Operations with 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?

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.

In your savings account for the end-of-year trip, you have an initial balance of $\$900$. During the week, you make the following transactions:

• You deposit $\$400$ given to you by your grandmother.

• You pay for a cinema ticket costing $\$180$.

• You buy a second-hand book for $\$120$.

• You are paid back $\$60$ that you had lent to a friend.

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

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

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

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?

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) $(+35) : (-7) =$

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

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.

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

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

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) $(-5) \times (-2) + (-6) =$

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

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

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

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) $( -10 + 2 ) \times ( -4 )$

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.

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) $-10 - ( \square ) = -4$

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 $-10$.

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

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

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.

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

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

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

• Monday: rises $3$ pesos.

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

• Wednesday: loses $4$ pesos.

• 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.

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.

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

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

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

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

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

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

1. Ascends $80$ meters 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 traveled.