Word problems with integers

Salma has $\text{€}\ 34$ in her money box. She decides to buy a book that costs $\text{€}\ 15$ and a backpack that costs $\text{€}\ 25$. 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?

Aoife has $\text{€}\ 45$ in her bank account. Over the weekend, she makes the following transactions: she pays for a streaming platform subscription for $\text{€}\ 12$, receives a bank transfer (Revolut) of $\text{€}\ 20$ from her father and buys a book costing $\text{€}\ 15$. If at the end of Sunday her bank charges her an account maintenance fee of $\text{€}\ 3$, what is the final balance of her account?

A food delivery courier gets into the elevator of a large shopping mall on the third basement floor, which is labelled floor $-3$. First he goes up $7$ floors to deliver an order and, right after that, he goes down $2$ floors to pick up another order. On which floor is the courier after these movements?

In the figure you can see the floors of the elevator and the starting point.

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

The summit of Mount Everest is at an altitude of $+8,848$ metres above sea level, while the deepest point of the Mariana Trench is at $-11,034$ metres. 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.

To find out how many years the Roman emperor Caesar Augustus lived, who was born in the year $63$ BC and died in the year $14$ AD, Seán writes the following operation in his notebook: $63 - 14 = 49$ years.

1. a) Explain in detail why Seán's approach is incorrect from the perspective of the number line and the passage of time.
2. b) What mistake did he make with the signs of the integers representing each historical date?
3. c) Write the correct operation using integers (indicating the sign of each year and the subtraction between them) and calculate the actual age at which Augustus died.

At a mountain viewpoint in the Wicklow Mountains, the thermometer showed a temperature of $-6\,^\circ\mathrm{C}$ at 8:00 in the morning. At 12:00 noon, the temperature had risen $9\,^\circ\mathrm{C}$, and at 20:00 in the evening it had dropped $7\,^\circ\mathrm{C}$ compared with the noon temperature. What temperature did the thermometer show at the end of the day?

An Amazon delivery driver starts their working day on floor $3$ of a large logistics centre. First, they go down $5$ floors to pick up a parcel 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 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?

An underwater cave explorer starts 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 upward current that lifts them $12$ meters.

• Finally, they go down to the bottom of the cave, whose position relative to sea level is three times the position they are at that moment.

Take sea level as $0$, so that positions above it are positive and those below it are negative.

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

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?

At a Met Éireann weather station in Birr (Co. Offaly), the thermometer reads a temperature of $-7\,^\circ\text{C}$ at 07:00 in the morning. Throughout the day, the following temperature changes are recorded: it rises by $11\,^\circ\text{C}$ until midday, drops by $4\,^\circ\text{C}$ in the late afternoon, and drops another $3\,^\circ\text{C}$ late at night.

1. a) Before calculating the exact result, make a reasoned estimate of whether the final temperature at midnight will be positive, negative, or exactly $0\,^\circ\text{C}$, mentally comparing the total rise with the cumulative total drops.
2. b) Write the chain of additions using integers, calculate step by step from left to right the temperature shown on the thermometer at the end of the day, and check whether your initial estimate was correct.

A freight elevator in a logistics distribution centre in Limerick moves between underground storage floors (negative numbers), the street-level loading area (floor $0$), and upper office floors (positive numbers). The manager does not remember on which floor they left the elevator first thing in the morning, but the digital log shows that it made the following consecutive movements: it went up $6$ floors, down $9$ floors, up $4$ floors, and finally down $5$ floors, ending up stopped at floor $-2$.

1. a) Set up an expression using integers where the starting floor is represented as a blank space or a question mark, and calculate in a reasoned way from which floor the elevator started.
2. b) Check your result by calculating step by step each of the intermediate floors the elevator passed through, and determine whether at any point during the trip it reached the office floors (above floor $0$).

A delivery driver uses a lift in a large logistics centre 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 parcel.

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 travelled in total (counting both up and down)?