Exercises: Word problems with integers

At a ski resort in the Pyrenees, the thermometer read $-5^{\circ}C$ at seven in the morning. By twelve noon, the temperature had risen by $8^{\circ}C$, but by midnight, it had dropped by $6^{\circ}C$ compared to the temperature at noon. What temperature did the thermometer read at the end of the day?

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.

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?

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.

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?

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

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?

Pablo has a balance of $-24\ €$ on his travel card because he forgot to top it up. If he opens the transport app and tops up by $30\ €$, what will his new available balance be for travelling?

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, Marcos writes the following operation in his notebook: $63 - 14 = 49$ years.

1. a) Explain in detail why Marcos'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 an AEMET weather station in Molina de Aragón (Guadalajara), 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.

At a ski resort in the Aragonese Pyrenees, 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.

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

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

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.

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?

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.

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 caving team explores a chasm system in the Picos de Europa (Cantabria). The cave entrance is located at $+1.240\text{ m}$ above sea level. During the expedition, they record the following successive vertical sections: they descend a vertical shaft of $180\text{ m}$, go down a rock ramp another $95\text{ m}$, climb up an internal chimney rising $40\text{ m}$, and, finally, descend down a pit of $115\text{ m}$ until reaching a chamber with an underground lake.

1. a) Model the expedition by writing a single expression with integers that includes the initial elevation and each of the vertical displacements with its corresponding sign.
2. b) Calculate the altitude of the underground chamber relative to sea level by performing the additions in order, from left to right.
3. c) What total vertical distance separates the cave entrance from the underground lake?

A freight elevator in a logistics distribution centre in Zaragoza 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 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 $1.200$ €.

• Pays platform rent and advertising: $450$ €.

• Buys materials worth $600$ €.

• Sells $45$ T-shirts at $15$ € 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 $12$ €.

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

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?

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?

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)?

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?

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. It goes up $120$ meters to avoid a mountain.

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

3. It moves horizontally (without changing altitude) and then goes down another $150$ meters.

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

Calculate:

1. a) How many meters did the drone descend in its last fall (step 4)?

1. b) If the drone were able to climb from the ravine directly to the initial base, how many meters would it have to ascend?

A high school sports club manages its funds for the term. They have an initial balance of $+180$ € in the school account. During the term, they record these six transactions: purchase of training equipment ($-95$ €), membership fees from registered members ($+140$ €), goalpost repairs ($-60$ €), revenue from a charity raffle ($+210$ €), payment to match referees ($-130$ €), and registration fee for a local tournament ($-80$ €).

Two board members propose two different calculation methods:

• Method 1 (chronological): Write the initial balance and add or subtract each transaction one by one, in the exact chronological order in which they occurred.

• Method 2 (by blocks): Add all income to the initial balance on one side, add all expenses in absolute value on the other side, and finally subtract the two amounts.

1. a) Apply both methods separately and verify that the same final balance is obtained.
2. b) Imagine that the bank charges a fixed penalty fee of $15$ € the exact moment the account goes into the red (balance strictly less than $0$ €). Justify which of the two methods is essential to determine whether the club will have to pay that penalty, and deduce whether the club had a negative balance at any point during the term.

A marine archaeology team in the Bay of Cádiz studies submerged remains at different depths relative to sea level ($0\text{ m}$) and across different historical periods:

• Phoenician remains: dated to the year $750$ BC and located at a depth of $-38\text{ m}$.

• Roman galley: dated to the year $125$ AD and located at a depth of $-54\text{ m}$.

1. a) Model the time elapsed between both historical events using a subtraction of integers and calculate how many years passed between the Phoenician and Roman shipwrecks, explaining why the distance in years is always a positive value.
2. b) An underwater drone departs from the Phoenician remains, descends in a straight line to the depth of the Roman galley to film it, and then ascends $21\text{ m}$ vertically to transmit the collected data. Write the expression using integers and determine the final depth of the drone.
3. c) Explain the mathematical difference between the drone's final position relative to sea level (a signed quantity) and the total distance travelled by the drone in its two vertical manoeuvres (a quantity that accumulates distances).

Topic challenge. Become a weather analyst! You are going to investigate the temperature range in Spain using official, real data from AEMET.

1. Data collection: Go to www.aemet.es and write down the maximum and minimum temperatures (to one decimal place, for example $-2,4\text{ ºC}$ and $14,8\text{ ºC}$) of four locations at contrasting altitudes over two consecutive days.

2. Operations and rationals: In a spreadsheet or your notebook, calculate the daily temperature range using signed subtraction: $\Delta T = T_{\text{max}} - T_{\text{min}}$. Next, find the mean temperature for each municipality, expressing it first as a rational number in fraction form and then as a decimal number with a comma.

3. Representation and conclusions: Design a bar chart (in GeoGebra or a spreadsheet) that clearly shows both positive and negative values on the axis. Write a brief report explaining which location experienced the most extreme temperature change and addressing how below-zero temperatures affect the calculation of these variations.

Assessment criteria:

• Technical accuracy in operations with integers and signed decimals.

• Appropriate use of rational numbers (fractions and decimal expressions).

• Quality and clarity in the graphical representation of data.

• Mathematical coherence and rigour in the report's conclusions.