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Elementary Algebra: | ||
Math Fundamentals: |
In just FIVE minutes you should learn to quickly multiply up to 20x20 in your head. With this trick, you will be able to multiply any two numbers from 11 to 19 in your head quickly, without the use of a calculator. I will assume that you know your multiplication table reasonably well up to 10x10. Try this:
May be out of print "Super Math-E-Magics" by V.A. Stephen Lenaghan The 11 RuleYou likely all know the 10 rule (to multiply by 10, just add a 0 behind the number) but do you know the 11 rule? It is as easy! You should be able to do this one in you head for any two digit number. Practice it on paper first!To multiply any two digit number by 11:
The only thing tricky to remember is that if the result of the addition is greater than 9, you only put the "ones" digit in the hole and carry the "tens" digit from the addition. For example 11 x 57 ... 5__7 ... 5+7=12 ... put the 2 in the hole and add the 1 from the 12 to the 5 in to get 6 for a result of 627 ... 11 x 57 = 627 Finger Math: 9X RuleTo multiply by 9,try this: (1) Spread your two hands out and place them on a desk or table in front of you. (2) To multiply by 3, fold down the 3rd finger from the left. To multiply by 4, it would be the 4th finger and so on. (3) the answer is 27 ... READ it from the two fingers on the left of the folded down finger and the 7 fingers on the right of it. This works for anything up to 9x10! Square a 2 Digit Number Ending in 5For this example we will use 25
Try a few more 75 squared ... = 7x8=56 ... put 25 behind it is 5625. Square 2 Digit Number: UP-DOWN MethodSquare a 2 Digit Number, for this example 37:
With practice this can easily be done in your head. Thanks To RyanMultiply By 4To quickly multiply by four, double the number and then double it again. Often this can be done in your head. Multiply By 5To quickly multiply by 5, divide the number in two and then multiply it by 10. Often this can be done quickly in your head.The 11 Rule ExpandedYou can directly write down the answer to any number multiplied by 11.
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To find out if a number is divisible by seven:
Take the last digit, double it, and subtract it from the rest of the
number.
If the answer is more than a 2 digit number perform the above
again.
If the result is 0 or is divisible by 7 the original number is also
divisible by 7.
Example 1 ) 259
9*2= 18.
25-18 = 7 which is divisible by 7 so 259 is also divisible by 7.
Example 2 ) 2793
3*2= 6
279-6= 273
now 3*2=6
27-6= 21 which is divisible by 7 so 2793 is also divisible by 7 .
Now find out if following are divisible by 7
1) 2841
2) 3873
3) 1393
4) 2877
TO FIND SQUARE OF A NUMBER BETWEEN 40 to 50
Sq (44) .
1) Subtract the number from 50 getting result A.
2) Square A getting result X.
3) Subtract A from 25 getting result Y
4) Answer is xy
EXAMPLE 1 : 44
50-44=6
Sq of 6 =36
25-6 = 19
So answer 1936
EXAMPLE 2 : 47
50-47=3
Sq 0f 3 = 09
25-3= 22
So answer = 2209
NOW TRY To Find Sq of 48 ,26 and 49
TO FIND SQUARE OF A 3 DIGIT NUMBER :
LET THE NUMBER BE XYZ
SQ (XYZ) is calculated like this
STEP 1. Last digit = last digit of SQ(Z)
STEP 2. Second Last Digit = 2*Y*Z + any carryover from STEP 1.
STEP 3. Third Last Digit 2*X*Z+ Sq(Y) + any carryover from STEP
2.
STEP 4. Fourth last digit is 2*X*Y + any carryover from STEP 3.
STEP 5 . In the beginning of result will be Sq(X) + any carryover
from Step 4.
EXAMPLE :
SQ (431)
STEP 1. Last digit = last digit of SQ(1) =1
STEP 2. Second Last Digit = 2*3*1 + any carryover from STEP
1.= 6
STEP 3. Third Last Digit 2*4*1+ Sq(3) + any carryover from STEP
2.= 2*4*1 +9= 17. so 7 and 1 carryover
STEP 4. Fourth last digit is 2*4*3 + any carryover (which is 1) . =
24+1=25. So 5 and carry over 2.
STEP 5 . In the beginning of result will be Sq(4) + any carryover
from Step 4. So 16+2 =18.
So the result will be 185761.
If the option provided to you are such that the last two digits are
different, then you need to carry out first two steps only , thus
saving time. You may save up to 30 seconds on each
calculations and if there are 4 such questions you save 2
minutes which may really affect UR Percentile score.
PYTHAGORAS THEROEM :
In any given exam there are about 2 to 3 questions based on pythagoras theorem. Wouldn’t it be nice that you remember some of the pythagoras triplets thus saving up to 30 seconds in each question. This saved time may be used to attempt other questions. Remember one more right question may make a lot of difference in UR PERCENTILE score.
The unique set of pythagoras triplets with the Hypotenuse less than 100 or one of the side less than 20 are as follows :
(3,4,5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), (28, 45, 53), (11, 60, 61), (33, 56, 65), (16, 63, 65), (48, 55, 73), (36, 77, 85), (13, 84, 85), (39, 80, 89), and (65, 72, 97).
(15,112,113), (17,144,145), (19,180,181), (20,99,101)
If you multiply the digits of the above mentioned sets by any constant you will again get a pythagoras triplet .
Example : Take the set (3,4,5).
Multiply it by 2 you get (6,8,10) which is also a pythagoras triplet.
Multiply it by 3 you get ( 9,12,15) which is also a pythagoras triplet.
Multiply it by 4 you get (12,16,20) which is also a pythagoras triplet.
You may multiply by any constant you will get a pythagoras triplet
Take another example (5,12,13)
Multiply it by 5,6 and 7 and check if you get a pythagoras triplet.
TIPS FOR SMART GUESSING :
You will notice that in any case, whether it is a unique triplet or it is a derived triplet (derived by multiplying a constant to a unique triplet), all the three numbers cannot be odd.
In case of unique triplet , the hypotenuse is always odd and one of the remaining side is odd the other one is even.
Below are the first few unique triplets with first number as Odd.
3 4 5
5 12 13
7 24 25
9 40 41
11 60 61
You will notice following trend for unique triplets with first side as odd.
Hypotenuse = (Sq(first side) +1) / 2
Other side = Hypotenuse -1
Example : First side = 3 ,
so hypotenuse = (3*3+1)/2= 5 and other side = 5-1=4
Example 2: First side = 11
so hypotenuse = (9*9+1)/2= 41 and other side = 41-1=40
Please note that the above is not true for a derived triplet for example 9,12 and 15, which has been obtained from multiplying 3 to the triplet of 3,4,5. You may check for other derived triplets.
Below are the first few unique triplets with first number as Even .
4 3 5
8 15 17
12 35 37
16 63 65
20 99 101
You will notice following trend for unique triplets with first side as Even.
Hypotenuse = Sq( first side/ 2)+1
Other side = Hypotenuse-2
Example 1. First side =8
So hypotenuse = sq(8/2) +1= 17
Other side = 17-2=15
Example 2. First side = 16
So hypotenuse = Sq(16/2) +1 =65
Other side = 65-2= 63
PROFIT AND LOSS : In every exam there are from one to three
questions on profit and loss, stating that the cost was first
increased by certain % and then decreased by certain %. How
nice it would be if there was an easy way to calculate the final
change in % of the cost with just one formula. It would really help
you in saving time and improving UR Percentile. Here is the
formula for the same :
Suppose the price is first increase by X% and then decreased
by Y% , the final change % in the price is given by the following
formula
Final Difference % = X- Y – XY/100.
EXAMPLE 1. : The price of T.V set is increased by 40 % of the
cost price and then decreased by 25% of the new price . On
selling, the profit for the dealer was Rs.1,000 . At what price was
the T.V sold.
From the above mentioned formula you get :
Final difference % = 40-25-(40*25/100)= 5 %.
So if 5 % = 1,000
then 100 % = 20,000.
C.P = 20,000
S.P = 20,000+ 1000= 21,000.
EXAMPLE 2 : The price of T.V set is increased by 25 % of cost
price and then decreased by 40% of the new price . On selling,
the loss for the dealer was Rs.5,000 . At what price was the T.V
sold.
From the above mentioned formula you get :
Final difference % = 25-40-(25*45/100)= -25 %.
So if 25 % = 5,000
then 100 % = 20,000.
C.P = 20,000
S.P = 20,000 – 5,000= 15,000.
Now find out the difference in % of a product which was :
First increased by 20 % and then decreased by 10 %.
First Increased by 25 % and then decrease by 20 %.
First Increased by 20 % and then decrease by 25 %.
First Increased by 10 % and then decrease by 10 %.
First Increased by 20 % and then decrease by 15 %.
TIPS TO IMPROVE UR PERCENTILE :
HOW ABOUT SOLVING THE FOLLOWING QUESTION IN JUST
10 SECONDS
Ajay can finish work in 21 days and Blake in 42 days. If Ajay,
Blake and Chandana work together they finish the work in 12
days. In how many days Blake and Chandana can finish the
work together ?
(21*12 )/(24-12) = (21*12)/9= 7*4= 28 days.
NOW CAREFULLY READ THE FOLLOWING TO SOLVE THE
TIME AND WORK PROBLEMS IN FEW SECONDS.
TIME AND WORK :
1. If A can finish work in X time and B can finish work in Y time
then both together can finish work in (X*Y)/ (X+Y) time.
2. If A can finish work in X time and A and B together can finish
work in S time then B can finish work in (XS)/(X-S) time.
3. If A can finish work in X time and B in Y time and C in Z time
then they all working together will finish the work in
(XYZ)/ (XY +YZ +XZ) time
4. If A can finish work in X time and B in Y time and A,B and C
together in S time then :
C can finish work alone in (XYS)/ (XY-SX-SY)
B+C can finish in (SX)/(X-S)
and A+ C can finish in (SY)/(Y-S)
Here is another shortcut
TYPE 1 : Price of a commodity is increased by 60 %. By how
much % should the consumption be reduced so that the
expense remain the same.
TYPE 2 : Price of a commodity is decreased by 60 %. By how
much % can the consumption be increased so that the expense
remain the same.
Solution :
TYPE1 : (100* 60 ) / (100+60) = 37.5 %
TYPE 2 : (100* 60 ) / (100-60) = 150 %