Exercises: Operations with integers

In a ski resort in the Rocky Mountains, the thermometer read $-4$ ºF at seven in the morning. If by midday the temperature has risen by $9$ ºF, what temperature does the thermometer show now? Express the operation 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) $(-6) + (+10) =$

Emma 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 sign rules to evaluate 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 $-15$ feet relative to sea level. If he decides to descend another $12$ feet, at what depth will he be now? Draw this on a vertical number line.

Evaluate the following operations by removing the parentheses first:

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

Evaluate the result of these simple combined operations, respecting the order of operations (remember: multiplications and divisions first, then additions and subtractions):

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 mall. You go up $3$ floors to go to the food court and then go down $6$ floors to reach the parking garage. On which floor is the parking garage? Indicate each movement with an integer.

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 Rockies recorded the following temperature changes over a winter's day:

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

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

3. At 6:00 p.m., it dropped by $4\ ^\circ C$ compared to the previous measurement.

4. At 10:00 p.m., it dropped by another $6\ ^\circ C$.

Write a single mathematical expression with 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 possible way (factor 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 $15$ feet above the level of a bridge. It makes the movements indicated in the graph (ascents in orange, descents in gray):

Calculate the final position of the drone with respect to the bridge using a sum of integers. Is it above or below it?

The following movements appear on the bank statement of a comic book store:

• Initial balance: $+450\ \$$

• Payment to supplier: $-620\ \$$

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

• Return of 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 into negative numbers (balance less than 0), will the store owner have to pay it?

Find the integer that simultaneously satisfies these three conditions:

1. Its absolute value 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, the result is $0$.

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

Evaluate the following combined operations with square brackets. Remember to work out what is 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 a sonar to measure the distance between a rescue helicopter and a submarine carrying out depth tests.

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

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

Place the necessary parentheses in the following expression so that the final result is exactly equal to $0$:

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

A caving team is exploring a deep chasm. They start at the entrance (level $0$ feet). They make the following consecutive movements:

1. They descend $45$ feet.

2. They go up $12$ feet to set up a base camp.

3. They descend three times what they have just gone up.

4. Finally, they go down another $18$ feet to the bottom.

Write a single expression with integers that reflects the whole process and calculate the depth at which the bottom of the chasm is located.

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 $500$ honor 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 penalized by doubling their negative score accumulated in 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) $x \cdot (-4 + 9) = -25$
2. b) $(-30) : x + (-12) = -18$
3. 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 box must be the result of multiplying the two numbers in the boxes it rests on.

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:

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