Back to the on-screen lesson ·
Adding and subtracting positives and negatives, including fractions and decimals.
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 add and subtract rational numbers: positives and negatives, whole numbers, fractions and decimals. You picture each number as a move along the number line, learn the two rules for adding numbers with the same sign and with different signs, and turn every subtraction into adding the opposite. You also find the distance between two numbers and use signed numbers to read time zones, tides and stock prices.
You can place negative numbers on a number line, and you know that $-5$ sits 5 units to the left of 0. You know that the absolute value of a number is its distance from 0, so $|-5| = 5$. You can add and subtract fractions and decimals when they are positive, finding a common denominator or lining up the decimal points. In this lesson you put these ideas together to add and subtract any rational numbers: whole numbers, fractions and decimals, positive or negative. You also meet the one idea that makes every subtraction easy: subtracting a number is the same as adding its opposite.
| Term | What it means |
|---|---|
| Rational number | Any number that can be written as a fraction of two integers, such as $\tfrac{2}{3}$, $-7$ or $-1.25$. |
| Integer | A whole number or its opposite: $\ldots, -2, -1, 0, 1, 2, \ldots$ |
| Opposite | The number the same distance from 0 on the other side: the opposite of $-8$ is 8. |
| Additive inverse | Another name for the opposite; a number plus its additive inverse is 0. |
| Absolute value | The distance of a number from 0, its size without its sign: $\vert -3.5\vert = 3.5$. |
| Sum | The answer to an addition. |
| Difference | The answer to a subtraction; it can be negative. |
| Distance between two numbers | The absolute value of their difference, which is never negative. |
Think of every rational number as a move along the number line. A positive number is a move to the right and a negative number is a move to the left. Its absolute value says how long the move is. To add, start at the first number and make the move the second number describes: $2 + (-5)$ starts at 2 and moves 5 left, landing on $-3$.
Subtraction needs just one extra idea. Subtracting a number is the same as adding its opposite:
$$a - b = a + (-b)$$
So $6 - 9 = 6 + (-9) = -3$, and $-4 - (-10) = -4 + 10 = 6$. Once every subtraction has been turned into an addition, you only ever need the adding rules, and they work the same way for fractions and decimals as they do for whole numbers.
A number and its opposite always add to zero: $-2.7 + 2.7 = 0$. Those two moves are the same length in opposite directions, so you end where you began.
Another way: picture
On a number line, $2 + (-5)$ is an arrow that starts at 2 and points 5 units to the left. Two of those units reach 0 and the other three go past it, so the arrow stops at $-3$.
Another way: story
An elevator in a building with parking levels below the ground floor: level 0 is the lobby and level $-2$ is two floors down. Riding up 6 floors from level $-2$ gets you to $-2 + 6 = 4$. The ride from level 4 down to level $-2$ is $-2 - 4 = -6$ floors: 6 floors in the down direction.
When both numbers are positive, you already know what to do. When both are negative, both moves go left, so the result is further left than either one. Add the sizes and keep the negative sign: $-6 + (-9) = -15$, and $-1.2 + (-0.5) = -1.7$.
Money makes this easy to believe. If you owe a friend 6 dollars and then borrow 9 more, you owe 15 dollars in all. Two debts make a bigger debt, never a gain. So a sum of two negatives is always negative, and its size is the sum of the two sizes.
When one number is positive and one is negative, the two moves pull in opposite directions, and part of one move cancels part of the other. Subtract the smaller size from the larger size, and give the answer the sign of the number with the larger size.
For $-13 + 8$, the sizes are 13 and 8. The difference of the sizes is 5, and the negative number has the larger size, so the sum is $-5$. For $13 + (-8)$ the sizes are the same, but now the positive number is larger, so the sum is $+5$.
On the number line you can see why: 8 of the 13 steps to the left are undone by the 8 steps to the right, and the 5 steps nobody undid decide where you land. If the two sizes are equal, every step is undone and the sum is 0.
Every subtraction can be rewritten as an addition. Change the minus sign to a plus sign and change the number after it to its opposite:
Why is this allowed? Subtraction asks how much you must add to the second number to reach the first. The answer to $-3 - (-8)$ is the number you add to $-8$ to get $-3$, and that is 5, because you move 5 to the right. The rule $a - b = a + (-b)$ always gives that same answer.
Notice that subtracting a negative makes a number bigger. Taking away a debt of 8 dollars leaves you 8 dollars better off. Subtracting a positive number makes it smaller, even when the answer goes below zero.
The difference $b - a$ tells you how far and which way to move to get from $a$ to $b$. If the difference is positive, you move right; if it is negative, you move left. The distance between the two numbers ignores the direction, so it is the absolute value of the difference:
$$\text{distance} = |b - a|$$
The distance between $-11$ and 4 is $|4 - (-11)| = |15| = 15$, and it does not matter which number you subtract from which: $|-11 - 4| = |-15| = 15$ too. You can check by counting: 11 units from $-11$ up to 0, then 4 more to reach 4, which makes 15.
The sign rules do not change when the numbers are not whole. First line up the pieces, then follow the same rules.
With decimals, line up the decimal points and then add or subtract the sizes: $-4.35 + 1.2$ has sizes 4.35 and 1.20, the difference of the sizes is 3.15, and the negative number is larger, so the sum is $-3.15$.
With fractions, rewrite them over a common denominator first. $\tfrac{1}{3} - \tfrac{5}{6} = \tfrac{2}{6} + \left(-\tfrac{5}{6}\right)$, and $2 + (-5) = -3$, so the answer is $-\tfrac{3}{6} = -\tfrac{1}{2}$.
With mixed numbers, remember that $-2\tfrac{1}{4}$ means $-\left(2 + \tfrac{1}{4}\right)$: the whole part and the fraction part are both negative. Writing it as an improper fraction, $-\tfrac{9}{4}$, or as a decimal, $-2.25$, avoids the mistake of treating only the 2 as negative.
When there are many numbers to add, you may add them in any order and group them however you like. This works because addition is commutative (order does not matter) and associative (grouping does not matter). It only works for additions, which is one more reason to turn every subtraction into adding the opposite first.
Look for opposites that cancel, and group the positives together and the negatives together. For $-17 + 9 + 17 + (-4) + 6$, the $-17$ and 17 cancel, leaving $9 + 6 + (-4) = 11$. Adding all the positives, adding all the negatives, and then combining the two totals means you only have to think about signs once, at the very end.
Use the same steps for any sum or difference of rational numbers.
How to check. Estimate first: $-8.9 + 3.1$ is about $-9 + 3 = -6$, so an answer of $-5.8$ is sensible and 12 is not. Check a subtraction by adding back: if $-3 - 8 = -11$, then $-11 + 8$ should give $-3$, and it does. Finally, sketch the moves on a number line and ask whether you should have ended left or right of where you started.
Clock times around the world are set as an offset from Coordinated Universal Time (UTC). In winter, New York is at UTC $-5$ hours, Honolulu is at UTC $-10$, Tokyo is at UTC $+9$, and Mumbai in India is at UTC $+5.5$, a half hour off from most of the world. To find how far ahead one city is of another, subtract the offsets. Tokyo is $9 - (-5) = 14$ hours ahead of New York, so a video call at 8 p.m. in New York is at 10 a.m. the next day in Tokyo. Mumbai is $5.5 - (-5) = 10.5$ hours ahead of New York, and Honolulu is $-10 - (-5) = -5$ hours from New York: the negative answer means Honolulu is 5 hours behind.
Tide tables measure the water level from a fixed zero line, and on some days the low tide drops below that line. A table for a beach might list a low tide of $-1.4$ feet at 6:10 a.m. and a high tide of $7.9$ feet at 12:25 p.m. The water rises $7.9 - (-1.4) = 9.3$ feet in about six hours. People who explore tide pools look for these negative tides, because the extra-low water uncovers rocks, sea stars and crabs that are usually hidden.
A share of stock changes price every trading day, and news reports give each change as a signed decimal. Suppose a share starts the week at 42.50 dollars and changes by $+1.25$, $-0.80$, $-2.15$, $+0.60$ and $-0.35$ dollars. The gains add to $1.25 + 0.60 = 1.85$ and the losses add to $-0.80 + (-2.15) + (-0.35) = -3.30$, so the week's total change is $1.85 + (-3.30) = -1.45$ dollars. The share ends the week at $42.50 + (-1.45) = 41.05$ dollars.
The most common mistake is using the multiplying rule when adding. "Two negatives make a positive" is a rule for multiplying. When you add, $-5 + (-4) = -9$: two debts make a bigger debt.
The second is reading the sign from the first number instead of the larger size. In $-3 + 10$ the first number is negative, but the positive number is larger in size, so the answer is $+7$.
The third is changing only one sign when you subtract. To subtract, you change the operation and the number after it. $4 - (-6)$ becomes $4 + 6$, not $4 + (-6)$ or $-4 + 6$.
The fourth is thinking a subtraction always gives a smaller answer. That is true only when you subtract a positive. $2 - (-5) = 7$, which is bigger than 2.
The last is making only the whole part of a mixed number negative. $-2\tfrac{1}{4}$ is $-2.25$, not $-2 + \tfrac{1}{4} = -1.75$.
Read the signs of the two numbers.
$-8.5 < 0, \quad 3.2 > 0$
Different signs mean the two moves pull in opposite directions.
Find the size of each number.
$|-8.5| = 8.5, \quad |3.2| = 3.2$
The sizes are the lengths of the two moves.
Subtract the smaller size from the larger.
$8.5 - 3.2 = 5.3$
The move to the right undoes 3.2 of the move to the left.
Give the answer the sign of the larger size.
$-8.5 + 3.2 = -5.3$
The negative number is longer, so the leftover move is to the left.
Check by adding back.
$-5.3 + (-3.2) = -8.5$
Undoing the move of 3.2 to the right returns you to the start.
Rewrite the subtraction as adding the opposite.
$\tfrac{2}{3} - \left(-\tfrac{3}{4}\right) = \tfrac{2}{3} + \tfrac{3}{4}$
The opposite of $-\tfrac{3}{4}$ is $\tfrac{3}{4}$.
Choose a common denominator.
$\text{LCM}(3, 4) = 12$
Twelfths can be made from both thirds and quarters.
Rewrite the first fraction in twelfths.
$\tfrac{2}{3} = \tfrac{8}{12}$
Multiply the top and the bottom by 4.
Rewrite the second fraction in twelfths.
$\tfrac{3}{4} = \tfrac{9}{12}$
Multiply the top and the bottom by 3.
Add the numerators.
$\tfrac{8}{12} + \tfrac{9}{12} = \tfrac{17}{12}$
Both numbers are positive now, so the sizes simply add.
Write the answer as a mixed number.
$\tfrac{17}{12} = 1\tfrac{5}{12}$
Seventeen twelfths is one whole and five twelfths more.
Turn the subtractions into additions.
$-3\tfrac{1}{2} + 5.25 + 1\tfrac{3}{4} + (-6)$
Subtracting $-1\tfrac{3}{4}$ adds $1\tfrac{3}{4}$; subtracting 6 adds $-6$.
Write every number as a decimal.
$-3.5 + 5.25 + 1.75 + (-6)$
One form for every number lets the pieces line up.
Group the positive numbers.
$5.25 + 1.75 = 7$
Addition can be done in any order, so collect the positives first.
Group the negative numbers.
$-3.5 + (-6) = -9.5$
Same signs: add the sizes and keep the negative sign.
Combine the two totals.
$7 + (-9.5)$
Now only one pair with different signs is left.
Subtract the sizes and keep the sign of the larger.
$7 + (-9.5) = -2.5$
The negative total is larger in size by 2.5.
Check with an estimate.
$-3.5 + 5 + 2 - 6 = -2.5$
Rounding each number gives the same result, so the answer is sensible.
Rewrite the subtraction as adding the opposite.
$-6.4 - 2.9 = -6.4 + (-2.9)$
Subtracting 2.9 is the same as adding $-2.9$.
The signs are the same, so add the sizes.
$6.4 + 2.9 = 9.3$
Two moves to the left add up.
Keep the negative sign.
You start at $3$ on a number line. Which move shows adding $-11$?
Complete the worked solution to find $-13 + 9 - (-45)$.
Rewrite subtracting the negative as adding its opposite.
$-13 + 9 - (-45) = -13 + 9 + 45$
Taking away a negative amount has the same effect as adding the positive amount.
Add the first pair: the signs are different, so subtract the sizes.
$-13 + 9 =$ p
The negative number has the larger size, so the sum is negative.
Add the last number to that sum.
$\text{that sum} + 45 =$ r
The positive number is now the larger in size, so the result is positive.
What is $-5 + (-12)$?
answer
What is $-13 - (-12)$?
answer
How far apart are $-9$ and $15$ on the number line? Also give the difference $-9 - 15$.
Difference: d. Distance: answer.
What is $-\dfrac{5}{9} + \dfrac{6}{9}$? Give a fraction or a decimal.
answer
A tide table measures the water level from a fixed zero line. One morning the low tide is $-2.4$ feet, and that afternoon the high tide is $9.2$ feet. How many feet does the water rise?
answer feet
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Find $-\dfrac{11}{4} - \dfrac{19}{6}$. Give your answer as a fraction.
answer
You can add and subtract rational numbers. Without looking: why is $-4 - (-9)$ positive, and what is the distance between $-2.5$ and $3.5$?
19. Your turn: $-6.4 - 2.9$, step 3
$-6.4 - 2.9 = -9.3$
Two negatives make a bigger negative when you add.