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Temperatures, bank balances and depths, where the sign carries the meaning.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you solve problems where negative numbers mean something real: a temperature below zero, an account that is overdrawn, a depth below the surface. You will turn each story into signed numbers, choose whether to add, subtract, multiply or divide, and say what the sign of your answer means. The arithmetic you already have; the new skill is reading the story and the answer correctly.
You can add, subtract, multiply and divide positive and negative numbers, including fractions and decimals. You know that a number line has 0 in the middle, with positive numbers to the right and negative numbers to the left, and that the absolute value of a number is its distance from 0. In this lesson you put all four operations to work on real problems: temperatures, money, depths and scores. The arithmetic is the part you already have. The new work is reading a story, deciding which operation it describes, and saying what the sign of your answer means.
| Term | What it means |
|---|---|
| Signed number | A number with a sign, positive or negative, such as $+12$ or $-7.5$. |
| Reference point | The value that counts as zero in a story: sea level, a balance of 0 dollars, par in golf. |
| Change | How much a quantity went up or down: positive for a rise or a gain, negative for a fall or a loss. |
| Balance | The amount of money in an account. A negative balance means money is owed. |
| Elevation | Height above sea level. A place below sea level has a negative elevation. |
| Deposit and withdrawal | Money put into an account (a positive change) and money taken out (a negative change). |
| Absolute value | The distance of a number from 0, written $\vert x\vert $: $\vert -45\vert = 45$. |
| Mean | The sum of the values divided by how many values there are. |
In a word problem, every negative number stands for something real. A temperature of $-8^\circ$F is 8 degrees below zero. A balance of $-30$ dollars is 30 dollars owed. An elevation of $-282$ feet is 282 feet below sea level. Each story has a reference point, the thing that counts as zero, and the sign tells you which side of it you are on.
Most problems then ask about changes. A change has a direction, so it has a sign too: a rise, a gain or a deposit is positive, and a fall, a loss or a withdrawal is negative. Once every quantity is written as a signed number, the story turns into arithmetic you already know:
| The story says | Operation |
|---|---|
| a value, then a change | add the change |
| how far apart, or how much it changed | subtract: end minus start |
| the same change, repeated | multiply |
| a total shared equally, or an average | divide |
The last step is always to read the answer back into the story. A final balance of $-14$ dollars means 14 dollars owed. A change of $-21$ degrees means the temperature fell by 21 degrees.
Another way: picture
Picture a thermometer standing on its end as a vertical number line. Zero is marked in the middle. A reading moves up the tube when it gets warmer and down when it gets colder, so every change is a move up or down from where the reading was, and the number next to the top of the liquid is the signed temperature.
Another way: story
An elevator in a tall building also goes below the ground floor. The lobby is floor 0, parking levels are floors $-1$, $-2$ and $-3$, and pressing a button changes the floor by a signed number. Going from floor $-3$ to floor 5 is a change of $5 - (-3) = 8$ floors up.
Before you calculate anything, go through the problem and give every number its sign. Words tell you which sign to use.
For example, "a submarine is 150 meters below the surface and then climbs 40 meters" becomes the position $-150$ and the change $+40$. "A store lost 1,200 dollars in March and made 800 dollars in April" becomes $-1{,}200$ and $+800$. Doing this first stops you from making the most common mistake in these problems, which is to do the right arithmetic with the wrong sign. It also shows you when a number has no sign at all: a count of days, hours or people is just a count, and it is the thing you multiply or divide by.
When a story gives a starting value and then a change, add the change to the start. A diver at $-20$ feet who rises 8 feet ends at $-20 + 8 = -12$ feet. A diver at $-20$ feet who goes 8 feet deeper ends at $-20 + (-8) = -28$ feet. Writing the second change as adding a negative, rather than as subtracting, keeps every step the same: start plus change equals end.
Several changes in a row are handled one at a time, keeping a running total, or all at once by adding the changes first. A temperature of $-5^\circ$ that rises 12 degrees and then falls 9 degrees ends at $-5 + 12 + (-9) = -2^\circ$. The changes alone add to $12 + (-9) = 3$, a net rise of 3 degrees, and $-5 + 3 = -2$ as well.
When a story asks how much something changed, or how far apart two values are, subtract: end minus start. If the temperature went from $-7^\circ$F to $15^\circ$F, the change was $15 - (-7) = 15 + 7 = 22$ degrees, a rise. If it went from $15^\circ$F to $-7^\circ$F, the change was $-7 - 15 = -22$ degrees, a fall. The order matters, because it decides the sign of the change.
When a problem asks only for a distance, such as how many feet separate two elevations, the answer is the absolute value of the difference, because a distance is never negative. Subtracting a negative number is the same as adding its opposite, and this is exactly where that rule earns its keep: the distance from $-282$ to $100$ is $100 - (-282) = 382$.
When the same change happens several times, multiply the change by the number of times. Five withdrawals of 20 dollars change a balance by $5 \times (-20) = -100$ dollars. A tank that drains 3.5 gallons an hour for 6 hours changes by $6 \times (-3.5) = -21$ gallons.
When a total change is shared equally, or when you want an average, divide. If a stock price changed by $-18$ dollars over 4 days, the average change was $-18 \div 4 = -4.5$ dollars a day. The mean of a list of signed numbers is found the usual way, the sum divided by how many there are, and the sum keeps every sign. The mean of $-6$, $2$, $-9$ and $1$ is $-12 \div 4 = -3$.
Many real problems need two or more operations. Break them into pieces and label each piece with what it means before you combine them. A club that starts with 60 dollars, buys 4 pizzas at 13.50 dollars each and collects 2 dollars from each of 15 members is doing three things: a repeated cost $4 \times (-13.50) = -54$, a repeated gain $15 \times 2 = 30$, and a start of 60. Then $60 + (-54) + 30 = 36$ dollars.
Follow the order of operations: multiply and divide before you add and subtract, unless parentheses say otherwise. Writing each piece on its own line makes the order clear, and it makes a mistake easy to find, because each line can be checked against the story.
Real measurements are rarely whole numbers. Money comes in dollars and cents, temperatures are often given to a tenth of a degree, and depths and heights can be fractions of a foot. The rules for signs do not change. Work out the size with the fraction or decimal methods you know, and decide the sign from the story.
With money, keep two decimal places all the way through and round only at the end. With fractions, it often helps to turn them into decimals when the question asks for a decimal answer. A river that falls $\tfrac{3}{4}$ inch a day for 10 days changes by $10 \times \left(-\tfrac{3}{4}\right) = -7.5$ inches.
Use the same five steps for every problem with negative numbers.
How to check. First, does the sign make sense? Money owed should be negative, a depth below the surface should be negative, and a time or a distance should be positive. Second, work backward: start from your answer and undo each change to see whether you get back to where the story began. Third, estimate: round the numbers and see whether your answer is about the right size. If a diver who started 30 feet down and rose 10 feet ends up at $-40$ feet, something went the wrong way.
On the morning of January 22, 1943, the temperature in Spearfish, South Dakota, was $-4^\circ$F. At about 7:30 a.m. a warm wind called a chinook swept down from the mountains, and two minutes later the thermometer read $45^\circ$F. The change was $45 - (-4) = 45 + 4 = 49$ degrees, a rise, in 2 minutes: an average of $49 \div 2 = 24.5$ degrees a minute. It is still the fastest temperature change ever recorded. By about 9:30 a.m. the wind had died and the temperature was back to $-4^\circ$F. That second change was $-4 - 45 = -49$ degrees, a fall of the same size. The two changes add to $49 + (-49) = 0$, which is why the town ended where it began.
Badwater Basin in Death Valley, California, is 282 feet below sea level, the lowest point in North America, so its elevation is $-282$ feet. Mount Whitney, about 80 miles away, rises to 14,505 feet, the highest point in the 48 neighboring states. The difference in height is $14{,}505 - (-282) = 14{,}505 + 282 = 14{,}787$ feet. Hikers who walk the "lowest to highest" route climb nearly three miles straight up in total, because $14{,}787 \div 5{,}280 \approx 2.8$ miles. Subtracting the negative elevation adds the 282 feet below sea level to the climb, exactly as it should.
In football, each play gains or loses yards, and a team keeps the ball if it gains at least 10 yards in four plays. Suppose a running back gains 7 yards, the quarterback is sacked for a loss of 8, and then a pass gains 6. The net change is $7 + (-8) + 6 = 5$ yards, so on the fourth play the team still needs $10 - 5 = 5$ more yards. Announcers call that "fourth and five." If the sack had been for a loss of 12 instead, the net would be $7 + (-12) + 6 = 1$ yard, and the team would need 9.
The most common mistake is dropping the sign of a quantity. A problem that says a balance is 40 dollars overdrawn means $-40$, and treating it as 40 gives an answer that is wrong by 80 dollars.
The second is subtracting in the wrong order when finding a change. The change from $-5$ to 3 is $3 - (-5) = 8$, a rise. Working out $-5 - 3 = -8$ gives the right size but the wrong direction.
The third is giving a negative distance or a negative time. A distance between two elevations and a number of hours are always positive. If you get a negative, you subtracted in the wrong order or divided a negative by a positive when the story had two negatives.
The last is stopping at the number. $-14$ is not a finished answer to a question about a bank account; "she owes 14 dollars" is.
Write the starting depth as a signed number.
$\text{start} = -120 \text{ m}$
Below the surface is below the reference point, sea level, so the depth is negative.
Write both moves as signed changes.
$\text{rise} = +45, \quad \text{dive} = -70$
Moving up toward the surface is positive; moving down is negative.
Add the first change to the start.
$-120 + 45 = -75$
The submarine is still below the surface, 75 meters down.
Add the second change.
$-75 + (-70) = -145$
Adding a negative moves further down the number line.
Interpret the answer and check with the net change.
$45 + (-70) = -25, \quad -120 + (-25) = -145$
The two moves together are 25 meters down, so the submarine ends 145 meters below the surface.
Write the two temperatures.
$\text{noon} = 12^\circ\text{F}, \quad \text{midnight} = -9^\circ\text{F}$
The midnight reading is below zero, so it is negative.
Write the change as end minus start.
$\text{change} = -9 - 12$
A change always subtracts the starting value from the ending value.
Rewrite the subtraction as adding the opposite.
$-9 - 12 = -9 + (-12) = -21$
Subtracting 12 is the same as adding $-12$.
Say what the change means.
$-21 \Rightarrow \text{a fall of } 21^\circ$
The sign gives the direction and the size gives how many degrees.
Divide by the 12 hours to find the average change per hour.
$-21 \div 12 = -1.75$
A negative divided by a positive is negative: on average the temperature fell 1.75 degrees an hour.
Check by multiplying back.
$12 \times (-1.75) = -21$
Twelve hours of that average change gives the whole fall.
Write the starting balance.
$\text{start} = 45.50$
The club begins with money in the account, so the balance is positive.
Write the cost of 6 banners at 12.75 dollars each as one change.
$6 \times (-12.75) = -76.50$
Spending is a negative change, repeated 6 times.
Write the income from 18 bake-sale items at 1.25 dollars each.
$18 \times 1.25 = 22.50$
Money coming in is a positive change.
Add the spending to the start.
$45.50 + (-76.50) = -31.00$
The club spent more than it had, so the balance dips below zero.
Add the income.
$-31.00 + 22.50 = -8.50$
The income moves the balance up, but not all the way back to zero.
Interpret the balance.
$-8.50 \Rightarrow \text{the club owes } 8.50 \text{ dollars}$
A negative balance is money owed.
Find how many more items must be sold to reach zero.
$8.50 \div 1.25 = 6.8$
Each item brings the balance 1.25 dollars closer to zero.
Round up and check.
$7 \times 1.25 = 8.75, \quad -8.50 + 8.75 = 0.25$
Six items are not quite enough, so the club must sell 7, which leaves 25 cents in the account.
Add the two negative temperatures.
$-6 + (-9) = -15$
Grouping numbers with the same sign makes the sum easier.
Add the two positive temperatures.
$4 + 3 = 7$
These are the readings above zero.
Combine the two groups.
Divide by 4 and interpret.
Leo had $25$ dollars in his account and paid for a class trip. Now his balance is $-41$ dollars. What does the balance tell you?
The temperature in a mountain town is $28^\circ\text{F}$ at sunset. It then falls $5$ degrees every hour for $7$ hours. Complete the worked solution to find the temperature at the end.
Write one hour's fall as a signed change.
$\text{change each hour} = -5$
A fall is a change in the negative direction.
Multiply the hourly change by the number of hours.
$7 \times (-5) =$ c
Equal changes repeated $7$ times are one multiplication, and a positive times a negative is negative.
Add the total change to the sunset temperature.
$28 + (\text{total change}) =$ f
The final temperature is where you start plus how much it changed.
Check that the sign makes sense.
$\text{total fall} > 28 \Rightarrow \text{final} < 0$
The temperature fell by more than it started at, so it must end below zero.
At 6 a.m. the temperature in Fairbanks, Alaska, was $-9^\circ\text{F}$. By noon it had risen $12$ degrees. By midnight it had fallen $5$ degrees from the noon reading. What were the temperatures at noon and at midnight?
Noon: n °F. Midnight: m °F.
Maya's account holds $24$ dollars. She makes $4$ withdrawals of $28$ dollars each, and then deposits $21$ dollars. What is the total change from the withdrawals, and what is her balance at the end?
Change from withdrawals: c dollars. Final balance: f dollars.
Find the sum and the mean of $2$, $-10$, $6$ and $-6$.
Sum: s. Mean: m
A scuba diver is at $-15$ feet, below the surface. She rises $9$ feet, then swims $3$ feet deeper. Mark her final position on the number line, which shows feet from the surface.
-40 |——————————| 0
Mark the position with a cross, then write the value:
In a quiz game, a correct answer scores $+3$ points, a wrong answer scores $-2$ points, and a question left blank scores 0. Sam answered $4$ questions correctly, $8$ wrongly and left $4$ blank. Find the points from his correct answers, the points from his wrong answers, and his final score.
Correct: r points. Wrong: g points. Final score: t
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A lemonade stand loses $4.50$ dollars on each rainy day and makes $11$ dollars on each sunny day. One week has $4$ rainy days and $4$ sunny days. What is the total change for the rainy days, and what is the result for the whole week, in dollars?
Rainy days: r dollars. Whole week: w dollars.
You can solve a problem with negative quantities and explain the answer. Without looking: if an account at $-40$ dollars receives a deposit of 25 dollars, what is the new balance and what does it mean? What is the change from $-7^\circ$F to $15^\circ$F?
19. Your turn: the mean of the low temperatures $-6$, $4$, $-9$ and $3$ degrees, step 3
$-15 + 7 = -8$
The negative group is larger, so the sum is negative.
19. Your turn: the mean of the low temperatures $-6$, $4$, $-9$ and $3$ degrees, step 4
$-8 \div 4 = -2$
The mean low temperature was 2 degrees below zero.