level. On the same island, there is an underwater cave whose base is at $85$ feet 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 $+8.848$ feet above sea level, while the deepest point of the Mariana Trench is at $-11.034$ feet. What is the total vertical distance between the highest and lowest points on the planet?

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

1. It goes up $120$ feet to avoid a mountain.

2. It goes down $300$ feet to analyze an area of low vegetation.

3. It moves horizontally and then goes down another $150$ feet.

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

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

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

Pythagoras, the famous Greek mathematician and philosopher, was born in the year $570$ BC and died in the year $495$ BC. At what age did Pythagoras die? Use integers (associating the years before Christ with negative numbers) to perform 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 what year was Hypatia born? (Express it as an integer).
2. b) In what year was her father Theon born?
3. c) How old was Theon when his daughter was born?

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

An underwater cave explorer begins his dive from a platform located at $4$ feet above sea level.

• First, he descends at a constant speed of $2$ feet per minute for $15$ minutes.

• Then, he encounters an updraft that lifts him $12$ feet.

• Finally, he decides to go down to the bottom of the cave, which is at a depth that is three times his current position.

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

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

1. He goes up from floor $0$ to floor $+12$.

2. He goes down $15$ floors to collect a package.

3. He goes up $7$ floors to deliver it.

4. He goes down twice as many floors as he has just gone up.

5. Finally, he goes up until he is halfway between the floor he is on and floor $+8$.

On which floor is the delivery driver at the end of his shift? How many floors has he traveled in total (counting going up and down)?

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