Exercises: Operations with integers

At a ski resort in the Aragonese Pyrenees, the thermometer read $-4$ ºC at seven in the morning. If by midday the temperature has risen by $9$ ºC, what temperature does the thermometer show now? Express the calculation using integers.

Calculate the following simple additions and subtractions of integers:

1. a) $(+7) + (+5) =$
2. b) $(-8) + (-3) =$
3. c) $(+9) + (-4) =$
4. d) $(-6) + (+10) =$

Nadia has $25$ € in her bank account. If the bank charges her an internet bill of $30$ €, what will her new balance be? Express the result as an integer.

Use the rule of signs to solve these multiplications and divisions:

1. a) $(+4) \cdot (-3) =$
2. b) $(-5) \cdot (-2) =$
3. c) $(-20) : (+5) =$
4. d) $(-18) : (-3) =$

A diver is at a depth of $-15$ metres relative to sea level. If he decides to descend another $12$ metres, what depth will he be at now? Draw this on the vertical number line.

Solve the following operations by removing the brackets first:

1. a) $12 + (-5) =$
2. b) $8 - (+10) =$
3. c) $(-3) - (-7) =$
4. d) $(-5) + (+5) =$

Calculate the result of these simple combined operations, respecting the order of operations (remember: multiplication and division first, then addition and subtraction):

1. a) $5 + 2 \cdot (-3) =$
2. b) $(-10) : 2 + 8 =$
3. c) $4 \cdot 3 - 15 =$

You are on floor $2$ of a shopping centre. You go up $3$ floors to get to the food court and then go down $6$ floors to reach the car park. Which floor is the car park on? State each movement using an integer.

Marcos has solved two exercises in his notebook and has made very common mistakes with integers:

• Exercise 1: He wrote $(-7) + (-4) = +11$, justifying that "minus times minus is plus".

• Exercise 2: He wrote $-5^2 = 25$, reasoning that "any number squared gives a positive result".

1. a) Reasonably explain to Marcos what conceptual mistake he made in Exercise 1, clarifying the difference between adding integers of the same sign and multiplying integers. Write the correct solution.
2. b) Explain to Marcos why $-5^2$ does not give the same result as $(-5)^2$ based on the meaning of the base of a power and the order of operations. Write the correct value of both expressions.

At the weather observatory in Molina de Aragón (Guadalajara), AEMET recorded a temperature of $-7$ ºC at 7:00. Throughout the day, the following consecutive changes were recorded:

• A rise of $+11$ ºC during the morning.

• A drop of $6$ ºC in the mid-afternoon with the arrival of north wind.

• An additional decrease of $3$ ºC during the night.

1. a) Before calculating, make an estimate: will the final temperature at the end of the night be positive (above zero) or negative (below zero)? Justify your estimate by comparing the total rise with the combined effect of the initial cold and the drops, without calculating the exact figure yet.
2. b) Set up a single chain of additions and subtractions with integers to find the exact temperature at the end of the night and check if your estimate was correct.

Calculate the result of the following combined operations, paying special attention to the order of operations and the rule of signs:

1. a) $15 - (8 - 12) + (-5) \cdot 2$

1. b) $(-3) \cdot [(-2) + (-4) \cdot (-3)]$

1. c) $24 : (-4) - (-6) : (+3)$

1. d) $(-10) + (-2) \cdot (-5) + (-20) : 5$

A weather station in the Pyrenees recorded the following temperature changes over the course of a winter's day:

1. At 8:00 h, the temperature was $-5\ ^\circ C$.

2. At 12:00 h, it had risen by $8\ ^\circ C$.

3. At 18:00 h, it dropped by $4\ ^\circ C$ compared to the previous reading.

4. At 22:00 h, it dropped again by another $6\ ^\circ C$.

Write a single mathematical expression using integers that reflects all these changes, and calculate the final temperature at the end of the day.

Apply the distributive property to calculate these expressions in the simplest way possible (factorise out a common factor when necessary):

1. a) $(-7) \cdot (+12) + (-7) \cdot (-2)$

1. b) $15 \cdot (-4) - 15 \cdot (+6)$

1. c) $(-10) \cdot (-3) + (-10) \cdot (+13)$

An inspection drone starts from a height of $15$ meters above the level of a bridge. It makes the movements shown in the graph (ascents in orange, descents in red):

Using a sum of integers, calculate the drone's final position relative to the bridge. Is it above or below it?

On the bank statement of a comic book shop, the following transactions appear:

• Starting balance: $+450\ €$

• Payment to supplier: $-620\ €$

• Sale of collectible figures: $+185\ €$

• Refund for a defective item: $-45\ €$

• Income from subscriptions: $+110\ €$

Calculate the final balance of the account. If the bank charges a fee of $15\ €$ if the account goes overdrawn (balance less than 0), will the shop owner have to pay it?

Find the integer that simultaneously satisfies these three conditions:

1. Its modulus is greater than $5$ and less than $10$.

2. When multiplied by $-3$, the result is a positive number.

3. If you add its opposite to it, the result is $0$.

Is there more than one possible solution? Justify your answer.

Solve the following combined operations with square brackets. Remember to perform the operations inside the brackets first:

1. a) $18 - [ 2 + (-5) - (4 - 7) ]$

1. b) $[ (-6) \cdot 2 - (-10) ] : (-2)$

1. c) $30 : [ (-4) + (-2) \cdot (-3) ]$

1. d) $(-5) \cdot [ 8 - (12 : (-3)) ]$

A marine research team uses sonar to measure the distance between a rescue helicopter and a submarine conducting depth tests.

If the helicopter is at a height of $+120$ metres and the submarine is at a depth of $-85$ metres:

1. a) What is the total vertical distance separating them?
2. b) If the submarine ascends $25$ metres and the helicopter descends $30$ metres, what will the new distance between them be? Express it using operations with integers.

The underground car park of a train station in Madrid has $5$ floors below street level (from $-1$ to $-5$) and the office building above has $6$ floors above ground (from $+1$ to $+6$). Street level is floor $0$.

A maintenance worker starts his day on the street and makes these consecutive movements:

1. Goes down to floor $-4$ to check the meters.

2. Goes up $5$ floors to leave supplies in the offices.

3. Moves a number of floors equal to the opposite of $+3$.

4. Finally, goes up twice the number of floors he descended in the previous step.

1. a) Model the entire journey using a single combined operation of integers that determines the worker's final floor.
2. b) Solve the expression step by step and indicate which floor he ends up on and whether he is above or below ground.
3. c) If from his final position he must go to the boiler room on floor $-5$, calculate the change in floors using a subtraction of integers in the form: $\text{destination floor} - \text{origin floor}$, and interpret the sign of the result obtained.

To open a mystery box in a maths workshop, you must solve two chained challenges using exclusively the integers $-6$, $+4$, $-2$, and $+3$:

• Challenge 1: Use the four numbers exactly once each, along with the operations of your choice ($+$, $-$, $\cdot$, $:$) and any parentheses you need, to form a combined expression whose result is exactly equal to $-12$. Write the complete expression and check its result step by step.

• Challenge 2: Without testing random combinations, reason whether it is possible to multiply any three of those four numbers and add the fourth to obtain a strictly positive result. Justify your answer using the sign rules of multiplication and write, if possible, a combination that satisfies it along with its calculation.

Place the necessary parentheses and brackets in the following expression so that the final result is exactly equal to $-5$:

$-15 + 3 \cdot -4 - 12 : -6 + 1$

A caving team is exploring a deep chasm. They start at the entrance (level $0$ metres). They make the following moves consecutively:

1. They descend $45$ metres.

2. They climb up $12$ metres to set up a base camp.

3. They descend three times as far as they have just climbed.

4. Finally, they descend another $18$ metres to the bottom.

Write a single expression using integers reflecting the whole process and calculate the depth of the bottom of the chasm.

Calculate the value of the following combined expression step by step, strictly respecting the order of operations:

$(-2)^4 + \sqrt{25} \cdot (-3) - [(-8 + 10)^3 : (-2^2)]$

Note: Pay attention to the difference between $(-2)^2$ and $-2^2$.

In a strategy video game, a player starts with $500$ honour points. The rules are:

• For each battle won: $+120$ points.

• For each battle lost: $-150$ points.

• If they lose three battles in a row, they are penalised by doubling their accumulated negative score from those three battles.

If a player wins $2$ battles and then loses $3$ battles in a row, write down the combined operation to calculate their final score and solve it.

Find the unknown integer $x$ in each of the following equalities:

1. a) $x \cdot (-4 + 9) = -25$

1. b) $(-30) : x + (-12) = -18$

1. c) $2 \cdot |x| = 14$ (Find all possible solutions for $x$).

A "trio of integers" is said to be balanced if the sum of the first two multiplied by the third results in $-24$.

Determine which of the following trios are balanced by performing the corresponding operations:

1. a) $(+5, -8, +8)$
2. b) $(-10, +2, +3)$
3. c) $(-4, -2, +4)$

Complete the following "Integer Wall". The number in each upper cell must be the result of multiplying the two numbers in the cells supporting it.

Make up a real-life problem that can be solved using the following mathematical expression and then calculate the final result to answer your problem:

$100 + 4 \cdot (-15) - 3 \cdot (10 - 4)$

Two students, Lucía and Marta, have to calculate the value of the following expression with integers:

• Lucía applies the standard order of operations: she first performs the two multiplications separately and then adds the obtained results.

• Marta observes the structure of the calculation and applies the distributive property in reverse (factoring out the common factor): $C = (-15) \cdot (74 + 26)$.

1. a) Calculate the result following in detail Lucía's method and Marta's method. Verify that both approaches reach the same solution.
2. b) Compare the two methods and justify which of them is more efficient for mental calculation and why it reduces the risk of making mistakes with signs and intermediate operations.
3. c) Design a new combined operation consisting of the addition or subtraction of three products of integers where applying Marta's strategy allows it to be solved mentally in a few seconds without the need for long calculations on paper. Show your expression and its justified resolution.

Three unknown integers, $a$, $b$, and $c$, simultaneously satisfy the following three conditions:

1. Their product is strictly negative: $a \cdot b \cdot c < 0$.

2. The division $a : b$ is exact and results in a positive integer: $a : b > 0$.

3. The sum of two of them is zero: $a + c = 0$, also knowing that $a \neq 0$.

1. a) Logically and reasonably deduce the sign (positive or negative) of each of the three numbers ($a$, $b$, and $c$). Justify each step based on the concept of opposite numbers and the sign rules of multiplication and division.
2. b) If it is known that $|b| = 2$ and that the product of the three numbers is $a \cdot b \cdot c = -72$, determine the exact numerical values of $a$, $b$, and $c$.
3. c) With the values found in the previous section, calculate the result of the following combined operation respecting the order of operations and the sign rules: