Basic Maths Shortcuts
1. Division-shortcuts
In division instead of direct division, use factoring method
Example:1848/264=(2*3*4*7*11)/(2*3*4*11)=7
2.Multiplication-shortcuts
SUM – 10 METHOD:
Example: 78 and 72. These two numbers, if we add the numbers in the unit's
place, the resultant is 10 and the numbers in the ten's place are both the
same. In such cases, we can have a simple solution.
Step1: multiply the numbers in the unit's place and write down the resultant.
(8*2 = 16)
Step2: say, the number in the ten's digit is a, then multi a*(a+1) and write
down the resultant. => (7*(7+1) = 56)
Step3: write the final result: 5616
Example:118*112 follow above steps
8*2 = 16; and 11*(11+1) = 11*12 = 132. And hence
the result is: 13216.
In short: ab*ac = (a*(a+1))(b*c)
Base Method:
Base numbers, in general, are nothing but multiples of 10. If the given
numbers are nearer to
base numbers, then you can follow this method to multiply them.
Example: 98*95 =?
Here 98 is ,2 less than the base number 100 and 95 is ,5 less than 100. We
can write them like this:
98 -2
95 -5
The first step will be deducting/subtracting the resultant of the diff between
the base number and the given number with the given number in a crossway!
That is, you need to subtract 98 and 5 (which is the resultant of
difference between the base number and 95) or you can also cross-subtract
95 and 2, the result will be same. This result forms the 1st part of the resultant
at the start. The last part of the resultant will be multiplication of the
differences from base numbers (i.e., 2 * 5 = 10)
98 -2
95 -5
(98 – 5) (-2 * -5)
Hence, the answer will be: 9310
Example: 998*997 =?
998 -2
997 -3
Observe carefully, in the second part, the multiplication of difference yield in
a single digit number, but no. of zeroes in the base number, here 1000, is
three. Hence add two zeroes before the result. Therefore, the answer will be:
(998-3) | (-2 * -3) = 995006
What if the numbers we get are like this? I mean, the base is 50 here. We will
follow the same procedure as above but a small difference that the resultant
in the first part will be halved. And if the base is 200, then the number will be
doubled and so on based on the base number.
Multiplication with 5,25,50 etc...
Substitute 5 by 10/2,25 by 100/4 and 50 by 100/2.
Examples:
1. 5*18=18*10/2=180/2=90
2. 24*25=24*100/4=2400/4=600
3. 73*50=73*100/2=7300/2=3650
Multiplication with 9, 99,999 etc..
Examples:
1. 13*9=13*(10-1)=130-13=117
2. 26*99=26*(100-1)=2600-26=2574
3. 350*999=350*(1000-1)=350000-350=349650
3.Square-Shortcut Tricks
Method1:Apply (a + b)2 = a2 + b2 + 2ab
Example1:182 = (10 + 8)2 = 102 + 82 + 2 × 10 × 8 = 100 + 64 + 160 = 324
Example2:1032 = 1002 + 32 + 2 × 100 × 3 = 10000 + 9 + 600 = 10609
Example3:562 = 502 + 62 + 2 × 50 × 6 = 2500 + 36 + 600 = 3136
Method2:Square of a number ending with 5
(𝑋5)2 = 𝑋 ∗ (𝑋 + 1) 𝑎𝑛𝑑 52
Example1:352 = 3 ∗ (3 + 1) 𝑎𝑛𝑑 52 = 12 25
Example2:652 = 6 ∗ (6 + 1) 𝑎𝑛𝑑 52 = 42 25
Example3:1152 = 11 ∗ (11 + 1) 𝑎𝑛𝑑 52 = 132 25
Method3:Squres of numbers from 51-59
Add 25 to unit digit and square unit digit
Example1:572 = 7 + 25 𝑎𝑛𝑑 72 = 32 49
Example2:532 = 3 + 25 𝑎𝑛𝑑 32 = 28 09
Example3:592 = 9 + 25 𝑎𝑛𝑑 92 = 34 81
Method4:square if you know square of previous number
(𝑛 + 1)2 = 𝑛2 + 𝑛 + (𝑛 + 1)
Example1:312 = 302 + 30 + 31 = 961
Example2:262 = 252 + 25 + 26 = 676
Example3:812 = 802 + 80 + 81 = 6561
(Bankaspire special)
Method 5:Square of a number if you know square of any other number.
Let X and Y be two numbers. You know the square of X then you can deduce
square of Y.
𝑋2 − 𝑌2 = (𝑋 + 𝑌)(𝑋 − 𝑌)
=> 𝑋2 = (𝑋 + 𝑌)(𝑋 − 𝑌) + 𝑌2
Or 𝑌2 = 𝑋2 − (𝑋 + 𝑌)(𝑋 − 𝑌)
Example1:1152 =?
Choose a nearby number whose square is known to you.
Suppose we choose 110 whose square is 12100
1152 = 1102 + (115 − 110)(115 + 110)
=> 12100 + 5 ∗ 225 = 13225
Example2:482 = 502 − [(50 − 48)(50 + 48)] = 2500 − 2 ∗ 98 = 2304
Example3:272 = 302 − [(30 − 27)(30 + 27)] = 900 − 3 ∗ 57 = 729
Example4:432 = 402 + 3 ∗ 83 = 1849
4.Cubes-Shortcut
Apply (a + b)3 = a3 + b3 + 3a2b + 3ab2
Example1:153 = (10 + 5)3 = 103 + 53 + 3 ∗ 102 ∗ 5 + 3 ∗ 10 ∗ 52 = 1000 + 125 +
1500 + 750 = 3375
Example2:233 = (20 + 3)3 = 203 + 33 + 3 ∗ 202 ∗ 3 + 3 ∗ 20 ∗ 32 = 8000 + 27 +
3600 + 540 = 12167
5.Square roots (applicable only for perfect squares)
Method 1
Example1:Square root of 2704
step1:Seperate number into group of two from right to left ie 27 04.
step2:What number can be squared and less than 27=5, with remainder 2
step3:Bringdown the second group of digits next to remainder to get 204
step4:Double first part of answer to get 5*2=10
step5:Find a number X so that 10 X * X= 204, we get X=2
Thus final answer=52
5 2
5 2704
25
102 204
204
000
Example2: Example1:Square root of 9604
step1:Seperate number into group of two from right to left ie 96 04.
step2:What number can be squared and less than 96=9, with remainder 15
step3:Bringdown the second group of digits next to remainder to get 1504
step4:Double first part of answer to get 9*2=18
step5:Find a number X so that 18 X * X= 1504, we get X=8
Thus final answer=98
9 8
9 9604
81
1881504
1504
0000
Method2:Square root by prime factorisation.
Example1: √44100= √(2 ∗ 2 ∗ 3 ∗ 3 ∗ 5 ∗ 5 ∗ 7 ∗ 7) =2*3*5*7=210
Exampl2:√254016 = √9 ∗ 9 ∗ 8 ∗ 8 ∗ 7 ∗ 7 = 9 ∗ 8 ∗ 7 = 504
6.Cube root(for perfect cubes only)
13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, 63 = 216, 73 = 343, 83 = 512, 93 = 729,
Memorize this.
Example1:√21952 3
step1: Divide digits into group of three from right to left 21 952
step2: Last digit of rightmost group is 2.That means number ends with 8
step3: Now consider leftmost group 21.Cube of 2=8 and cube of 3=27 ,since
21 is between them we must use smaller one,2. Thus final answer is 28
Example2:√32768 3
step1: Divide digits into group of three from right to left 32 768
step2: Last digit of rightmost group is 2.That means number ends with 8
step3: Now consider leftmost group 32. Cube of 3=27 and cube of 4=64,since
32 is between them we must use smaller one,3. Thus final answer is 32.
1. Division-shortcuts
In division instead of direct division, use factoring method
Example:1848/264=(2*3*4*7*11)/(2*3*4*11)=7
2.Multiplication-shortcuts
SUM – 10 METHOD:
Example: 78 and 72. These two numbers, if we add the numbers in the unit's
place, the resultant is 10 and the numbers in the ten's place are both the
same. In such cases, we can have a simple solution.
Step1: multiply the numbers in the unit's place and write down the resultant.
(8*2 = 16)
Step2: say, the number in the ten's digit is a, then multi a*(a+1) and write
down the resultant. => (7*(7+1) = 56)
Step3: write the final result: 5616
Example:118*112 follow above steps
8*2 = 16; and 11*(11+1) = 11*12 = 132. And hence
the result is: 13216.
In short: ab*ac = (a*(a+1))(b*c)
Base Method:
Base numbers, in general, are nothing but multiples of 10. If the given
numbers are nearer to
base numbers, then you can follow this method to multiply them.
Example: 98*95 =?
Here 98 is ,2 less than the base number 100 and 95 is ,5 less than 100. We
can write them like this:
98 -2
95 -5
The first step will be deducting/subtracting the resultant of the diff between
the base number and the given number with the given number in a crossway!
That is, you need to subtract 98 and 5 (which is the resultant of
difference between the base number and 95) or you can also cross-subtract
95 and 2, the result will be same. This result forms the 1st part of the resultant
at the start. The last part of the resultant will be multiplication of the
differences from base numbers (i.e., 2 * 5 = 10)
98 -2
95 -5
(98 – 5) (-2 * -5)
Hence, the answer will be: 9310
Example: 998*997 =?
998 -2
997 -3
Observe carefully, in the second part, the multiplication of difference yield in
a single digit number, but no. of zeroes in the base number, here 1000, is
three. Hence add two zeroes before the result. Therefore, the answer will be:
(998-3) | (-2 * -3) = 995006
What if the numbers we get are like this? I mean, the base is 50 here. We will
follow the same procedure as above but a small difference that the resultant
in the first part will be halved. And if the base is 200, then the number will be
doubled and so on based on the base number.
Multiplication with 5,25,50 etc...
Substitute 5 by 10/2,25 by 100/4 and 50 by 100/2.
Examples:
1. 5*18=18*10/2=180/2=90
2. 24*25=24*100/4=2400/4=600
3. 73*50=73*100/2=7300/2=3650
Multiplication with 9, 99,999 etc..
Examples:
1. 13*9=13*(10-1)=130-13=117
2. 26*99=26*(100-1)=2600-26=2574
3. 350*999=350*(1000-1)=350000-350=349650
3.Square-Shortcut Tricks
Method1:Apply (a + b)2 = a2 + b2 + 2ab
Example1:182 = (10 + 8)2 = 102 + 82 + 2 × 10 × 8 = 100 + 64 + 160 = 324
Example2:1032 = 1002 + 32 + 2 × 100 × 3 = 10000 + 9 + 600 = 10609
Example3:562 = 502 + 62 + 2 × 50 × 6 = 2500 + 36 + 600 = 3136
Method2:Square of a number ending with 5
(𝑋5)2 = 𝑋 ∗ (𝑋 + 1) 𝑎𝑛𝑑 52
Example1:352 = 3 ∗ (3 + 1) 𝑎𝑛𝑑 52 = 12 25
Example2:652 = 6 ∗ (6 + 1) 𝑎𝑛𝑑 52 = 42 25
Example3:1152 = 11 ∗ (11 + 1) 𝑎𝑛𝑑 52 = 132 25
Method3:Squres of numbers from 51-59
Add 25 to unit digit and square unit digit
Example1:572 = 7 + 25 𝑎𝑛𝑑 72 = 32 49
Example2:532 = 3 + 25 𝑎𝑛𝑑 32 = 28 09
Example3:592 = 9 + 25 𝑎𝑛𝑑 92 = 34 81
Method4:square if you know square of previous number
(𝑛 + 1)2 = 𝑛2 + 𝑛 + (𝑛 + 1)
Example1:312 = 302 + 30 + 31 = 961
Example2:262 = 252 + 25 + 26 = 676
Example3:812 = 802 + 80 + 81 = 6561
(Bankaspire special)
Method 5:Square of a number if you know square of any other number.
Let X and Y be two numbers. You know the square of X then you can deduce
square of Y.
𝑋2 − 𝑌2 = (𝑋 + 𝑌)(𝑋 − 𝑌)
=> 𝑋2 = (𝑋 + 𝑌)(𝑋 − 𝑌) + 𝑌2
Or 𝑌2 = 𝑋2 − (𝑋 + 𝑌)(𝑋 − 𝑌)
Example1:1152 =?
Choose a nearby number whose square is known to you.
Suppose we choose 110 whose square is 12100
1152 = 1102 + (115 − 110)(115 + 110)
=> 12100 + 5 ∗ 225 = 13225
Example2:482 = 502 − [(50 − 48)(50 + 48)] = 2500 − 2 ∗ 98 = 2304
Example3:272 = 302 − [(30 − 27)(30 + 27)] = 900 − 3 ∗ 57 = 729
Example4:432 = 402 + 3 ∗ 83 = 1849
4.Cubes-Shortcut
Apply (a + b)3 = a3 + b3 + 3a2b + 3ab2
Example1:153 = (10 + 5)3 = 103 + 53 + 3 ∗ 102 ∗ 5 + 3 ∗ 10 ∗ 52 = 1000 + 125 +
1500 + 750 = 3375
Example2:233 = (20 + 3)3 = 203 + 33 + 3 ∗ 202 ∗ 3 + 3 ∗ 20 ∗ 32 = 8000 + 27 +
3600 + 540 = 12167
5.Square roots (applicable only for perfect squares)
Method 1
Example1:Square root of 2704
step1:Seperate number into group of two from right to left ie 27 04.
step2:What number can be squared and less than 27=5, with remainder 2
step3:Bringdown the second group of digits next to remainder to get 204
step4:Double first part of answer to get 5*2=10
step5:Find a number X so that 10 X * X= 204, we get X=2
Thus final answer=52
5 2
5 2704
25
102 204
204
000
Example2: Example1:Square root of 9604
step1:Seperate number into group of two from right to left ie 96 04.
step2:What number can be squared and less than 96=9, with remainder 15
step3:Bringdown the second group of digits next to remainder to get 1504
step4:Double first part of answer to get 9*2=18
step5:Find a number X so that 18 X * X= 1504, we get X=8
Thus final answer=98
9 8
9 9604
81
1881504
1504
0000
Method2:Square root by prime factorisation.
Example1: √44100= √(2 ∗ 2 ∗ 3 ∗ 3 ∗ 5 ∗ 5 ∗ 7 ∗ 7) =2*3*5*7=210
Exampl2:√254016 = √9 ∗ 9 ∗ 8 ∗ 8 ∗ 7 ∗ 7 = 9 ∗ 8 ∗ 7 = 504
6.Cube root(for perfect cubes only)
13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, 63 = 216, 73 = 343, 83 = 512, 93 = 729,
Memorize this.
Example1:√21952 3
step1: Divide digits into group of three from right to left 21 952
step2: Last digit of rightmost group is 2.That means number ends with 8
step3: Now consider leftmost group 21.Cube of 2=8 and cube of 3=27 ,since
21 is between them we must use smaller one,2. Thus final answer is 28
Example2:√32768 3
step1: Divide digits into group of three from right to left 32 768
step2: Last digit of rightmost group is 2.That means number ends with 8
step3: Now consider leftmost group 32. Cube of 3=27 and cube of 4=64,since
32 is between them we must use smaller one,3. Thus final answer is 32.
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