Operations with Integers
In a survival video game, you start with $150$ energy points. Every hour in a dark cave costs you $25$ points, and every light crystal you collect gives you $40$. How much energy do you have after $3$ hours in the caves and $2$ crystals? To answer, you need to add, subtract, and multiply integers without sign mistakes, and then combine several operations in the right order. Let's build that up step by step.
Opposites and Absolute Value
Two numbers that are the same distance from $0$ but on opposite sides are opposites. The opposite of $7$ is $-7$, and the opposite of $-7$ is $7$.
The absolute value of a number is its distance from $0$ on the number line. Distance is never negative, so an absolute value is never negative. We write it with vertical bars: $|-12| = 12$, $|5| = 5$, $|0| = 0$.
Both $-7$ and $7$ are $7$ units from $0$, so $|-7| = |7| = 7$.
Find the absolute value of the following numbers and order them from least to greatest: $|-12|, |+5|, |0|, |-3|, |+1|$
Find the absolute value of the following numbers and order them from least to greatest:
$|-12|, |+5|, |0|, |-3|, |+1|$
Adding and Subtracting Integers
Picture a number line. Adding a positive number moves you right. Adding a negative number moves you left. Here is a cold morning: it is $-6\,^\circ\text{F}$ at 7 a.m., it rises $14$ degrees by noon, then falls $11$ degrees by night.
Sum pyramids use this skill. Each box is the sum of the two boxes directly below it, so if you know the top and one box below it, you subtract to find the other.
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…
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…
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.
- a) What number does it reach after the first jump of $+4$?
- b) What number does it finally land on after the second movement?
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…
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.
Multiplying and Dividing: The Rule of Signs
Look at what happens when the second factor stays $-2$ and the first factor goes down by $1$ each time:
| Product | Result |
|---|---|
| $3 \times (-2)$ | $-6$ |
| $2 \times (-2)$ | $-4$ |
| $1 \times (-2)$ | $-2$ |
| $0 \times (-2)$ | $0$ |
| $(-1) \times (-2)$ | $2$ |
| $(-2) \times (-2)$ | $4$ |
The results go up by $2$ each row, so the pattern forces $(-1) \times (-2) = +2$. That is the rule of signs.
Evaluate the following products and quotients, applying the rule of signs correctly: a) $(+6) \cdot (-3) =$ b) $(-20) : (-5) =$ c) $(-8) \cdot (-4) =$ d) $(+35) : (-7) =$
Evaluate the following products and quotients, applying the rule of signs correctly:
- a) $(+6) \cdot (-3) =$
- b) $(-20) : (-5) =$
- c) $(-8) \cdot (-4) =$
- d) $(+35) : (-7) =$
Complete the following multiplication number pyramid. The value in each box is the product of the two integers located directly below it.
Complete the following multiplication number pyramid. The value in each box is the product of the two integers located directly below it.
Signs in Front of Parentheses
A sign in front of parentheses tells you what to do when you remove them. A plus sign leaves everything inside unchanged. A minus sign flips every sign inside.
Absolute value bars and opposites often show up together. Remember the bars act like parentheses, and a minus sign outside them is applied last.
Remove the parentheses and evaluate the following expressions: a) $+(+7) + (-3) - (+5) =$ b) $-(-4) - (+8) + (-2) =$ c) $+(-10) - (-6) - (-1) =$
Remove the parentheses and evaluate the following expressions:
- a) $+(+7) + (-3) - (+5) =$
- b) $-(-4) - (+8) + (-2) =$
- c) $+(-10) - (-6) - (-1) =$
Simplify the following expression by removing parentheses and using the properties of absolute value and the opposite:…
Simplify the following expression by removing parentheses and using the properties of absolute value and the opposite:
$- | -8 + 3 \cdot (-2) | - [ - ( - 10 ) + ( - 4 ) \cdot ( - 3 ) ]$
Order of Operations with Grouping Symbols
When an expression has several operations, everyone must get the same answer, so we agree on an order. Grouping symbols are parentheses $( \ )$, brackets $[ \ ]$, braces $\{ \ \}$, and absolute value bars. When they are nested, work from the inside out.
Some problems write division with a colon, $a : b$. It means $a \div b$.
Evaluate these multi-step operations using the order of operations: a) $10 + 2 \cdot (-3) =$ b) $(-15) : 3 - 4 =$ c) $8 - (-10) : 2 =$ d) $(-5) \cdot (-2) + (-6) =$
Evaluate these multi-step operations using the order of operations:
- a) $10 + 2 \cdot (-3) =$
- b) $(-15) : 3 - 4 =$
- c) $8 - (-10) : 2 =$
- d) $(-5) \cdot (-2) + (-6) =$
Calculate the result of the following mixed operations, respecting the order of operations and paying special attention to the signs: a)…
Calculate the result of the following mixed operations, respecting the order of operations and paying special attention to the signs:
- a) $-5 \cdot [ -3 + 2 \cdot (4 - 7) ]$
- b) $( -12 + 16 ) : ( 1 - 5 ) - 3 \cdot ( -2 )$
Solve the following mixed operation, paying special attention to the order of operations and the use of parentheses, brackets, and braces:…
Solve the following mixed operation, paying special attention to the order of operations and the use of parentheses, brackets, and braces:
$15 - \{ -3 \cdot [8 - (12 - 15) \cdot (-2)] + (-18) : (5 - 11) \}$
The Distributive Property
When a number multiplies a sum or difference, you can multiply it by each term separately.
The property also works backwards. If every term has the same factor, factor it out and the arithmetic gets easier.
Apply the distributive property to evaluate the following expressions in two different ways (evaluating inside the parentheses first and distributing the factor): a)…
Apply the distributive property to evaluate the following expressions in two different ways (evaluating inside the parentheses first and distributing the factor):
- a) $-3 \cdot ( 8 - 12 )$
- b) $5 \cdot [ -4 + ( -6 ) ]$
- c) $( -10 + 2 ) \cdot ( -4 )$
Use the distributive property in reverse (factor) to calculate the result of the following expression in the quickest and simplest way possible:…
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.
Finding a Missing Number
Sometimes one number is hidden: $\square$ or $x$. To find it, undo the operations. Each operation has an inverse, and what you do to one side you do to the other.
| Equation | Undo it | Missing number |
|---|---|---|
| $a + \square = c$ | subtract $a$ | $c - a$ |
| $\square - a = c$ | add $a$ | $c + a$ |
| $a - \square = c$ | subtract from $a$ | $a - c$ |
| $a \times \square = c$ | divide by $a$ | $c \div a$ |
| $\square \div a = c$ | multiply by $a$ | $c \times a$ |
| $a \div \square = c$ | divide $a$ by $c$ | $a \div c$ |
Always check by putting your answer back in.
For a nested equation, simplify what you can, then peel off the operations from the outside in. Here the equation is $36 \div \{-3 \times [x - (4 - 9)]\} = -4$, and the diagram shows each step forward and the undo step below it.
You can also build an expression by working backward from the answer you want.
Fill in the blanks with the corresponding integer to make the equations true: a) $-15 + ( \square ) = -32$ b) $4 \cdot ( \square ) = -36$ c) $( \square ) : (-3) = +12$…
Fill in the blanks with the corresponding integer to make the equations true:
- a) $-15 + ( \square ) = -32$
- b) $4 \cdot ( \square ) = -36$
- c) $( \square ) : (-3) = +12$
- d) $-10 - ( \square ) = -4$
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…
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.
Calculate the value that the unknown $x$ must have for the following equality to hold. Show all the intermediate steps of your reasoning:…
Calculate the value that the unknown $x$ must have for the following equality to hold. Show all the intermediate steps of your reasoning:
$24 : \{ -2 \cdot [x - (12 - 17)] \} = -3$
Stories with Gains and Losses
Stories become math when you decide on a sign for each change.
-
Up, deposit, gain, rise: positive.
-
Down, payment, loss, fall: negative.
Write the starting amount first, then each change in order. Equal repeated changes are a product: three charges of $\$9$ is $3 \times (-9)$.
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 $\$20$…
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.
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…
A delivery driver enters a building and parks his van in the second basement level (floor $-2$). He makes the following trips:
-
He goes up 5 floors to deliver a package.
-
He goes down 4 floors to pick up another box.
-
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:
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…
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.
Multi-Step Stories
Many stories describe a change in terms of an earlier amount, such as "twice what it gained" or "a third of its depth." Compute that amount first (it is a plain positive size), then apply the sign of the move.
| Phrase | Amount | Sign |
|---|---|---|
| goes up / gains / rises | the size | $+$ |
| goes down / loses / falls | the size | $-$ |
| twice, three times | $2 \times$, $3 \times$ the earlier amount | as above |
| half of, a third of | $\div 2$, $\div 3$ of the earlier amount | as above |
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…
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.
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…
An oceanographic research bathyscaphe is submerged at $-450$ feet with respect to sea level. It performs a maneuver in three stages:
-
It descends a distance equivalent to a third of its current depth.
-
It ascends $120$ feet to avoid a thermal current.
-
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.
A technology company listed on the stock exchange started the week with a share value of $\$45…
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.
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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.
Distance Traveled and Absolute Value
Where you end up and how far you traveled are different questions. Your position is the sum of the signed moves. The distance traveled adds up the absolute values of the moves, because walking down $36$ feet is still $36$ feet of walking.
Absolute value also tells you how far a position is from $0$. "Go up the absolute value of your depth" means go up exactly as far as you are below the surface.
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…
An underwater exploration drone is submerged at a depth of $-150$ feet with respect to sea level. It makes the following movements:
-
It ascends $80$ feet to avoid a current.
-
It descends three times what it has just ascended.
-
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.
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…
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?
Back to the start: After $3$ hours in the caves and $2$ crystals, the player has $150 - 3 \times 25 + 2 \times 40 = 150 - 75 + 80 = 155$ energy points.