Speed mathematics
First rule
of the mathematics:
The easier
the method you use to solve a problem, the faster you will solve it with less
chance of making a mistakes.
Why
mathematics is important?????
§ People equate mathematical ability
with general intelligence.
§ If you are good at maths, you are
generally regarded as highly intelligent.
§ High achieving maths students are
treated differently by their teachers and colleagues.
§ Mental calculations improves
concentration, develops memory, and enhances the ability to retain several
ideas at once
§ Mathematical knowledge boosts your
confidence and self-esteem.
§ These methods will give you
confidence in your mental faculties, intelligence and problem-solving
abilities.
1. MULTIPLYING NUMBERS UP TO 10:
We will
begin by learning how to multiply numbers up to 10x10
E.g. 1 7X8=????
3 x 2
Ø Go to the first number 7. How many
more do you need to make 10? The answer is 3.
Ø Go to the next number 8. How many
more do you need to make 10? The answer
is 2
Ø Then subtract diagonally. Take either
one of the number. ( 7-2 =5 8-3= 5)
Ø 7
x 8 =5(3
x 2)( put 5 on the left side
and next to tat 5 put the multiplication of 3 and 2) and we get
Ø 7
x 8 =56.
2.
MULTIPLYING NUMBERS GREATER THAN 10.
(MULTIPLYING WITH A REFERENCE NUMBER)
Using 100 as a reference number.
Steps involved:
Ø Just take an example of multiplying
96 X 97
Ø Take 100 as a reference number.
Ø Just subtract 96-100 = -4 and 97-100
= -3
Ø Then subtract diagonally and take
either of the number, we will get 93 and it was multiplied by 100.
Ø You will get 9300 and -4 X-3 gives
you +12. Add this 12 with 9300 we will get 9312.
3. MULTIPLYING NUMBERS ABOVE 100:
11,024
=>11000+24=11024
Ø It’s similar to the previous
technique.
Ø Take 100 as reference number, 106-100
= +6, 104-100 = 4.
Ø This time just add diagonally. We
will get 110 and it was multiplied by 100 we get 11000.
Ø Then multiply +6 and +4 we get 24 and
it was added to 11024.
4.
COMBINATIONAL METHOD:
10
-2 -3
8556
Ø Suppose on subtracting the given
numbers with the reference number we getting higher number, we need to do the
same procedure for the number (which we will get on subtracting the main number
from the reference number.
5. MULTIPLYING NUMBERS ABOVE AND BELOW THE REFERENCE NUMBERS:
13230
Ø The same principle is to be followed.
Ø Here 98 is lower than 100 and 135 is
more than 100.
Ø Just subtract the multiplying number
with the reference number. We will get -2 and +35.
Ø Then just follow the previous
technique we will get 13300,
Ø Then multiply -2 and +35 we get -70.
Ø Then just add 13300-70 we will get
13230.
6.
MULTIPLICATION BY FACTORS:
Ø It’s easy to multiply by 20, as 20 are
2 times 10. And it is simple to multiply by 10 and by 2. This is called
multiplication by factors, as 10 and 2 are factors of 20.
713
Ø Just do the same procedure.
Ø (23+11=31+3=34, )
Ø (34 x 20=> 34x 10=340, 440x 2=
680)-> multiplication by factors
Ø 3*11 gives 33 and 680+33 gives you
713 tats the answer.
7.
MULTIPLYING BELOW 20
Ø Here the only difference we need to
take 20 as reference number.
Ø And follow the same procedure.
8.
MULTIPLYING USING DOUBLE REFERENCE
NUMBER:
8 X
37=????
-8 296
Ø In this problem we have two reference
nos. , for 8, the reference no is 10 and for 37, the reference no is 40.
Ø Its shows the reference no of 8 is
four times the reference number of the 37.
Ø Just subtract the no from its
reference no. (8 - 10= -2, 37- 40=
-3)
Ø Then multiply the first difference (2)
by 4 ( the multiplies of the reference no 10) , -> -2 x 4 = -8,
Ø Then -8+37 = 29 , and multiply by the
main reference no 10., 29x10 = 290.
Ø Then -2 x -3 = 6, 290+6 = 296.
Ø The answer is 296.
EASY MULTIPLICATION BY 9
9X486 =
?
-49 4374
Ø The reference no for 9 is 10, and for
486 is 490,
Ø And the second reference no is 49
times the first reference no.
Ø Just the subtract the respective no
from their reference no.
Ø We will get 9-10 = -1, 486-490= -4
Ø Multiply the difference of first
reference no (10) by the multiple of the second reference no. => -1 x 49 =
-49.
Ø Subtract -49 from 486, => 486-50+1
= 437 and multiply by the first reference no. we will get -4370.
Ø Then -1 x -4 we will get 4 and it was
added to 4370 we will get 4374... That’s the answer.
Shortcut for
subtraction
Ø What is the easiest way to take 90
from a number?
Take 100 and give back 10
Ø What is the easiest way to take 80
from a number?
Take 100 and give back 20
Ø What is the easiest way to take 70
from a number?
Take 100 and give back 30
-2 35
30
Ø How do we take 70 from 13,330?
Ø Take away 100 and give back 30
Checking the
answers by using substitute number:
Ø 13 x 14 = 182
Ø The first number is 13. Add its
digits together to get the substitute (1+3=4: four becomes our substitute for
13)
Ø The next number we are working with
is 14. The substitute is ( 1+4=5)
Ø Then multiply the substitute numbers
( 4 x 5=20: twenty is a two digit number so we add its digits together to get
our check answer : 2+0=2).
Ø Do the same principle in the RHS Side
too ( 1+8+2=11, 1+1= 2)
Ø So both LHS and RHS are equal. Hence
our answer is correct.
DOUBLING AND HALVING
NUMBERS
To use 20 and 50 as reference
numbers, we need to be able to double and halve numbers easily.
Ø To halve 78, 78 = (80-2). and half of
(80-2) is 40-1 = 39
Ø To double 78, do it the reverse
manner. (80-2). Double of (80-2 ) is 160-4 = 156
MULTIPLYING THE DECIMALS:
1.3 X 1.4 =?
Ø Ignore the decimal point in the
calculation.
1.
10
13 14 =
170
Ø In order to put the decimal points,
count the number of digits after the decimal point in the given question.
Ø Here two digits after the decimal
points. Hence place the decimal point after two digits ( 1.82 is the answer)
2. 8 x 68 =?
Let take 8.0 x
68 =?.
-20 -32 640
Ø Let same principle for
multiplying we got the answer 5440
Ø Let count the no of digits, its 1
Ø Then answer is 544.0.
Ø Lets try 9 x 83 = , 9 x 67 =
MULTIPLICATION
BY 11
Ø To multiply a two digit number by 11,
simply add the two digits together and insert the result in between.
Ø For example, to multiply (23 x 11),
add (2+3= 5), and insert the 5 in between 2 and 3, the answer is 235.
MULTIPLYING
MULTIPLES OF 11.
EG: 330
x 12 = ?
Ø 330 = 3 x 10 x 11 ( ignore the zero
now)
Ø 3 x 11 x 12 = 11 x 36 = 396
Ø Now keep the zero now. We will get
3960.
CONVERTION
OF KILOGRAMS INTO POUNDS:
Ø DOUBLE THE NUMBER, MULTIPLY BY 11,
AND DIVIDE BY TEN TO ACCOUNT FOR THE DECIMAL PLACE. ( TO CONVERT KG TO POUNDS
YOU MULTIPLY BY 2.2)
1. 80 KG
TO POUNDS = ?
Ø 80 kg = (80 x 2.2) = 160x 11/10 = 176
pounds.
MULTIPLYING
LARGER NUMBERS BY 11:
2. 12345 X 11 = ?
Ø Write zero in front of the number we
are multiplying. ( 012345 x 11 )
Ø Beginning with the units digit, add
each digit to the digit on its immediate right. In this case add 5 to the digit
on its right. There is no digit on its right, so add nothing
Ø 5+0=5, (write 5 as the last digit of
your answer)
012345 x 11
Ø Now go to the 4. Five is the digit on
the right of the 4: (4+5 =9 )
Ø 012345 x 11, ( continue the same
procedure)
Ø 95
Ø 3+4 =7
Ø 2+3=5
Ø 1+2=3
Ø 0+1=1
Ø The answer is 135795
CHECKING
YOUR ANSWERS:
1. 012345 x 11
Ø Write a cross under every second
digit of the answer, beginning from the right hand end of the number. The
calculation will now look like
Ø 012345 x 11
1 3 5 7 9 5
X
X X
Ø Now add the digits with the cross
under them ( 1+5+9 = 15)
Ø Add the digits without the cross (
3+7+5 = 15)
Ø We are getting both the answers are
equal hence our answer is correct.
2. 0217475 x 11
2 3 9 2 2 2 5
X x
x
Ø Add the digits in the cross( 3+2+2 =
7)
Ø Add the digits in the without cross (
2+9+2+5 = 18)
Ø The difference between two answers is
11.
Ø If the difference is
0,11,22,33,44,55,66 etc., then the answer is correct.
MULTIPLICATION
BY 9 :
Ø There is an easy shortcut for
multiplication by 11 because 11 is one more than 10, so there is an easy
shortcut for multiplication by 9 because 9 is one less than 10.
Ø Instead of adding each digit to the
digit on the right, we subtract each digit from the digit on the right.
Ø Because subtraction involves
borrowing and carrying, we can make the following short cut..
Ø We subtract the units digit from 10,
then subtract each successive digit from 9 and add the neighbor. We subtract 1
from the first digit of the number for the first digit of the answer.
ADDITION
The basic rule for mental addition:
To add 9, add 10 and subtract 1: to
add 8, add 10 and subtract 2; to add 7 add 10 and subtract 3, and so on.
Ø If you wanted to add 47, you would
add 50 and subtract 3,
Ø To add 196, add 200 and subtract 4.
Ø To add 38 to a number, add 40 and
subtract 2,
TWO DIGIT MENTAL ADDITIONS:
If the
units digit is high, round off to the next ten and then subtract the
difference. If the units’ digit is low, add the tens then the units.
Ø With two digit mental addition you
add the tens digit of each number first, then the units. If the unit’s digit is
high, round off the number upwards and then subtract the difference. If you are
adding47, add 50, and then subtract 3.
Ø To add 35, 67, and 43 together you
would begin with 34, add 70 to get 105, subtract 3 to get 102, add 10 to get
142 then the 3 to get your answer of 145.
ADDING THREE DIGIT NUMBERS:
355+752+694
=?
Ø 355+700 = 1055
Ø 1055+50+2 = 1107
Ø 1107+700-6 = 1807-6 = 1801
OR
Ø You may prefer to add from left to
right; adding the hundreds first, then the tens and then the units.
ADDITING LARGER NUMBERS:
8461
Ø We begin with the thousands
column.8+5 = 13, since we are dealing with thousands, our answer is 13
thousand.
Ø Observe that the numbers in the
hundreds column conveniently add to 10, so that gives us another thousand. Then
answer is 14000.
Ø Then add 61 to 14000, we getting
14061.
Ø Add 80 to and subtract 2. To add 80
add 100 and subtract 20, (14061+100-20-2) = 14161-20-2=14141-2=14139 is the
answer.
Ø An easy rule is: when adding a column
of numbers add pairs of digits to make tens first, then add the other digits.
SUBTRACTION:
To
subtract mentally, try and round off the number you are subtracting and then
correct the answer.
To
subtract 9, take 10 and add 1: to subtract 8, take 10 and add 2; to subtract 7,
take 10 and add 3,
1. Eg: 56-9 =
-1
(To
take 9 from 56 in your head, the easiest and fastest method is to subtract 10,
(46) and add1 we get 47.)
2. 54-38 = 16
+2
Ø 44-40, plus 2 makes 16
3. 436-87 =
+13
Ø Take 100 to get 336. Add 13 and we
will get 349 easy.
Ø SUBTRACTING ONE NUMBER BELOW A HUNDREDS VALUE FROM ANOTHER WHICH IS JUST
ABOVE THE SAME HUNDREDS NUMBERS.
Three digit SUBTRACTIONS:
1. 461 -275 =
25
161+25 = 160+20+5+1 = 186
2. 834 – 286 =
14
534+14 = 530+10+4+4 = 540+8 = 548
Subtraction from
a power of 10:
The
rule is : SUBTRACT THE UNITS DIGIT FROM 10, THEN EACH SUCCESSIVE DIGIT FROM 9,
THEN SUBTRACT 1 FROM THE DIGIT ON THE LEFT.
1. 1000
-574
Ø 10-4=6,
Ø 9 - 7 = 2,
Ø 9 - 5 = 4,
Ø 1 - 1 = 0
The answer is 0426
Subtracting
smaller numbers:
If the number we are subtracting has fewer digits than the
one you are subtracting from, then add zeros before the number (at least,
mentally) to make the calculation:
For instance:
23 000 – 46 =
23 000
Use the same principle as subtraction method 2.
Squaring
the numbers:
1.
Squaring
the numbers ending with 5.
352 =
Ø Separate the 5 from the digits in
front. in this case there is only a 3 in front of the 5. Add 1 to 3 get 4 (3+1=
4)
Ø Multiply these numbers together: 3 x
4 = 12
Ø Write the square of 5 (25) after 12.
We will get 1225.
135 2 = ??
Ø Take 13, add 1 to it we will get 14.
Ø Then 13 x 14 = 182
Ø Add the square of 5 next to it. We
will get 18225.
SQUARING THE NUMBERS
NEAR TO 50:
1.
462
=
Ø Forty six squared means 46 x 46.
Rounding upwards, 50 x 50 = 2500.
Ø Take 50 and 2500 as our reference
points.
50
462
- 4
46 = 50-4, so 4 is a minus number.
Ø So we take 4 from the 25 hundreds.
Ø (25-4) x 100= 2100
Ø To get the rest of the answer, we
square the number in the minus. ( 4 2= 16)
Ø Add 2100 and 16 we will get 2116 is
the answer.
562 =
Ø Fifty six squared means 56 x 56.
Rounding upwards, 50 x 50 = 2500.
Ø Take 50 and 2500 as our reference
points.
50
562
+6
56 = 50+4, so 6 is a positive number.
Ø So we add 6 to 25 hundreds.
Ø (25+6) x 100= 3100
Ø To get the rest of the answer, we
square the number in the minus. ( 62= 36)
Ø Add 3100 and 36 we will get 3136 is
the answer.
SQUARING NUMBERS NEAR
TO 500:
Ø This is similar to our strategy for
squaring numbers near 50.
Ø Five hundred times 500 is 250000, we
take 500 and 250000 as our reference number.
1. 5062 =
500 5062
+6
5002 = 250000
Ø Five hundred and six is greater than 500,
Ø Square of 500 is 250000
Ø The number 6 is added to the
thousands
Ø (250+6) x1000 = 256000
Ø Square 6 is 36.
Ø 256000+36 =256036 is the answer.
Square the number ends
with 1:
1.
312
=
Ø First, subtract 1 from the number.
The number now ends in zero and should be easy to square. (302 = 3 x 3 x 10 x 10) = 900
Ø Add 30 and its next number 31
(30+31). We will get 61)
Ø Add (900 + 61) = 961.
2.
3512=?
Ø 3502 = 122500
Ø 350 +351 = 701
Ø 122500 +701 = 123201
We
can also use the method for squaring numbers ending in 1 for those ending in 6.
3.
862
=
Ø 852 =7225
Ø 85+86 = 171
Ø 7225+171 = 7396
Squaring numbers ending with 9
1.
292
=
Ø Add 1 to the number. The number now
ends in zero and is easy to square.
Ø 302 = 900
Ø Now add 30 with given number29 (30+29
=59)
Ø Then sub (900 - 59 = 841)
2.
3492
=
Ø 3502 = 122500
Ø 350+349 = 699
Ø Sub (122500 - 699 = 121801)
MULTIPLYING NUMBERS
WHEN THE UNITS DIGITS ADD TO 10 AND THE TENS DIGITS DIFFER BY 1.
THERE IS RULE IN
MATHEMATICS THAT IF YOU MULTIPLY TWO NUMBERS THAT DIFFER ABOVE AND BELOW A SQUARE
1.
67
x 73 =
Ø 67 = 70 - 3
Ø 73 = 70 + 3
Ø 67 x 73 = 702 – 32
Ø = 4900 – 9 = 4891
MULTIPLYING NUMBERS
NEAR 50
1.
54
x 58 =
50 54
x 58
4 8
Ø Add 4 + 8 = 12
Ø Halve the answer 12/2 = 6
Ø Add that answer to 25 , we will get
25+6 = 31 (31 hundreds)
Ø Multiply ( 4 x 8 = 32)
Ø Add (3100 + 32 = 3132)
2.
46
x 48 =
50 46
x 48
-4 -2
Ø Add (-4 -2 = -6)
Ø Halve the answer (-6/2 = -3)
Ø Subtract ( 25 – 3 = 22) ( 22 hundreds
)
Ø 4 x 2 = 8
Ø Add ( 2200 + 8 = 2208)
TEMPERATURE CONVERSION:
To
change Fahrenheit to Celsius, subtract 30 degree and then halve.
To
change Celsius to Fahrenheit, double and then add 30.
Ø Then answer you will get using this
formula will not be exact, but its close enough for practical purposes.
Ø If you are told the temperature will
be 8 o C, double it and add 30. We will get 46 o F. but
the actual temperature would be 46.4oF
Ø Easy temperature conversion formulas
: (o C x 2) +30 = o F, (o F -30) /
2 = o C,
Ø Exact temperature conversion formulas
: (o C x 9/5) +32 = o F, (o F -32)
x 5/9 = o C,
TIME AND DISTANCES:
1.
A
common ruler is 30 cm. thirty divided by 12 gives approximately 2 ½. ( 2.5 cm is
equal to 1 inch)
2.
100
km = 60 miles.
POUNDS TO KILOGRAMS
1.
1
KG = 2.2 POUNDS
65 kg = 65 x 11 x 2/10
=143
pounds
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