160 - Multiplication and Long Division

Multiplication Tricks

Multiply by 9,99,999,etc...

56*99=5544

Step 1:Place a zero at the end for each 9 :5600

Step 2 : Subtract the original number from Step 1 like this 5600-56=5544

5425×99

5425×(100-1)

542500-5425 = 5307075

6289×99

6289×(100-1)

628900-6289 = 56611

Multiply by 125

68*125=8500

Step 1 :Place three zeros at the end of the number :68000

Step 2: Divide the number from Step 1  by 8:68000/8=8500

64*125 is the same as :

32*250 is the same as

16*500 is the same as 

8*1000

64*125

Step 1. Each time you just need to pick 125 multiply it by 8 will get 1000

Step 2. Pick 64 and divide it by 8 will get 8

Step 3. Multiply the results with each other 8* 1000 

Hence Solution is 8000

[Hint: Just remember 125*8=1000]

Multiply two digits numbers ending in 1

51*31=1581

Step 1: Multiply the left most digits : 5*3=15

Step 2: Add the left most digits:5+3=8

Step 3: Places the result from Step 2 next to the result from Step 1:158

Step 4 : Places 1 next to the result from  Step 3 : 1581

61×41=2501

6×4=24

6+4=10

2400+100+1 = 2501

21×11=231

2×1=2

2+1=3

231

51*61=3111

5*6=30

5+6=11

3000+110+1=3111

Multiply two digit number by 11

53*11=583

Step 1: Add the both digits of the two digit number:5+3=8

Step 2: Place the result in between both digits : 583

59*11=649

Step 1: 5+9=14

Step 2 : Carry the 1 when the result is greater than 9:5+1=6

Step 3: 649

35×11 = 3,(3+5),5 =385

78×11 =7,(7+8),8 = 858

Multiply by 5

 1234 *5 =6170

Step 1 : Divide  the number by 2,   1234/2=617

Step 2: Multiply  the result from Step 1 by 10 :  617*10=6170

Multiply by 25

 18*25=450
Step 1: Divide  the number by 4,    18/4
Step 2: Multiply  the number from Step 1 by 100:    4.5 * 100 = 450

Multiply by 9

56*9=504

Step 1: Multiply the number by 10,:     56*10=560

Step 2: Subtract the original number from Step 1:     560-56=504

Factorization

By Factoring number, you can break down problems into simpler multiplication tasks. Also, you may be able to apply some techniques you learned.

21*33

step 1 : 21*11*3

Step 2: 231*3

Step 3 :693

67*81

Step 1: 67*9*9

Step 2:603*9=5427

28*125=3500

Step 1:  28*125

Step 2:  28*25*5

Step 3: 28*(100/4)*5

Step 4: 28/4*100*5

Step 5: 7*500=3500

Some Special type

When sum of unit digit is 10 and remaining digit is same.

43×47 = 4 × ( 4+1 ) / 3×7

= 4×⁵/21

= ²⁰/21
Ans = 2021

72×78 = 7×8 / 2×8
= ⁵⁶/16
Ans = 5616

104 × 106 = 10× ( 10+1 ) / 4×6
= 10 × ¹¹/24
= ¹¹⁰/24
Ans = 11024

When sum of ten's digit is 10 and unit digit is same

46 × 66
= ( 4×6 ) +6 / 6×6
= 24 + ⁶/36
= ³⁰/ 36
Ans = 3036

83 × 23
= ( 8×2 ) +3 / 3×3
= ¹⁹/09
Ans = 1909

92 × 12
= ( 9×1 ) + 2/2 × 2
= ¹¹/04
Ans = 1104

When unit digit is 5 in both the numbers and difference between both number is 10.

75 × 65

=  6 × ( 7+1 ) / 75
=  ⁴⁸/75
Ans = 4875

45 × 35
= 3 × ( 4+1 )/75
= 15 /75
Ans = 1575

105 × 95
= 9 × ( 10+1 ) / 75
= 99/75
Ans = 9975

Find the unit digit  of 147128 * 138148 ?

7128 ð (74)28  ð 1 (because 74 =2401)…….(1)

6148 ð (63)49 × 61 ð 1 (because 63 =216)……. (2)

ð 6×6 = 6

ð 6×1 = 6

Multiply by 11,111,1111....so on

Ques 1. 111111111 ✘ 111111111 = ?

Sol:
No of digits in multiplier = 9
Write in ascending order from left side like this:
987654321
and now 9-1=8
write it in descending order just after it
12345678
now you will get like this:
12345678987654321
hence
111111111✘111111111 = 12345678987654321

Ques 2. 1111111111 ✘ 1111111111 = ?
Sol:
No of digits in multiplier =10
Write in ascending order from left side like this:
10 9 8 7 6 5 4 3 2 1
and now 10-1=9
write it in descending order just after it
1 2 3 4 5 6 7 8 9
and after it just add the carry
1 2 3 4 5 6 7 8/ 9/ 10 9 8 7 6 5 4 3 2 1
                  8+1/   9+1 / 0
1 2 3 4 5 6 7 9 0 0 9 8 7 6 5 4 3 2 1
now you will get like this:
1234567900987654321
hence
1111111111✘1111111111 = 1234567900987654321

11×22 = 242

2×1 = 2, 2×2=4 ð 242

1111×2222 = 2468642

2×1 =2,

2×2=4,

2×3=6,

2×4=8

Ascending Order = 2468

Descending Order = 8642

1111×2222 = 2468642

Ques 3. 1111111✘2222222 = ?
Sol:
No of digit in the multiplier is 7 then let n=7;
Now Just multiply the digit 2 from 1 to 7 time & arrange them from extreme left to right in ascending order,you will get like this:
14 12 10 8 6 4 2
and now just subtract one from n.like this n=7,so n-1=6.
Multiply the digit 2 from 1 to 6 time & arrange them from just right after it,you will get like this:
2 4 6 8 10 12
Now placing both outcome like this & add the carry
2 4 6 8 10 12 14 12 10 8 6 4 2
        8+1/0+1/2+1/4+1/2+1
You will get the answer:
2 4 6 9 1 3 5 3 0 8 6 4 2

Ques 4. 1111111✘5555555 = ?
Sol:
No of digit = 7
Now Just multiply the digit 5 from 1 to 7 time & arrange them from extreme left to right in ascending order,you will get like this:
35 30 25 20 15 10 5
Just right after it perform same action but in descending order & till 6 times only.like this:
5 10 15 20 25 30
Now placing together ,just add the carry
5 10 15 20 25 30 35 30 25 20 15 10 5
6 1 7 2 8 3 8 2 7 1 6 0 5
1111111✘5555555=6172838271605

3 Step Multiplication Trick - A shortcut method

§   

In the series of providing quantitative shortcuts and tricks, today I come up with multiplication trick.

While doing multiplication of a two digit number with another two digit number, we take at least 6 steps. Try yourself. Multiply 62 with 32. 

Now let's do this with a trick

Step 1:  First step is same as conventional method; here we multiply 2 with 2.

62×32

2×2 = 4

Step 2: This is an interesting step. Now multiply last digit first value and first digit of second value and vice-versa. Then we add outcomes. But we need the last number that is 8 here.

62×32

2×3 + 6×2 = 18

Step 3 : This is the last step, in this step we do multiplication ten's digit of both value and add the remainder from previous calculation. That's it, we completed the calculation in 3 steps instead of six steps.

62×32

6×3 = 18 + 1 = 19

Now put the products 19  8  4  = 1984

Multiplication of 3 digit numbers

In this example I will multiply 432 with 346. Now the 3 step multiplication method will become 5 step. This method can be used for 4 and even 5 digit numbers but as in bank exams there is lack of time available for calculations I recommend you to use approximation for long calculations.

432×346

STEP 1:  4 3 2×3 4 6

2×6 = 12, put 2 in digit place carry forward 1

STEP 2:  4 3 2×3 4 6

3×6 + 4×2 = 26,   2 6 +1 (cf) =27, put 7 in tenth place carry forward 2

STEP 3:  4 3 2×3 4 6

4×6 + 3×2 + 3×4 = 42,   42 +2(cf) =44, put 4 in hundredth place carry forward 4

STEP 4:  4 3 2×3 4 6

3×3 +4×4 = 25,   25 +4(cf) =29, put 9 in hundredth place carry forward 2

STEP 5:  4 3 2×3 4 6

3×4 = 12,   12 +2(cf) =14, put 4 in thousandth place and 2 in ten thousandth place

Final result = 149472

In case you find any difficulty to understand the above multiplication method then ask your question in the comments. 

Long Division Tricks

One of the biggest problem with IBPS/Bank, SSC, RAILWAY, LIC  type of exams is  long division and percentage (%) calculation every now and then, directly or indirectly (in Data interpretation questions). And If you’re not good with speed maths, you’ll waste a lot of time in such stupid calculations.

For the aptitude questions of Bank/MBA exams, Knowing the  answer approach is not sufficient. You also need to get the answer quickly and accurately. Otherwise someone else will tick more answers and he’ll get the rank.

Consider these two questions

Question:  In a warehouse there are 230kg of wheat initially. But rats ate away 34 kg. How much % of wheat is left?

Answer:               (a).  49.3%  (b). 60.1% (c). 85.2%  (d). 85.7%         

Approach            {(230-34)/230} x100=(196/230)x 100=answer       196/23= answer

Now the Questions is : What is the answer of 196/23=?

(a) 4.93 (b) 6.01 (c) 8.52 (d)  8.57

Everyone, even with half hearted preparation, knows the approach. But the problem is actually in ‘doing’ that calculation or division (196/23). Here I’ll show a single method, to solve both types of calculations. As long as you know how to add two numbers, and how to multiply a number with 5, you can execute this method effortlessly.
Just for easy understanding  create a “Master Table” (don’t just read it, do this simultaneously using your own pen and paper)

100%

230

Now make a new row for 50%

100%	230
50%

So either divide 230 by 2=115 or multiply 230 with 5 and then shift one decimal point leftwards. (that is 1150 ==> to 115.0). In either case you get 50% of 230=115

100%	230
50% (half of 230)	115

Now make a new row for 20% but keep it empty right now.

100%	230
20%

Create one more row for 10% and simply shift one decimal leftwards. i.e. 230–> 23.0

100%	230
50% (half of 230)	115
20%
10% (one decimal point less)	23.0

Now double the 10% number (i.e.by adding 23 into 23 again) so you get 23+23=46. That’s our 20%. Fill up the table.

100%	230
50% (half of 230)	115
20% (double of 10%)	46
10% (one decimal point less)	23.0

Our master table is ready, now Imagine there is a big water tank with total capacity of 196 lit.
We can fill it with buckets of size 10%, 20% and 50% only.
We want to fill up the tank with minimum effort. So first take 50% (115), some space will be left.
By this time you get the idea that

1.     answer is more than 50% (if % value of 196/230 is asked)

2.     answer is more than 5 (if absolute value 196/23 is asked)

so eliminate answer options that do not meet these criteria.
Move on

Tank	Filled	Buckets
196	115	50%
Total	115	50%

There is still some space left in the tank so let’s throw a 20% bucket

Tank	Filled	Buckets
196	115	50%
	046	20%
Total	161	70%

Or you can add 10% bucket two times, you’ll get same result.
It’s clear that our answer is bigger than 70%. So eliminate any options less than 70%
Hmm, so far we’ve filled 161, It can still accommodate another 10% bucket

Tank	Filled	Buckets
196	161	70%
	023	10%
Total	184	80%

Now we are very close, only 196-184=12 lit. remains. But no bucket is that small!
Solution= move the decimal numbers, to create new small sized buckets.

Master Table		Moving decimal numbers
100%	230
50% (half of 230)	115	5%	11.5
20% (double of 10%)	23×2= 46	2%	4.6
10% (one decimal point less)	23.0	1%	2.3

In the exam, you don’t have to actually write new columns of 5%, 2% and 1%, just visualize them in your head, by shifting the decimal to one point leftwards.
Recall that 12 lit is empty and Now we’ve a new 5% bucket that can almost fill it up.

Tank	Filled	Buckets
196	184	80%
	011.5	5%
Total	195.5	85%

By this time you get the idea that

1.     answer is just a little higher than 85% (if % of 196/230 is asked)

2.     answer is just a little higher than 8.5 (if absolute value 196/23 is asked)

so eliminate any answer options that are not meeting this criteria.
Still if two or more options remain. For example

a.     8.52

b.     8.57

This situation usually happens in Data Interpretation questions. Now what to do?
Well, Total capacity is 196 lit. and so far we filled up 195.5 so, 0.5 lit is still empty. But no bucket is small enough to carry water in this scale. Solution= create more buckets, by shifting decimal points in the “Master Table”.

Master Table		Moving decimal numbers
100%	230	Phase I		Phase II
50% (half of 230)	115	5%	11.5	0.5%	1.15
20% (double of 10%)	23×2= 46	2%	4.6	0.2%	0.46
10% (one decimal point less)	23.0	1%	2.3	0.1%	0.23

Recall that 0.5 lit is empty and from above table, it is clear that 0.2% bucket (0.46 is very close) so let’s use it.

Tank	Filled	Buckets
196	195.50	85%
	000.46	0.2%
Total	195.96	85.2%

So the final answer is

• 196/23=8.52
• 196/230=85.2%

If you want even more accurate answer, create more buckets and proceed in the same manner.

Important side notes

1.     Whenever you have to do long-division e.g. 256/29, always make the denominator (bottom number i.e. 29) very close to the top number (256) and take that as 100%. That is 290=100%. And then rephrase question: “256 is how much % of 290”, then proceed according to the method you just learned. You’ll get 88.27%, but our question was 256/29. Recall that you’ve added one zero more. (290)

So, 1%=1/100
Therefore, 88.27 %=( 88.27/100)
And from the ‘bottom’ we take back one zero that we had added earlier. So instead of 100, there remain only 10
88.27/10=8.827 is our answer for 256/29

2.     If there is 7526/67 then? Again same method, 7526 is how much % of 6700? You’ll get 112.3%, this time we’ve added two zeros more (i.e. we used 6700 instead of 67).

So, 1%=1/100
Therefore 112.3%=112.3/100
But take back those two zeros we had added earlier. So, instead of 100, there remains only 1
112.3/1=112.3 is our answer for 7526/67

• This method looks awkward and tiresome initially, but once you’ve enough practice of doing mental addition then it’s way easier than the Vedic Maths’ concept of double or triple digit division (because in Vedic method, many a times you’ve to adjust and carry over the numbers= not very convenient).
• This method can be used for three-digit, four digit divisions also.
• You can do any division as long as you can find out 10%, 20% and 50% of a number (and consequently 1%, 2%, 5%, by shifting decimal places.)

Percentage calculation or Long Division has direct or indirect applications in following topics:

1.     Profit Loss

2.     Data Interpretation, especially those based on Pie-charts.

3.     Compound interest, Simple Interest Rate, Population Growth: by the way, they can be solved without mugging up formulas.

4.     Mixture-Alligiation, Wine-water, Metal alloys

5.     Time-Speed-Distance, Time and Work, Pipes and Cisterns, Boats and trains.