4. General Tips for Studying Mathematics
1. Go To Class regularly
2. Get to Class On Time.
3. LISTEN During Class.
4. Take Good Notes.
5. Ask Questions.
6. Listen When Others Ask Questions.
7. Review Notes After Class.
8. Make a Set of Index Cards.
9. Learn The (Proper) Notation.
10.Get Into A Study Group.
11.Note Due Dates.
12.Budget Adequate Time For Studying/Homework.
13.Do Homework After Each Class.
14.Do Homework Without Notes and Book.
15.Do More Homework.
16.Practice, Practice, Practice.
17.Persevere Keep Old Homework and Exam Papers.
18.Don’t Forget Your Textbook.
19.Seek Help If You Need It.
20.Seek Help If You Need It . You should always do the best that you can
and strive for the best grade that you can possible get.
5. Study Tips for Math
1. Always read math problems completely before beginning any calculations.
If you "glance" too quickly at a problem, you may misunderstand what really
needs to be done to complete the problem.
2. Whenever possible, draw a diagram. Even though you may be able to
visualize the situation mentally, a hand drawn diagram will allow you to label
the picture, to add auxiliary lines, and to view the situation from different
perspectives.
3. Do not feel that you must use every number in a problem when doing your
calculations. Some mathematics problems have "extra" information. These
questions are testing your ability to recognize the needed information, as
well as your mathematical skills.
4. Remain confident! Do not get flustered! Focus on what you DO know, not
on what you do not know. You know a LOT of math!!
5. If you are "stuck" on a particular problem, go on with the rest of the test.
Oftentimes, while solving a new problem, you will get an idea as to how to
attack that difficult problem.
6. In certain problems, you may be able to "guess" at an approximate (or
reasonable) answer. After you perform your calculations, see if your final
answer is close to your guess.
6.
7. Fear of Maths is only mental
I suggest:
1.Instead of saying DIVIDE BY 2, say HALF/HALVE IT.
2.Instead of saying MULTIPLY it by 2, say DOUBLE IT.
3.Never use more than two digit numbers to prove the
working of a method.
4 .Show the more interesting sides of maths, for example,
show the beauty of the table of nine (which really looks
cute, simple and well arranged).
After these small things, leave the person to grow up
inside herself, by herself. They’ll start with small victories,
and keep gathering courage for bigger ones. Maths is easy
and beautiful up to a certain level. Let’s all enjoy this
beautiful, universal language.
8. Tough Multiplication
If you have a large number to multiply and one of the
numbers is even, you can easily subdivide to get to
the answer:
32 x 125, is the same as:
16 x 250 is the same as:
8 x 500 is the same as:
4 x 1000 = 4,000
Tough Multiplication
11. When solving absolute value inequalities: if the
absolute value is greater than a number you must
use the conjunction (OR),
when the absolute value is less than a number you
must use the conjunction AND.
To remember this just remember two words
"GOR"-"LAND," which translate into "G(greater)OR"
and "L(less than)AND."
When I introduce this topic I tell students that
we are about to enter "GOR-LAND." (no political
implications intended)
12. Simple Multiplication Verification Method
How do you verify your multiplication? Here is a simple method.
Always reduce computations to a single digit.
43
x 92
3956
Add the digits of multiplicand i.e. 4 + 3 = 7
Add the digits of multiplier i.e. 9 + 2 = 11
then reduce to a single digit 1 +1 = 2
Multiply 2 x 7 = 14 then reduce to a single digit 1 + 4 = 5
Add 3+9+5+6 = 23 then reduce to a single digit 2 + 3 = 5
Both numbers (5) are equal, therefore multiplication is correct
13. A=(pi)r^2 ⇒ Apple pies r square
A=(pi)r*r ⇒ Apple pies r round
C= (pi)d ⇒ Cherry pie delight
I = p r t ⇒ I "am" p-r-t
pronounced I am pretty
rt = d ⇒ rt are d
pronounced retard
Quadratic formula : "X equals to
negative b Plus or minus the square root,
Of b squared minus four a c All over 2 a"
Formulas For Easy Remberence
14. supplementary and complimentary angles
I teach middle school students. My students know that
Supplementary and complimentary angles are angles that equal 90
degrees and 180degrees, but they get confused as to which is which.
They also know that 90 degree angles are right angles.
So I tell them that a compliment is the right thing to do, and right angles
equal 90degrees, therefore complimentary angles are two angles that
equal 90 degrees.
A. Then they know that 180 degrees is the other one, supplementary.
"Complementary" - early in the alphabet, so = 90degrees.
B."Supplementary“ - later in the alphabet, so = 180degrees.
15. Triangle Names
o Equilateral triangles have 3 sides and 3
angles equal.
o Isosceles triangles have 2 sides and
o 2 angles equal.
o Scalene triangles have 0 sides and 0 angles
equal.
o So, to remember them in that order, EIS, "Eat
ice slowly"
16. just see whether the product of coefficients
of x in both the equations is equal to that
of coefficients of y.
If the given lines are
ax+by+c=0 and bx+ay+d=0
they cut the axes in concyclic points.
17.
18. Convert 10 decameters to centimeters.
Set up the columns as shown below
so that the ones column comes under deca.
Move the decimal point to the
right of the column with centi.
Add zeros until you are under centimeters.
That is your answer.
Kilo hecta deca unit deci centi milli
1 0
Kilo hecta deca unit deci centi milli
1 0 0 0 0
i.e. 10 dam = 10 000 cm
Example
19. Km Hm Dam M Dm Cm Mm
To convert...3.75 Hm = ______ Cm
It is 4 jumps to the right from Hm to Cm,
Simply move the decimal 4 jumps to the right.
3.75 Hm = 37,500. Cm
0.59 Dm = _______ HmI
It's 3 jumps to the left from Dm to Hm,
Simply move the decimal 3 jumps to
the left .
0.59 Dm = 0.00059 Hm
“King Henry Died Monday Drinking Chocolate Milk"
20. 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 step1
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 STEP3
STEP 5 . In the beginning of result will be Sq(X) + any
carryover from Step 4.
TO FIND SQUARE OF A 3 DIGIT
NUMBER
21. TO FIND SQUARE OF A 3 DIGIT
NUMBER :
EXAMPLE :
SQ (431)
STEP 1). Last digit = last digit of SQ(1) =1
STEP 2). Second Last Digit = 2*3*1 + any carryover from STEP1
= 6
STEP 3). Third Last Digit 2*4*1+ Sq(3) + any carryover from
STEP2
= 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.
22. Special Numbers
e to 15 decimal places e=2.718281828459045...
Andrew Jackson was the 7th president, elected in
1828 to two terms. Then tack on the 45-90-45 right triangle.
• pi - first eight digits of pi by K.Mahadevan,PGT
to get the first eight digits of pi,
count the number of letters in each word of this phrase:
• May(3) I(1) have(4) a(1) large(5)container(9) of(2)
coffee(6)?
23. PROFIT AND LOSS :
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.
24. 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.
EXAMPLE 2 :
25. TRY THESE
Now find out the difference in % of a product which was :
1)First increased by 20 % and then decreased by 10 %.
2)First Increased by 25 % and then decrease by 20 %
3)First Increased by 20 % and then decrease by 25 %.
4)First Increased by 10 % and then decrease by 10 %.
5)First Increased by 20 % and then decrease by 15 %.
26. 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)
27. TYPE 1 : Price of a commodity is increased by r%. By how
much % should the consumption be reduced so that the
expense remain the same.
TYPE 2 : Price of a commodity is decreased by r %. By how
much % can the consumption be increased so that the expense
remain the same.
Solution :
TYPE1 : (100* r ) / (100+r)
TYPE 2 : (100* r ) / (100-r)
PERCENTAGE
28. Example
Solution :
TYPE1 : (100* 60 ) / (100+60) = 37.5 %
TYPE 2 : (100* 60 ) / (100-60) = 150 %
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.
29. 1) Apollonius theorem could be applied to the 4 triangles
formed in a parallelogram.
2) Area of a trapezium = 1/2 * (sum of parallel sides) * height
= median * height
where median is the line joining the midpoints of the
oblique sides.
3)Let W be any point inside a rectangle ABCD . Then
4)Let ‘ a’ be the side of an equilateral triangle.
Then if three circles be drawn inside this triangle touching
each other then each’s radius =
2WA2WC2WB2WD +=+
132 +
a
Geometry
30. Successive discounts
Suppose in 1999 population increases by x% and then in
2000 by y%
so the population in 2000 now is
more that was in 1999.
Suppose in 1999 population decreases by x% and then in
2000 by y%
so the population in 2000 now is
less that was in 1999.
In 1999 population increases by 10% and then in 2000 by
5% so the population in 2000 now is 10+5+(50/100)=+15.5%
more that was in 1999.
If there is a decrease then it will be preceded by a negative
sign and likewise.
100
)(
xy
yx ++
100
)(
xy
yx −−
31. Fibonacci Addition Trick
Step 1: Choose two numbers
Step 2: Form a Fibonacci sequence for ten numbers
Example, I choose number 5 for my first number
and 6 for my second number.
Then I add the numbers to get a Fibonacci sequence.
5+6 gives my 3rd
number which is 11;
6+11 gives me my 4th
number which is 17.
The entire sequence is as follows:
1st – 5,2nd – 6, 3rd – 11, 4th – 17,5th – 28,6th - 45
7th – 73,8th – 118, 9th – 191,10th – 309.
What is the sum of all these 10 numbers? (6 seconds)
The answer will also be 803.
Trick: Multiply 7th
number by 11 and the answer is 803.
(It is true is any set of ten Fibonacci numbers)
33. Labeling Right Triangles
Let’s put it all together.
Given that angle B is the reference angle, here is how
you must label the triangle:
A
B (ref. angle)
C
hypotenuse
opposite
adjacent
34. Labeling Right Triangles
Let’s put it all together.
Given that angle C is the reference
angle, here is how you must label the
triangle:
A
B
C (ref. angle)
hypotenuse
opposite
adjacent
35. There are three ratios that you need to
learn:
sin
o
yp
pp
h
θ = cos
a
yp
dj
h
θ = tan
o
dj
pp
a
θ =
θ
Where are the hypotenuse, adjacent and opposite lengths.
This is opposite the right-angle
This is next to the angle
This is opposite the
angle.
TRIGONOMETRY
36. A PHARSE TO REMEMBER THE ABOVE
DEFINITION OF TRIGONOMETRIC RATIOS
SOME OLD HORSES ⇒
CAN ALWAYS CAN HEAR ⇒
TREIR OWNER’S APPORACH
sin
o
yp
pp
h
θ =
cos
a
yp
dj
h
θ =
tan
o
dj
pp
a
θ =⇒
37. Math Magic – Trick 1
Pick a number… any number! (keep it a secret
though) say x
Add 1 to that number i.e. x+1
Multiply by 3 i.e.3x + 3
Subtract your ‘secret’ number i.e. 2x +3
Add 5 i.e. 2x + 8
Divide by 2 i.e. x + 4
Subtract your secret number. i.e. 4
The answer is always 4
38. Everyone needs a little humor in their life
Especially mathematicians
Trick1.Viral Math
Challenge : Can you find the answer to this problem?
Trick:2.
Write down a three digit number. The first and third digits must
differ by more than one. Example: 264
Now reverse the digits to form a second number. we get 462.
Subtract the smaller number from the larger one. 462 - 264 = 198
Now reverse the digits in the answer you got in step 3 and add it to
that number. : 891 + 198 = 1089
39. Fast Multiplication
Select a four-digit number Example : 2345
Any 4 digit number is multiplied by 10001.what will be the answer ?
Do you know the answer is amazing?
The answer will be the given four digit number is written twice in
the same order .
2345 × 10001 = 23452345
Since 10001 = 73 x 137, 2345 × 73 × 137 = 23452345
abcd × 73 x 137 = abcdabcd.
Entering a four-digit number twice (23452345) will be divisible by
73, 137, and the original four-digit number.
1234 × 73 × 137 = 12341234
Can you tell me 26852685 is divisible by 73?
Can you tell me 81948194 is divisible by 137?
Can you tell me 38293829 is divisible by 3829?
40. Select a three-digit number Example : 234
Any 4 digit number is multiplied by 1001.what will be the answer ?
Do you know the answer is amazing?
The answer will be the given three digit number is written twice in the
same order .
234× 1001 = 234234
Since 1001 = 7 x 11 ×13, 234 × 7 x 11 ×13 = 234234
abc × 73 x 137 = abcabc.
Entering a three-digit number twice (234234) will be divisible by 73, 137,
and the original three-digit number.
123 × 7 x 11 ×13 = 12341234
Can you tell me 685685 is divisible by 7?
Can you tell me 194194 is divisible by 13?
Can you tell me 829829 is divisible by 829?
Can you tell me 529529 is divisible by 11?
Fast Multiplication/Division
41. Fast Multiplication/Division
Select a two-digit number Example : 23
Any 2 digit number is multiplied by 10101.what will be the answer ?
Do you know the answer is amazing?
The answer will be the given two digit number is written thrice in the same
order .
23 × 10101 = 23 2323
Since 10101 = 21 x 37 ×13, 23 × 3 × 7 x 37 ×13 = 232323
ab × 37 x 13 × 21 = ababab.
Entering a two-digit number thrice (232323) will be divisible by 37, 13, 7, 3
and the original two-digit number.
12 × 3 × 7 x 37 ×13 = 121212
Can you tell me 686868 is divisible by 7?
Can you tell me 191919 is divisible by 13?
Can you tell me 828282 is divisible by 82?
Can you tell me 525252 is divisible by 7?
Can you tell me 757575 is divisible by 21?
42. STEP 1: Ask participants to write down their mobile number .
STEP 2 :Ask them to add the digits.
STEP 3 :Ask them to subtract this number from the original one.
STEP 4 :Ask them to select any digit from this new number and strike it
out, without showing you.
STEP 5 :Ask them to add the remaining digits and write down the answer
they get. Example: 8+3+9+7+0+7+0 = 34
STEP 6 :Ask them to tell you the number they get (34)
and you will tell them which number they struck out.
MISSING DIGIT TRICK
43. TRICK Hate 8 *
Ask your friend to choose a number between 1 and 9 except 8
Multiply the number by 9.
7 x 9 = 63
•Multiply the answer by 12345679 (no 8)
63 x 12345679 = magic!
Select another number say 6
6 x 9 = 54
54 x 12345679 = 666666666
1 x 9 x 12345679 = 111111111
2 x 9 x 12345679 = 222222222
• Can you find the value of the following?
3 x 9 x 12345679
4 x 9 x 12345679
5 x 9 x 12345679
9 x 9 x 12345679
44. Dice Magic*
Find three dice and a friend. Turn your back.
Ask your friend to roll the three dice so that you can't see the
resulting numbers.
Multiply the number on the first die by 2.Add 5.Multiply by 5
* Add the number on the second die .Multiply by 10
*Add the number on the third die. Subtract 250 .
Now you are able to tell numbers on the top of three dice.
Now for your magic prognostication.
Example
(5 x 2 + 5) x 5 + 3 x 10 + 4 -250 = 534
Now you are able to tell numbers on the bottom of three dice.
i.e. 243
Magic happens...all the time!
45. Calculating Dice*
1. Find three dice, a calculator and a friend.
Roll the three dice and write down this number.
Repeat this number.
2. Now roll one dice and multiply your number by this roll.
3. Divide this number by 11
Divide this number by 13
Divide this number by your single dice roll and divide by 7
4. Magic happens!
5. The magic is the three digit number appeared on three dice
6. Example: 123, 123123
7. 123123 x 5 = 615615
8. 615615/11= 55965, 55965/13 = 4305. 4305/5=861 , 861/7=123
46. Roll Them*
Ask your friend to roll the dice without revealing to you the numbers
Example : Your friend rolls a 4 and a 6.
Ask your friend to multiply the number on the first die by 2
4 x 2 = 8. Add 5 ⇒8 + 5 = 13⇒Multiply by 5⇒13 x 5 = 65
Add the number on the second die. i.e. 65 + 6 = 71
You can now predict the numbers on the two dice.
Here is what you must do:
Subtract 25
71 - 25 = 46
4 = first die
6 = second die.
47. Mobile number Trick
1)Insert in the first five digits of your phone number (not the area code)
Example : 94456
2)Multiply these three numbers by 80 ⇒ 94456 x 80 = 7556480
Add 1 ⇒ 7556480 +1 = 75564801
3)Multiply by 2500 ⇒ 75564801 x 2500 = 18891202500
4)Add to this the last 4 digits of your phone number
18891202500 + 06533 = 18891209033
5)Add again the last 4 digits of your phone number.
18891209033 + 06533= 18891215566
6)Subtract 2500
18891215566 - 2500 = 18891213066
7&. Divide number by 2
18891213066 / 2 = Magic phone number = 9445606533
48. Math mentalism to amaze everyone
1- Ask a participant to choose a four digit number
2- Write down your prediction.
3- Ask participant to choose another four digit number
4- Write down your own number under the participants number.
5- Ask participant to choose another four digit number.
6- Write down your own number under the participants number.
7- Total the five numbers.
8- Show your prediction and exchange high fives!
You are now a Mathalism specialist.
Example:
i.2345
ii.22343
iii.7123
iv.2876
v.5690
vi.4309
vii.2345 + 7123 + 2876 + 5690 + 4309 = Answer in step ii
49. i.
1)Use three dice.
2)Have a friend roll the dice and then stack them one on top of the other.
3)Tell your friend that you can not see five faces of the dice.
4) The 5 numbers you cannot see are :
i)the bottom of the top dice; the top of the second dice
ii) the bottom of the second dice; the top of the last dice
iii) the bottom of the last dice
You will now predict the total of the five hidden numbers.
5)In your head subtract the very top face of the three dice from 21.
6)The answer will be the total of the five hidden numbers.
Reason is
7) The sum of the numbers of opposite faces on each die is 7
Dice Math Trick
50.
51. The Locker Problem
As 500 students enter a school, they pass lockers
that are numbered from 1 to 500. The first student
opens every locker; the second student closes
every second locker; the third student changes the
position of every third locker (by opening the
closed lockers and closing the open lockers); and
the fourth student changes the position of every
fourth locker. This pattern continues for all 500
students. Which lockers are open after all
students enter the school?
52.
53. The Hat Problem
There are 100 people lined up on the steps of a stadium, each on
a different step, all looking down toward the field so that they
can see everyone in front of them, but no one behind them.
Each person will be given either a red or black hat. We do not
know the total number of red or black hats. Each person will
not be able to see the color of his own hat (or the ones behind
him), but will be able to see the colors of all the hats in front of
him.
Starting in the back, the last person will be asked what color hat
he is wearing. If he guesses correctly, he will live; if he guesses
incorrectly, he will be shot immediately. The second to last will
be asked, and so on, until we reach the person on the bottom
step. Each person will be able to hear what all the people
behind him say, and will also be able to hear which people
behind him were shot.
54. The Hat Problem
Before we begin this process, the 100 people may meet to
discuss a strategy. They can plan whatever they want, but
once the line-up begins, they may no longer confer. At
each person's turn, he may only say "black" or "red," and
no other words -- if he says anything else, all 100 people
will be executed. He may also not use tone of voice,
volume, etc., to convey any meaning -- this will be
detected and they will all be shot.
What strategy will guarantee saving the maximum
number of people? What is this number?
55.
56. A Familial Math Equation
A mother is 21 years older than her son
In 6 years, she will be 5 times older than him
Where is the father?
57. A Familial Math Equation
Let x = the age of the mother (in years)
Let y = the age of the son (in years)
x = 21 + y
x + 6 = 5(y + 6)
Therefore,
(21 + y) + 6 = 5(y+6)
58. A Familial Math Equation
y + 27 = 5y + 30
5y – y = 27 – 30
4y = -3
y = -3/4
Remember that y is expressed in years…
60. Math Quiz Answers
1) There are 8 apples on the table, you take away 3. How
many do you have?
3 – The other 5 are still on the table
61. Math Quiz Answers
2) There are 10 birds in a field. If 2 are shot, how many
are left?
2 – The others have flown away
62. Math Quiz Answers
3) Take away the first letter, take away the last letter,
then take away all the other letters. What do you have
left?
The mailman
http://www.curiousmath.com
63. Math Quiz Answers
4) If you have 4 melons in one hand, and 7 apples in the
other - What do you have?
Big hands
http://www.curiousmath.com
64. Math Quiz Answers
5) A box holds nine ears of corn. A squirrel carries out
three ears a day, but it takes him nine days to carry
out all the corn. Why?
He carries out one ear of corn
in addition to his own two ears
65. Math Quiz Answers
6) Why do white sheep eat more than black sheep?
There are more white sheep than black sheep.
66. Math Quiz Answers
7) It takes 7 men 2 hours to build a wall. How long does it
take 3 men to build the same wall?
Why bother? The 7 men have already built it.
67. Math Quiz Answers
8) I have 2 coins in my hand that add up to 60 cents.
One of the coins is not a half dollar. What are the
coins?
A half dollar and a dime
one (the dime) is not a half dollar
68. Math Quiz Answers
9) A man wanted to plant 4 trees, but all 4 had to be
equal distances from each other. How did he do it?
69. Math Quiz Answers
10) A fisherman was asked the length of the fish he had
caught. He said "it is 30 cms plus half its length.“
How long was the fish?
60 cms
70. Math Quiz Answers
11) What comes next in the following sequence ?
1, 4, 5, 6, 7, 9, 11,...
100 – the next number spelled without a t
71. Math Quiz Answers
12) In a scientific context, what could the following
phrase mean?
“How I want a drink, alcoholic of course, after the
heavy chapters involving quantum mechanics…”
π = 3.14159265358979…
