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ESO Y1 · Mathematics
Numbers · Integers

Problems with whole numbers

Bank balance, degrees on the thermometer, floor in the lift, year BC. All these problems look alike: there is a starting point and a series of movements, positive or negative, that must be added in order.

Money and temperatures

Bank balance diagram with a starting value of 45 € and four successive movements giving a final balance of 35 €. Balance: 45 € −12 € +20 € −15 € −3 € 45 − 12 + 20 − 15 − 3 = 35 €
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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…

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?

Solve
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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…

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?

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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…

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?

0€ 10€ 20€ 30€ 40€ 50€ 60€ +45€ −12€ +20€ −15€ −3€ ?€ Start TV series Bizum Book Fee Balance
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Floors, heights and depths

Lifts, divers and drones move "upwards" (positive) or "downwards" (negative) from a level $0$.

Vertical axis with sea level at zero, a peak at $+1.200$ m, a cave at $-85$ m and the vertical distance between them, $1.285$ m. sea level (0) peak: +1.200 m cave: −85 m distance = |1.200 − (−85)| = 1.285 m
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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…

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.

Vertical scale of elevator floors, from −4 to +5, with the starting point marked on floor −3 +5 +4 +3 +2 +1 0 −1 −2 −4 −3 Start
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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 $4$…

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.

Sea level (0 m) 0 −10 −20 −18 +7 m −4 m ? ?
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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…

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?

+10000 +5000 −5000 −10000 −15000 Sea level (0 m) +8 848 m Everest −11 034 m Mariana Trench ? m Total distance
Solve

Dates before and after Christ

The year $0$ (approximately) separates the years BC (which are treated as negative, and the higher the number the older the date) from the years AD (positive).

Timeline with year zero in the centre, the year 570 BC as a negative number on the left and 415 AD as a positive number on the right. 0 570 a.C.→ −570 415 d.C.→ +415 before Christ: negativeafter: positive
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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…

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?

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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…

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.

Towards year 0 ➔ −600 −550 −500 −570 Birth −495 Death Age?
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Multi-step problems

Longer problems chain several movements together: it helps to write down the intermediate result after each step.

Diagram of the floors of a building with the ground floor at zero, upper floors positive and basements negative, showing successive movements. start0 up 12+12 down 15−3 up 7+4 down 14final result: −10
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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…

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?

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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…

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.

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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…

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.

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Summary

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