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Subtracting
integers is the second operations on integers, among the four fundamental
operations on integers.
Subtraction is the inverse process of addition.
We have learnt that subtraction is the inverse process of addition.
Therefore, subtracting 5 from 9 means adding -5 to 9, i.e., 9 + (-5) = 4.
For example:
(i) (- 5) – (- 2) = – 5 + 2 = – 3
(ii) (7) – (2) = 7 – 2 = 5
(iii) (-6) -4 = -6 – 4 = -10
Examples:
For example, subtracting 6 from 24 is the same as adding the additive inverse of 6 to 24.
Therefore, 24 – 6 = 24 + (additive inverse of 6) = 24 + (- 6) = 18
The rule can be extended to subtraction of a negative integer from a positive or negative integer.
For example, subtracting -7 from 16 is the same as adding the additive inverse of -7 to 16.
Therefore, 16 – (-7) = 16 + (additive inverse of -7) = 16 + (7) = 23
Thus, to subtract an integer ‘y’ from another integer ‘x’, the additive inverse of ‘y’ is added to ‘x’,
i.e., x – y = x + (additive inverse of y) = x + (- y)
and x – (- y) = x + (additive inverse of -y) = x + (y)
Change the sign of the integer to be subtracted and then add.
(i) Subtract: +3 from +9
(+9) – (+3)
Here the +3 is subtracted from +9
= +9 – 3, [(+ 3) changes to (-3)]
Now add +9 and -3 using the addition of integers rule.
= +6
(ii) Subtract:
-7 from +12
(+12) –
(-7)
Here the
-7 is subtracted from +12
= +12 +
7, [(-7) changes to (+7)]
Now add
+12 and +7 using the addition of integers rule.
= +19
(iii) Subtract:
+8 from -18
(-18) –
(+8)
Here the
+8 is subtracted from -18
= -18 –
8, [(+ 8) changes to (-8)]
Now add
-18 and -8 using the addition of integers rule.
= -26
(iv) Subtract:
-4 from -11
(-11) –
(-4)
Here the
-4 is subtracted from -11
= -11 +
4, [(-4) changes to (+4)]
Now add
-11 and +4 using the addition of integers rule.
= -7
(v) Subtract:
+44 from +99
(+99) –
(+44)
Here the
+44 is subtracted from +99
= +99 –
44, [(+ 44) changes to (-44)]
Now add
+99 and -44 using the addition of integers rule.
= +55
(vi) Subtract:
-63 from +87
(+87) –
(-63)
Here the
-63 is subtracted from +87
= +87 +
63, [(-63) changes to (+63)]
Now add
+87 and +63 using the addition of integers rule.
= +150
(vii) Subtract:
+41 from -76
(-76) –
(+41)
Here the
+41 is subtracted from -76
= -76 –
41, [(+ 41) changes to (-41)]
Now add
-76 and -41 using the addition of integers rule.
= -117
(viii) Subtract:
-16 from -46
(-46) –
(-16)
Here the
-16 is subtracted from -46
= -46 +
16, [(-16) changes to (+16)]
Now add
-46 and +16 using the addition of integers rule.
= -30
Solved Examples Subtraction of Integers:
1. Find the value of -5 – (-13) using a number line.
Solution:
-5 – (-13) = -5 + (additive inverse of -13) = – 5 + (13)
On the number line, starting from -5 move 13 units to the right to reach point P, which represents 8.
Thus, -5 – (-13) = 8
2. Subtract the following.
(i) 23 from 9
(ii) -6 from 4
(iii) 4 from -8
Solution:
(i) 9 – 23 = 9 + (additive inverse of 23) = 9 + (- 23) = -14
(ii) 4 – (-6) = 4 + (additive inverse of -6) = 4 + 6 = 10
(iii) -8 – 4 = (-8) + (additive inverse of 4) = (-8) + (-4) = -12
3. Subtract the following.
Solution:
(i) 936 from -53
(ii) -386 from 121
(iii) -2150 from -897
(iv) -2645 from +5189
Solution:
(i) -53 – 936 = -53 + (additive inverse of 936) = – 53 + (-936) = – 989
(ii) 121 – (-386) = 121 + (additive inverse of -386) = 121 + 386 = 507
(iii) -897 – (-2150) = -897 + (additive inverse of -2150) = -897 + 2150 = 1253
(iv) 5189 – (-2645) = 5189+ (additive inverse of-2645) = 5189 + 2645 = 7834
4. Subtract the sum of 225 and -299 from the sum of -342 and 151
Solution:
225 + (- 299) = – (299 – 225) = – 74
– 342 + 151 = – (342 – 151) = – 191
Required difference (-191) – (-74) = (-191) + (additive inverse of -74)
= (-191) + 74
= -117
5. Subtract:
(i) -8 from 15
(ii) 11 from -9
(iii) -13 from -9
(iv) -245 from -517
Solution:
(i) 15 – (- 8) = 15 + 8 = 23
(ii) – 9 – 11 = (- 9) + (- 11) = – (9 + 11) = – 20
(iii) -9 – (-13) = -9 + 13 = 4.
(iv) – 517 – (- 245) = – 517 + 245 = – (517 – 245) = – 272
6. Evaluate:
(i) – 730 – (- 515)
(ii) -210 – 715
(iii) 345 + (-271)
(iv) (-314) + 25
Solution:
(i) – 730 – (- 515) = – 730 + 515 = – (730 – 515) = – 215
(ii) – 210 – 715 = – (210 + 715) = – 925
(iii) 345 + (- 271) = 345 – 271 = 74
(iv) (- 314) + 25 = – 314 + 25 = – (314 – 25) = – 289
7. Evaluate:
(i) 8 + (- 4) – (- 2)
(ii) 6 – 5 + 3 + 4 – 1 – 6 – 7
Solution:
(i) 8 + (- 4) – (- 2) = (8 – 4) – (- 2)
= 4 + 2
= 6
(ii) Group the positive and the negative numbers separately.
6 – 5 + 3 + 4 – 1 – 6 – 7 = (6 + 3 + 4) + (- 5 – 1 – 6 – 7)
= 13 + (-19)
= 13 – 19
= -6.
8. The sum of two integers is -225. If one of them is 75, find the other.
Solution:
To get the other integer, we subtract the given integer from the sum.
Therefore, Other integer = Sum of two integer – Given integer
= -225 – 75
= -300.
9. Fill in the blanks with the symbols >,
(i) (-3) + (-6) _____ (- 3) – (- 6)
(ii) 45 – (-11) _____ 57 + (-4)
(iii) -5 + 5 _____ 8 + (-8)
Solution:
(i) The sum of -3 and -6 is -9.
The difference of -3 and 6 is 3.
Since 3 > – 9
Therefore, (-3) + (-6)
(ii) The difference of 45 and -11 is 56.
The sum of 57 and -4 is 53.
Since 56 > 53
Therefore, 45 – (-11) > 57 + (- 4)
(iii) The sum of -5 and 5 is 0. The sum of 8 and 8 = 0.
Since both the sums are same.
Therefore, -5 + 5 = 8 + (-8).
10. What are the successor and predecessor of 66?
Solution:
The successor of 66 is (66 + 1) = 67, and the predecessor of 66 is (66 – 1) = 65
11. (i) What must be added to -49 to get 713?
(ii) What must be subtracted from 31 to get -904?
Solution:
(i) To get answer, we shall subtract -49 from 713.
i.e., 713 – (-49) = 713 + 49 = 762
Hence, the required number that must be added is 762.
(ii) To get answer, we shall subtract -904 from 31.
i.e., Required number = 31 – (-904) = 31 + 904 = 935.
Hence, the required number that must be subtracted is 935.
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