Exercises: Operations with integers

At a ski resort in Sierra Nevada, the thermometer read $-5^{\circ}C$ at 7:00 a.m. At noon, the temperature rose by $12^{\circ}C$, and when night fell, it dropped again by $9^{\circ}C$. What temperature did the thermometer read at the end of the day? Write the complete expression using integers.

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

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

Find the absolute value of the following numbers and order them from least to greatest:

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

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

• You deposit $\$20$ that your grandmother gave you.

• You pay for a movie ticket that costs $\$9$.

• You buy a used book for $\$6$.

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

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

Evaluate the following products and quotients, applying the rule of signs correctly:

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

A delivery driver enters a building and parks his van in 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.

Which floor is the delivery driver on when he finishes his work? Use the following diagram to help you:

Evaluate these multi-step operations using the order of operations:

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

Remove the parentheses and evaluate the following expressions:

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

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 mixed operations, respecting the order of operations and paying special attention to the signs:

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

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

An oceanographic research bathyscaphe is submerged at $-450$ feet with respect to sea level. It performs a maneuver in three stages:

1. It descends a distance equivalent to a third of its current depth.

2. It ascends $120$ feet to avoid a thermal current.

3. It descends twice what it ascended in the previous step.

At what depth is the bathyscaphe after finishing the maneuver? Express it using a mixed operation of integers.

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

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

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

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

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

Apply the distributive property to evaluate the following expressions in two different ways (evaluating inside the parentheses first and distributing the factor):

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

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

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

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

If at the beginning of the week the balance was $\$-15$, what is the final balance? Write your work as a sum of positive and negative numbers.

Complete 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 using at least one multiplication, one division, one addition, and parentheses, such that the final result is $-10$.

Hint: You can start with the result and work backward.

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

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

Calculate the value that the unknown $x$ must have for the following equality to hold. Show all the intermediate steps of your reasoning:

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

A technology company listed on the stock exchange started the week with a share value of $\$45$. During the five working days, it experienced the following daily variations:

• Monday: goes up by $\$3$.

• Tuesday: goes down by twice what it went up on Monday.

• Wednesday: loses $\$5$.

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

• Friday: goes up by three times Monday's variation.

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

Use the distributive property in reverse (factor) to calculate the result of the following expression in the quickest and simplest way possible:

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

Once the common factor has been extracted, simplify inside the parentheses and obtain the final result.

An underwater exploration drone is submerged at a depth of $-150$ feet with respect to sea level. It makes the following movements:

1. It ascends $80$ feet to avoid a current.

2. It descends three times what it has just ascended.

3. It ascends again by half the depth it was at after the second movement.

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

Simplify the following expression by removing parentheses and using the properties of absolute value and the opposite:

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

In a very tall skyscraper, a maintenance worker is in basement $-4$. He goes up $12$ floors to fix an air conditioning unit. Then, he goes down twice the number of floors he initially went up. Finally, he goes up the absolute value of the floor he is currently on to reach his last task.

On which floor does the worker finish his workday?

Complete the empty gaps in this integer pyramid knowing that the value of 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.