Chapter 6 · Rational numbers
Signed Numbers in Word Problems and Spotting Mistakes
Solve word problems with positive and negative numbers by finding what zero means and which direction is positive. Then find and fix mistakes in signed numbers.
You can already write a real-world quantity as a positive or negative number. You can also say what zero means in a situation. Now you will use both skills on full word problems. You will pull signed numbers out of a story, turn signed numbers back into words, and check someone else's work for mistakes.
A signed number is a number with a sign that shows its direction from zero: for positive and for negative. You can write a positive number with no sign, so and mean the same thing. Zero is neither positive nor negative, so you write it with no sign.
On its own, a number such as tells you very little. You need to know what it measures and where the counting starts. So before you write or read any sign, settle three things:
- What does the number measure, and in what unit? For example, a change in a score, in points.
- What does zero mean? Zero is the reference point: the place or amount that you measure every other value from.
- Which direction is positive? The opposite direction is then negative.
Suppose a game records the change in your score, in points. Zero means no change, and a gain is positive. Now tells you two things. The sign says points lost. The says how many: 15 points.
The next table gives the usual answers for five situations. A problem can set its own zero or its own positive direction. When it does, follow the problem.
| Quantity and unit | Zero means | Positive means | Negative means |
|---|---|---|---|
Elevation (height compared with sea level), in meters | at sea level | above sea level | below sea level |
Temperature, in degrees Celsius | at the mark on the scale | above zero (warmer) | below zero (colder) |
Account balance (money in an account), in dollars | no money and no debt | money you have | money you owe |
Change in a score, in points | no change | points gained | points lost |
Amount over or under a target | exactly on target | over the target | under the target |
Key concept
Zero first, then the sign
In a real-world situation, a signed number has meaning only after you know what zero means and which direction is positive. Once you know both, a number that is not zero tells you two things. Its sign tells you which side of zero the quantity is on. The number without its sign tells you how far from zero it is. A quantity that is exactly at the reference point is . To check a signed number, read it back as words and compare the words with the situation.
Worked example
From a story to numbers, and back
A coastal map gives elevation in meters. Sea level is , and heights above sea level are positive. A lighthouse lamp is meters above sea level. A shipwreck lies meters below sea level. The map also labels a reef with the number . Write a signed number for the lamp and for the shipwreck. Then say what the reef's label means.
Settle the three questions. The quantity is elevation, in meters. Zero means sea level. Above sea level is positive, so the opposite direction, below sea level, is negative.
Find which side of zero the lamp is on. It is above sea level, which is the positive direction. It is 85 meters from zero, so its elevation is positive 85.
Find which side of zero the shipwreck is on. It is below sea level, which is the negative direction. It is 40 meters from zero, so its elevation is negative 40.
Now go from a number to words. The reef's label is negative 6. The negative sign says the reef is below sea level. The 6 says the reef is 6 meters from sea level.
Check the two numbers you wrote by reading them back as words. Positive 85 says 85 meters above sea level, which matches the lamp. Negative 40 says 40 meters below sea level, which matches the shipwreck.
Answer
The lamp is at meters and the shipwreck is at meters. The label means the reef is meters below sea level.
Zero does not always mean sea level, and it does not always mean nothing. A word problem can set its own reference point, such as a target amount, a starting score or the ground floor of a building. The three questions still work. You may need one extra step: use the numbers in the problem to find how far a quantity is from the reference point.
Worked example
When zero means on target
A machine should put milliliters of juice in each bottle. An inspector records how far each bottle is from this target, in milliliters. Amounts over the target are positive and amounts under the target are negative. Bottle A holds milliliters, Bottle B holds milliliters and Bottle C holds milliliters. What number does the inspector record for each bottle?
Settle the three questions. The quantity is how far a bottle is over or under the target, in milliliters. Zero means the bottle holds exactly 500 milliliters. It does not mean an empty bottle. Over the target is positive and under the target is negative.
Bottle A holds more than 500 milliliters, so it is over the target and the recorded number is positive. To find how far over, subtract the smaller amount from the larger one.
Bottle B holds less than 500 milliliters, so it is under the target and the recorded number is negative. To find how far under, again subtract the smaller amount from the larger one.
Bottle C holds exactly 500 milliliters. It is at the reference point, not over or under it, so the recorded number is zero.
Read the numbers back as words and test them. Positive 4 says 4 milliliters over the target, and 500 plus 4 is 504. Negative 3 says 3 milliliters under the target, and 500 minus 3 is 497. Zero says exactly on target.
Answer
The inspector records for Bottle A, for Bottle B and for Bottle C.
Now switch roles. Someone else has written a signed number, and you decide if it is right. Settle the same three questions, then work backward:
- Take the number they wrote.
- Read it back as words, using the problem's zero and positive direction. Which side of zero does it say? How far?
- Compare your words with the original situation.
If your words and the situation say different things, the number does not fit the situation. The way they differ shows you what kind of mistake it is.
| Mistake | Example | What goes wrong | Correct version |
|---|---|---|---|
Dropping the sign | Jo owes dollars. Someone writes her balance as . | means , which says Jo has dollars. | |
Using the wrong zero | The target is milliliters. Someone records a milliliter bottle as . | says milliliters over the target. They measured from empty, not from the target. | |
Giving the direction twice | Someone describes as " degrees below zero." | The sign already means below zero. Adding the words gives the direction twice, which confuses the reader. | " degrees" or " degrees below zero" |
Worked example
Finding and fixing mistakes
Sea level is meters, and above sea level is positive. Maya wrote a signed number for the elevation in each sentence. (1) A fish swims meters below sea level. Maya wrote . (2) A balloon drops to meters above sea level. Maya wrote . (3) A kite flies meters above sea level. Maya wrote . Which of Maya's numbers are wrong? Fix each wrong one.
Settle the three questions. The quantity is elevation, in meters. Zero is sea level. Above sea level is positive, so below sea level is negative.
Read the first number back as words. A 12 with no sign is positive 12, which says 12 meters above sea level. The fish is below sea level, so Maya dropped the negative sign.
Read the second number back. Negative 50 says 50 meters below sea level, but the balloon ends up 50 meters above sea level. The word "drops" tells how the balloon moved. It does not tell which side of zero the balloon is on.
Read the third number back. Positive 20 says 20 meters above sea level. That matches the kite, so this number is correct.
Answer
Numbers (1) and (2) are wrong. The fish is at meters and the balloon is at meters. Maya's meters for the kite is correct.
Here is the idea to keep. A signed number counts from zero. So settle three things first: what the number measures, what zero means and which direction is positive. Then the side of zero gives the sign, and how far the quantity is from zero gives the number that follows the sign. To check a signed number, yours or anyone else's, read it back as words, using the problem's zero and positive direction. If your words say the same thing as the situation, the number fits.