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Adding and Subtracting Integers

Compare, order and solve problems involving addition and subtraction of integers.

Two sign rules cover everything here. Learn them properly once and this topic stops being frightening.

Builds onYr 6 · Negative numbers

Why this works

Every integer question is a journey along a number line. The first number is where you start; the operation tells you which way to face and how far to walk.

Adding a positive moves you right. Adding a negative moves you left.
5+(−3)=25 + (-3) = 2
Subtracting a positive moves you left. Subtracting a negative moves you right.
5−(−3)=85 - (-3) = 8
That last one is the strange-looking case. Subtracting a negative means removing a debt — and removing a debt leaves you better off, so you move right.

Both rules collapse into one line: two signs the same make a plus; two signs different make a minus. Simplify the double sign first, then it is ordinary arithmetic.

Worked example

e.g. −6−(−9)-6 - (-9)
  1. Find the two signs sitting next to each other.−6 − (−)‾ 9-6 \ \underline{- \ (-)} \ 9
  2. Two different... no — both are minus signs, so they are the same. Same signs become a plus.−6+9-6 + 9
  3. Now it is a normal calculation. Start at −6-6, move 9 to the right.−6+9=3-6 + 9 = 3
  4. Sanity-check on a number line: 9 to the right of −6-6 passes zero at 6 steps, with 3 left over.33

Common mistakes

Wrong−5+3=−8-5 + 3 = -8
Right−5+3=−2-5 + 3 = -2
Adding the digits and keeping the negative. Adding a positive always moves you towards zero from the negative side, so the answer must be closer to zero than −5-5.
Wrong4−(−6)=−24 - (-6) = -2
Right4−(−6)=4+6=104 - (-6) = 4 + 6 = 10
Missing the double negative. Two minus signs together become a plus, so this gets bigger, not smaller.
Wrong−3−7=−4-3 - 7 = -4
Right−3−7=−10-3 - 7 = -10
Here the signs are not adjacent — it is simply −3-3 then move 7 further left. You are going deeper into negative territory, not back towards zero.

Practice

FluencyGet quick and accurate at the method.
1
−5+3-5 + 3
Answer−2-2
2
4+(−9)4 + (-9)
Answer−5-5
3
−7−2-7 - 2
Answer−9-9
4
6−(−4)6 - (-4)
Answer1010
5
−8−(−3)-8 - (-3)
Answer−5-5
6
−2+(−6)-2 + (-6)
Answer−8-8
7
12+(−15)12 + (-15)
Answer−3-3
8
−10−(−10)-10 - (-10)
Answer00
9
−4+11-4 + 11
Answer77
10
3−8−(−2)3 - 8 - (-2)
Answer−3-3
11
Order smallest first: −3, 5, −8, 0\text{Order smallest first: } -3,\ 5,\ -8,\ 0
Answer−8, −3, 0, 5-8,\ -3,\ 0,\ 5
12
Which is larger: −2 or −7?\text{Which is larger: } -2 \text{ or } -7?
Answer−2 — it lies further right on the number line-2 \text{ — it lies further right on the number line}
AppliedThe same maths, inside a real question.
1
The temperature in Canberra is −3∘-3^\circC at 6 am and rises 11∘11^\circ by midday. What is the midday temperature?
Answer8∘8^\circC. −3+11=8-3 + 11 = 8.
2
A submarine at −180-180 m descends a further 75 m. What is its new depth?
Answer−255-255 m. −180−75=−255-180 - 75 = -255. Descending means moving further from the surface, so further into negatives.
3
Leah's account is −$85-\$85. She is refunded a $30\$30 fee that was charged in error. What is her balance?
Answer−$55-\$55. Removing a charge is subtracting a negative: −85−(−30)=−85+30=−55-85 - (-30) = -85 + 30 = -55.
4
The temperature drops from 4∘4^\circC to −9∘-9^\circC. By how many degrees did it fall?
Answer13∘13^\circ. The change is 4−(−9)=4+9=134 - (-9) = 4 + 9 = 13. Counting on a number line: 4 down to 0, then 9 more.
Next step
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