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.
One-step problems
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…
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?
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.
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.
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…
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 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.
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.
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?