Exercises: Operations with integers

At a ski resort in Sierra Nevada, the thermometer read $-5^{\circ}C$ at seven in the morning. At noon, the temperature rose by $12^{\circ}C$ and, by nightfall, it dropped again by $9^{\circ}C$. What temperature did the thermometer read at the end of the day? Write the complete calculation using integers.

Look at the following number line and the moves made by a grasshopper starting from point $0$. State the integer corresponding to each final position after making the jumps shown.

1. a) Which number does it reach after the first jump of $+4$?
2. b) Which number does it finally land on after the second move?

Calculate the absolute value of the following numbers and order them from smallest to largest:

$|-12|, |+5|, |0|, |-3|, |+1|$

In your savings account for the end-of-year trip, you have an initial balance of $\text{€}\ 45$. During the week, you make the following transactions:

• You deposit $\text{€}\ 20$ given to you by your grandmother.

• You pay for a cinema ticket costing $\text{€}\ 9$.

• You buy a second-hand book for $\text{€}\ 6$.

• You are paid back $\text{€}\ 3$ that you had lent to a friend.

Calculate the final balance using an expression with positive and negative numbers.

Solve the following multiplications and divisions, correctly applying the rule of signs:

1. a) $(+6) \times (-3) =$
2. b) $(-20) : (-5) =$
3. c) $(-8) \times (-4) =$
4. d) $(+35) : (-7) =$

A delivery driver enters a building and parks his van on the second basement level (floor $-2$). He makes the following trips:

1. He goes up 5 floors to deliver a package.

2. He goes down 4 floors to pick up another box.

3. He goes up 7 floors for the last delivery.

Calculate which floor the delivery driver is on when he finishes his work. Use the following diagram, which shows the starting floor; you can mark each trip on it.

Calculate the result of these combined operations, respecting the order of operations:

1. a) $10 + 2 \times (-3) =$
2. b) $(-15) : 3 - 4 =$
3. c) $8 - (-10) : 2 =$
4. d) $(-5) \times (-2) + (-6) =$

Expand the brackets and calculate the final result of the following expressions:

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

Cian has solved the following combined operation with integers on the board:

His step-by-step working was as follows:

1) $12 - 4 = 8$

2) Inside the bracket: $(-3) - 5 = -8$

3) Multiplies inside the bracket: $(-8) \times (-2) = 16$

4) Final result: $8 \times 16 = 128$

Identify and explain in a reasoned manner the two mistakes made by Cian regarding the order of operations. Next, solve the operation correctly step by step yourself, carefully applying the rules of signs.

Consider the following expression with integers:

1. a) Before doing any calculations on paper, reasonably estimate whether the final result will be a positive number, negative, or zero. To do so, mentally analyze the accumulated size of the terms that subtract versus those that add.
2. b) Calculate the exact value step by step, respecting the order of operations and the rules of signs.
3. c) Compare your result with the initial estimate and argue why your prediction was or was not correct.

In a survival video game, a player starts with $150$ energy points. For every hour spent exploring dark caves, they lose $25$ points, but for every "light crystal" they collect, they gain $40$ points.

If the player spends $4$ hours in the caves and manages to collect $3$ crystals, write an expression with integers that represents their final energy and calculate it.

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

1. a) $-5 \times [ -3 + 2 \times (4 - 7) ]$

1. b) $( -12 + 16 ) : ( 1 - 5 ) - 3 \times ( -2 )$

An oceanographic research bathyscaphe is submerged at $-450$ metres relative to sea level. It performs a three-stage manoeuvre:

1. Descends a distance equal to one third of its current depth.

2. Ascends $120$ metres to avoid a thermal current.

3. Descends double what it ascended in the previous step.

At what depth is the bathyscaphe after completing the manoeuvre? Express it using a combined operation with integers.

Fill in the blanks with the corresponding integer to make the equations true:

1. a) $-15 + ( \square ) = -32$

1. b) $4 \times ( \square ) = -36$

1. c) $( \square ) : (-3) = +12$

1. d) $-10 - ( \square ) = -4$

Apply the distributive property to solve the following expressions in two different ways (first calculating inside the brackets and then distributing the factor):

1. a) $-3 \times ( 8 - 12 )$

1. b) $5 \times [ -4 + ( -6 ) ]$

1. c) $( -10 + 2 ) \times ( -4 )$

In a young saver's bank account, the following transactions are recorded in one week: a deposit of $45$ €, a card payment of $-12$ €, a transfer received of $20$ €, and three automatic subscription charges of $-9$ € each.

If the balance at the start of the week was $-15$ €, what is the final balance? Write down the process using an addition of positive and negative numbers.

Solve the following multiplication number pyramid. The value in each box is the product of the two integers located directly below it.

Write a mathematical expression that uses at least one multiplication, one division, one addition, and a set of brackets, such that the final result is $-10$.

At the Met Éireann station in Valentia Observatory (Co. Kerry), the following temperature variations were recorded on a winter day:

• At 4:00 h, the thermometer read $-9\,^{\circ}\text{C}$.

• Between 4:00 h and 10:00 h, the temperature rose at a constant rate of $2\,^{\circ}\text{C}$ per hour.

• Between 10:00 h and 16:00 h, the temperature rose by a total of another $5\,^{\circ}\text{C}$.

• From 16:00 h to 23:00 h, the arrival of a cold front caused a constant drop of $3\,^{\circ}\text{C}$ every hour.

1. a) Write a single combined mathematical expression that includes all these variations to calculate the temperature at 23:00 h.
2. b) Solve it step by step, applying the order of operations.
3. c) Justify, based on the sign of the result obtained, whether at 23:00 h outdoor water would be frozen or in a liquid state.

To calculate the numerical value of the expression:

two 2nd Year students propose two different strategies:

• Cian says: "The best way is to apply the distributive property: multiply $-7$ by each term inside the bracket and then add and subtract the results."

• Aoife says: "It is much faster to first solve the operations grouped inside the bracket and then multiply the result by $-7$."

1. a) Solve the expression following Cian's procedure in detail.
2. b) Solve the expression following Aoife's procedure in detail.
3. c) Compare both paths: which of the two methods was more advantageous in this case? Explain what numerical feature the terms inside the bracket had that justifies your choice.

Solve the following combined operation, paying special attention to the order of operations and the use of brackets, square brackets, and braces:

$15 - \{ -3 \times [8 - (12 - 15) \times (-2)] + (-18) : (5 - 11) \}$

Find the value that the unknown $x$ must have for the following equation to hold true. Show all intermediate steps of your working:

$24 : \{ -2 \times [x - (12 - 17)] \} = -3$

A tech company listed on the stock market started the week with a value per share of $45$ €. During the five working days, it underwent the following daily changes:

• Monday: rises $3$ €.

• Tuesday: falls twice as much as it rose on Monday.

• Wednesday: loses $4$ €.

• Thursday: recovers half of what was lost between Tuesday and Wednesday.

• Friday: rises three times the change on Monday.

What is the final value of the share at the close on Friday? Express the whole process with a single operation using integers.

Use the distributive property in reverse (factorising out a common factor) to calculate the result of the following expression as quickly and easily as possible:

$(-12) \times 35 + (-12) \times (-18) + (-12) \times 33$

Once the common factor is taken out, solve the expression inside the brackets and obtain the final result.

An underwater exploration drone is submerged at a depth of $-150$ metres relative to sea level. It makes the following moves:

1. Ascends $80$ metres to avoid a current.

2. Descends three times as far as it just ascended.

3. Ascends again by half the depth at which it was located after the second move.

Calculate the final depth of the drone and express it using a combined operation that includes absolute values for the distances travelled.

Simplify the following expression by expanding brackets and using the properties of absolute value and the opposite:

$- | -8 + 3 \times (-2) | - [ - ( - 10 ) + ( - 4 ) \times ( - 3 ) ]$

In a very tall skyscraper, a maintenance worker is in basement $-4$. He goes up $12$ floors to fix an air conditioner. Then, he goes down twice as many floors as he initially went up. Finally, he goes up by the absolute value of the floor he is currently on to reach his final job.

On which floor does the worker end his working day?

Fill in the empty boxes in this integer pyramid, knowing that the value in each box is equal to the sum of the two numbers directly below it. You must first calculate the boxes at the base to be able to complete the rest.

You have the following six integers: $\{-6, -4, -2, 3, 5, 8\}$.

You must choose exactly four distinct numbers from the list to complete the empty boxes in this structure, so that the final result is exactly $-1$:

1. a) Find which number must be placed in each box and write the complete expression without repeating any of the chosen numbers.
2. b) Solve the resulting expression step by step, justifying each transformation according to the order of operations.
3. c) Without changing the position or replacing any of the four numbers you selected, explain what single sign change in one of them would allow the final result to be $+1$, relying on the properties of integer division.

Two underwater robots from the Marine Institute are exploring a marine fault off the Cliffs of Moher (Co. Clare):

• Robot Alpha starts from sea level ($0\text{ m}$) and descends at a constant speed of $5\text{ m/min}$ for $6\text{ minutes}$. Then, due to a current, it ascends $8\text{ metres}$ and, finally, descends twice the metres it just ascended.

• Robot Beta is located at an altitude/depth of $-18\text{ metres}$ and receives the order to move vertically until it is positioned at exactly half the final depth reached by robot Alpha.

1. a) Write a single combined operation with integers that determines the final depth of robot Alpha and calculate its value.
2. b) Determine the final depth of robot Beta and set up a subtraction of integers that expresses how many metres it must move from its initial depth ($-18\text{ m}$) to the final depth. Interpret the direction of movement based on the sign of that subtraction.
3. c) The distance between two depths $x$ and $y$ is defined by the absolute value of their difference: $|x - y|$. If the mission safety regulations require that the distance between both robots never exceeds $15\text{ metres}$ at any time, calculate the final distance between Alpha and Beta and justify whether the regulations are met or breached.