Problems with integers
You already know how to add, subtract, multiply and divide integers, and how to combine them in a fixed order. The hard part of a problem is almost never the calculation: it is translating the story into numbers with the correct sign. Here you are going to practise this with real problems.
Imagine a cold January evening in Athlone. Met Éireann says the thermometer reads $-3$ °C at 6 p.m. and the temperature will fall by another $5$ °C by midnight. What will it read? You will find out in the first section.
One-step problems
On a winter morning in the Wicklow Mountains, the thermometer read $-5^\circ\text{C}$ at eight in the morning. By midday, the temperature had risen by $9^\circ\text{C}$.…
On a winter morning in the Wicklow Mountains, 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?
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…
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.
Cian 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…
Cian 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.
Multi-step problems
When there are several movements in a row, it is best to note down the position after each one, rather than working everything out in your head in one go. You can also add the positive moves and the negative moves separately and then combine them: the final position is the same whatever order you use.
In a board game, each correct answer adds $10$ points and each mistake deducts $15$ points. If Aoife has $45$ points and gets the next three questions wrong, what will…
In a board game, each correct answer adds $10$ points and each mistake deducts $15$ points. If Aoife has $45$ points and gets the next three questions wrong, what will her score be at the end?
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…
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.
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…
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.
Distances and differences with integers
Many problems ask "how far apart" two things are: this is the distance between two integers, $|a - b|$.
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.
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.
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…
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.
- a) Represent with integers the position of the bird and that of the submarine after rising.
- b) Calculate the vertical distance between the bird and the submarine in its final position.
On a winter day in Kilkenny, Met Éireann recorded a minimum temperature of $-4\ ^\circ\text{C}$ and a maximum of $11\ ^\circ\text{C}$. Aoife calculated the temperature…
On a winter day in Kilkenny, Met Éireann recorded a minimum temperature of $-4\ ^\circ\text{C}$ and a maximum of $11\ ^\circ\text{C}$. Aoife 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 Aoife 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.
Problems with multiplication: rates of change
When something changes "by a fixed amount each time" (every hour, every round, every object), you multiply that amount by the number of times, with its sign.
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 $75$…
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 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…
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 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…
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?
Dates, counting steps and unknowns
Some problems use dates, ask how many times something happens, or hide a number you have to find.
For dates, treat years BC as negative and years AD as positive. The time between two years is the later year minus the earlier year. (If a problem says to allow for there being no year $0$, it will tell you.)
To find how many times a change happens, divide the total change by the change each time.
When the starting value is the unknown, call it $x$, write the whole story as an equation, then work backwards.
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).
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).
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…
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.
A school athletics club in Galway 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,…
A school athletics club in Galway 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.
Back to Athlone: $-3 - 5 = -8$ °C at midnight.