Exercises: Word problems with integers

At a ski resort in the Pyrenees, 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 $14$ € in his bank account. If he makes a purchase in an app store for his mobile phone that costs $18$ €, what will his new balance be? Express it as an integer.

An elevator in a shopping center is on floor $2$. First it goes down $4$ floors to reach the parking garage and then it goes up $3$ floors to reach the movie theater. On which floor is the movie theater?

The floors below the ground floor (floor $0$) are numbered $-1$, $-2$, $-3$…, as shown in the figure.

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

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

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).

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?

On a winter day in Teruel, the State Meteorological Agency (AEMET) recorded a minimum temperature of $-4\ ^\circ\text{C}$ and a maximum of $11\ ^\circ\text{C}$. Lucía calculated the temperature range (the difference between both temperatures) by writing in her notebook: “$11 - 4 = 7\ ^\circ\text{C}$ of variation”. Identify the reference point ($0\ ^\circ\text{C}$), explain in detail what conceptual error Lucía made by ignoring the sign of the minimum temperature, and write the correct operation using the concept of distance between two integers, $|a - b|$, to determine the actual temperature range.

In a shopping mall in Seville, street level is considered floor $0$. You leave your car in the underground car park, on floor $-3$. From there, you go up $5$ floors to visit some clothing stores, then you go down $4$ floors to access the cinema, and finally, you go up $1$ floor to eat on the terrace. a) Before performing the step-by-step operations, reasonably estimate whether you will end up above street level, at street level itself, or underground, comparing the total sum of floors you go up with the sum of those you go down. b) Write a single mathematical expression with integers that reflects the entire journey, calculate the exact final floor, and check whether your initial estimation was correct.

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 food delivery driver enters a building and leaves his scooter in the third basement level. He goes up $7$ floors to deliver an order, then goes down $4$ floors to collect another package from a resident, and finally goes up $2$ more floors to make the final delivery.

Which floor is the delivery driver on when he finishes his work? Express the basement level as a negative number.

Lucía has a balance of $120\ €$ in her bank account. During the morning, the bank charges her an internet bill of $45\ €$ and she makes a purchase in a clothing shop worth $82\ €$. In the afternoon, her grandfather transfers her $20\ €$ via Bizum.

Calculate the final balance of Lucía's account and explain whether she is in the red (negative balance) or not.

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

1. a) Represent with integers the position of the bird and that of the submarine after rising.

1. b) Calculate the vertical distance between the bird and the submarine in its final position.

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 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 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?

A school athletics club in Zaragoza keeps track of its bank account, where a balance of $0$ € indicates that there are no funds and no debts. During the month of October, the club pays for $4$ competition registrations at $35$ € each, receives a municipal grant of $220$ €, and buys sports equipment worth $150$ €. At the end of the month, the bank statement shows that the final balance is $-15$ € (they are in the red). Set up an expression with integers where the unknown initial balance appears as an unknown or a blank to fill in, and deduce with reasoning what the club's balance was at the beginning of the month.

The Cuatro Caminos metro station in Madrid has one of the deepest platforms in the network, located at an elevation of $-45\text{ m}$ relative to the surface (street level, $0\text{ m}$). A maintenance worker is on that platform and goes up in a freight elevator at a constant speed of $2\text{ m}$ per second for $12$ seconds. Afterwards, he walks down a service staircase for a distance of $7\text{ m}$ and, finally, goes up $15\text{ m}$ more to repair a ventilation panel. a) Write a single combined operation with integers that determines the worker's final elevation relative to the street, representing the ascent in the freight elevator with a signed multiplication. b) At what depth or height is the ventilation panel located? c) Using the absolute value in each section, calculate the total distance the worker has covered in his movements.

A content creator on a streaming platform starts the month with a balance of $-125$ € due to the purchase of a new microphone. During the first week, they receive donations worth $450$ €. However, the platform charges a commission of $2$ € for each of the $15$ donations received. In addition, they have to pay for $3$ software subscriptions of $12$ € each. Calculate the creator's final balance at the end of the week and explain, using the concept of absolute value, how much total money has "moved" (both inflows and outflows) in their account.

The diagram summarizes the data of the problem in the order in which they occur.

A weather exploration drone takes off from a base located on a mountain at $+1200$ metres above sea level. The temperature at the base is $+8$ $^\circ C$. The drone ascends $600$ metres, where the temperature drops by $5$ $^\circ C$ for every $200$ metres of ascent. It then descends rapidly to an underwater chasm located at $-150$ metres 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 underwater scientific station has its command centre 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 analyse 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 $4$ metres high, at what depth (in metres) relative to sea level is the researcher at the end of their journey?

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

1. How many years did this civilisation last?

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

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?

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?

Year 8 is organising the end-of-year trip and is keeping a joint account of income and expenses by term. In the first term, they made $12000$ € from calendar sales. In the second term, they spent $15000$ € 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 $0$ €), how much will they need to earn from fourth-term activities? Write down the complete calculation using integers to represent income and expenses.

In the Picos de Europa National Park, a group of hikers analyzes the changes in elevation relative to the starting mountain hut (which is set as the reference point $0\text{ m}$). During the route, they complete four consecutive sections with the following altitude variations: they descend $160\text{ m}$, ascend $420\text{ m}$, descend $290\text{ m}$, and ascend $130\text{ m}$. To find out what altitude relative to the hut they are at the end of the day and to plan their effort, two hikers propose different methods: Marcos' method consists of operating from left to right in chronological order: $[((-160) + 420) - 290] + 130$; Sofía's method consists of grouping all ascents on one side and all descents on the other: $(420 + 130) + [(-160) + (-290)]$. a) Calculate the final result with both methods and name the properties of operations with integers that ensure both results coincide. b) If the mountain guide also asks them to calculate the “total accumulated elevation gain” (the sum of all meters traveled both uphill and downhill using absolute value $|a|$), justify which of the two methods facilitates this calculation without having to restate the problem from scratch, and calculate that elevation gain.

A reservoir in the Ebro river basin uses its optimal reserve capacity as a reference level ($0\text{ hm}^3$). On a winter Sunday at 08:00, the reservoir is at $+18\text{ hm}^3$ relative to that level. Due to snowmelt on the peaks, the reservoir continuously receives an inflow of $+14\text{ hm}^3$ per day. At the same time, for urban water supply and environmental flow, $-21\text{ hm}^3$ per day are released in a controlled manner. a) Model the state of the reservoir after $5$ full days (Friday at 08:00) using an expression with combined operations. Explain whether during those $5$ days the reservoir has gotten closer to or further from the optimal level, justifying your answer with the concept of absolute value. b) The river basin authority needs the reservoir to reach exactly the optimal level ($0\text{ hm}^3$) by Sunday at 08:00 (exactly $2$ days later). Assuming that the snowmelt inflow remains unchanged at $+14\text{ hm}^3$ daily over the weekend, calculate using an expression with integers how many cubic hectometers in total must be allowed to be released over those two days to achieve this goal.

Unit challenge. How does the cold hit your pocket? Take on the role of an energy manager and prepare an emergency action report to tackle energy poverty.

1. Real data: Visit the AEMET website and record the maximum and minimum temperatures of 5 inland or mountain towns with sub-zero readings. Identify them as elements of $\mathbb{Z}$ and calculate the thermal oscillation ($T_{\text{max}} - T_{\text{min}}$) for each one.

2. Expense model: The daily base cost of heating is $14,25$ €. For each degree below zero of the minimum temperature ($T_{\text{min}} < 0$), a surcharge of $1,30$ € is added, and for each degree above zero of the maximum temperature ($T_{\text{max}} > 0$), a rebate of $0,65$ € is applied. Formulate and solve the combined operations with integers and decimals to find the actual daily cost for each municipality.

3. Representation: Enter the records into a spreadsheet, represent the temperatures on a number line, and calculate the mean temperature of each town, $\frac{T_{\text{max}} + T_{\text{min}}}{2}$ (with decimals).

4. Proposal: Write a reasoned proposal prioritising which town to allocate the financial aid budget to.

Submit a brief report with your table, calculations, and graph.

The following will be assessed:

• Rigorous use of the set $\mathbb{Z}$ and correct calculation of thermal oscillations.

• Precision in combined operations with decimal numbers.

• Adequate representation of data on the number line.

• Critical and consistent justification of the aid proposal.