- which is greater 12/19 or 2/3?
Approach is (cross multiply the numerator of one fraction with denominator with other.
i.e. (left hand fraction) 12*3 and (right hand fraction) 19*2
since 36 <38 that means 2/3 > 12/19 - which one is greater 100/101 or 300/670?
Left hand fraction will be 100*670 = 67000
Right hand fraction will be 101*300 = 30300
since 30300 < 67000 that means 100/101 > 300/670
Capsule ::The great pleasure in life is doing what people say you cannot do.
______________________________________________
Short-cut for any two/three digit number whose unit digit is 5.
E.g. 25*25 in this case put 5 square as it is and add 1 to previous digit and multiply it with original previous digit.
Step 1 : 5*5 = 25
Step 2: 2*(2+1)=2*3=6 , so final answer is 25*25=625
Look below examples.
35*35=[(3*(3+1))]25=1225 (always put 25 as a last two digit or say 5 square)
65*65=(6*7)25=4225; 105*105=11025; 125*125=(13*12)25=15625
25*35 in this case unit digit are same and tens digits are continus like 2,3. In this case product always end with 75 and for other digits compare 2 and 3, select greater digit-square it and deduct 1 from that square.
25*35 = [(3*3)-1]75 = 875. (here 3 is greater than 2,so take square and deduct 1)
35*45=[(4*4)-1]75= 1575 (here 4 is greater than 3,so take square and deduct 1)
105*115=[(11*11)-1]75=12075.
26*26 in this case unit digit and tens digits are same so product always end with 6, for other digits follow below steps
Step 1 : unit digit are same so multiply 6*6=36, so take 6 as a last digit and 3 as a carry.
Step 2: addition of ten digits + available carry => (2+2)+carry(3)
=> 4+3
=> 7, so second digit is 7.
Step 3: Now add 1 to previous digit and multiply it with original previous digit.
2*(2+1)=2*3=6 , so final answer is 26*26=676
E.g. 36*36 = 1296 ; 46*46=2116 , same method work for three digit as well.
Take an example of 126*126
Step 1 : unit digit are same so multiply 6*6=36, so take 6 as a last digit and 3 as a carry.
Step 2: addition of ten digits + available carry => (12+12)+carry(3)
=> 24+3
=> 27, so second digit is 7 and take 2 as carry
Step 3: Now add 1 to previous digit and multiply it with original previous digit.
{[12*(12+1)]+carry(2)} => {[12*13]+2} => [156+2]
=>158
so final answer is 126*126 = 15876
Capsule :: A champion is someone who gets up when he can't.
Capsule::"You were created to be a CHAMPION,don't settle for anything less."
Capsule::"A brave man will fall but he cannot yield".
________________________________________________
* Some Percentage Funda *
How to find out 2 1/2 percent of a number.
Step 1 Chosse any number.
Step 2 Divide by 4.
Step 3 Move the decimal point one place to the left.
E.g. If we select two digit number 75.
Step 2 divide 75 with 4 we get 18.75
Step 3 Move decimal point one place to the left we get 1.875
So,final answer is 2 1/2% of 75 returns 1.875.
How to find out 5 percent of a number.
Step 1 Choose any number
Step 2 Move the decimal point one place to the left.
Step 3 Divide by 2.
E.g. If we select 780$ amount.
Step 2: after moving one decimal point left we get, 78$.
Step 3: now divide 78 with 2, we will get 39.
So, finally 5% of 780$ is 39
How to find out 15 percent of a number.
Step 1 Select any number.
Step 2 Multiply the number by 3.
Step 3 Divide result by 2.
Step 4 Move the decimal point one place to the left.
E.g. If we select 67 number.
Step 2. multiply number with 3, 67*3 = 201.
Step 3. divide the result by 2, we get 201/2 = 100.5
Step 4. Move the decimal point one place left we get 10.05
so, finally 15% of 67 is 10.05
How to find out 45 percent of a number.
Step 1 Select any number.
Step 2 Multiply by 9.
Step 3. divide the result by 2.
Step 4. Move the decimal point one point to the left.
E.g. If we select 77 number.
Step 2 multiply 77 with 9, 77*9=630+63=693
Step 3 divide the result by 3, we get 693/2, 346.5
Step 4 move the decimal point one left, we will get 34.65
so, finally 45% of 77 is 34.65
______________________________________________
Multiply two numbers close to 10,100,1000 etc.
E.g Multiply 92 by 98.
Step -1 Here our base is 100,so offsets are 8 and 2. Offsets are differences from base (in this case 100).
our base 100,which has 2 zeros,the product of offsets must have also 2 digits. Hence we write 16 and write as last two digits of our answer.
92 -8
98 -2
_____
16
Step-2 Now for the previous digits, just add any two numbers crosswise i.e. either (92-2) or (98-8) we get 90. This is written before 16 as follows.
92 -8
98 -2
_______
90/16
That gives 9016.That is our answer, so finally 92*98 = 9016
Now take an example of three digit numbers.
E.g. Multiply 888 by 920.
Step -1 Here our base is 1000,so offsets are 112 and 80. Offsets are differences from base (in this case 1000).
our base 1000,which has 2 zeros,the product of offsets must have also 2 digits. Hence we write 16 and write as last two digits of our answer.
888 -112
920 -80
_____
8960
Step-2 Now for the previous digits, just add any two numbers crosswise i.e. either (888 -80) or (920-112) we get 808. This is written before 960 as follows.
888 -112
920 -80
_______
808/8960 now last there digits are 960 and previous digits are (808+8)=816
That gives 81696 .That is our answer, so finally 92*98 = 816960
Capsule :: When the going gets tough, the tough get going
______________________________________________
Find out square of two digit numbers.
E.g 25*25
Step 1) 5*5 = 25, put 5 as a last digit and 2 as a carry
Step 2) (2*5)*2+2(carry) = 22. put 2 as a digit and 2 as a carry
Step 3) (2*2)+2 (carry)=6
So in this case final answer is 25*25 = 625
Take another example so you will get more idea.
E.g. 43*43
Step 1) 3*3 =09,put 9 as a last digit and o as a carry.
Step 2) (4*3)*2+0(carry)=24,put 4 as a digit and 2 as a carry.
Step 3) (4*4)+2(carry)=18
So in this case final answer is 43*43=1849
How to find the units digit.
- to find the unit digit of x^y we only consider the units digits of the number x.
- to calculate units digit of 237^234 we only consider digit of 7^234. Hence we find the units digit of 7^234.
- to find the units digit of a*b, we only consider the units digits of the numbers a and b.
- to calculate units digit of 233*254, we only consider the units digit of 233 and 254 i.e. 3 and 4,respectively.
- Hence, we find the units digit of 3*4,respectively.To calculate units digit of x^y where x is a single digit number.
To calculate units digit of numbers in the form x^y such 7^253,8^93,3^74 etc.
Case 1:: When y is NOT a multiple of 4.
Case 2:: When y is a multiple of 4.
Find the units digit of 7^33. Ans. The remainder when 33 is divided by 4 is 1. Hence the units digit 7^33 is the unit digit of 7^1=7.
Find the units digit of 43^47. Ans. The units digit of 43^47 can be found by finding the units digit of 3^47. 47 gives a remainder of 3 when divided by 4. Hence units digit= units digit of 3^3=7.
Find the units digit of 43^43 - 22^22. Ans. Unit digits of 43^43 is 7 and units digit of 22^22 is 4. Hence the units digit of expression will be 7-4=3.
Find the units digit of 3^3^3. Ans. Again, we find the remainder when the power is divided by 4. Therefore, we find the remainder when 3^3 is divided by 4. Now, 3^3=27, remainder by 4 = 3.Therefore, units digit of 3^3^3 = units digit of 3^3 = 7.
Find the unit digit of 7^11^13^17. Ans. Again, we find the remainder when the power is divided by 4. Therefore, we find the remainder when 11^13^17 is divided by 4. Now 11 = 12 -1 => Remainder [11Odd] = Remainder[(-1)Odd] = -1 = 3.Therefore, units digit of 7^11^13^17 = units digit of 7^3 = 3.
Ans. Again, 16^17 means 6^17 so finally we get 6 remainder, now we get 15^6 so when we divide 5^6 we get 25,means 5 is a unit digit,so when we divide 7^5 by 4 we get 1 remainder,finally we get 7^1=7. Therefore, unit digit of 7^11^13^17 is 7. Find the units digit of 28^28 - 24^24. Ans. We have to find the units digit of 8^28 - 4^24. Since 28 and 24 are both multiple of 4, the units digits of both 8^28 and 4^24 will be 6. Hence the units digit of the difference will be 0. How to find last two digits of a number. We are dividing this method into four parts and we will discuss each part one by one. a. Last two digits of numbers which end in one. b. Last two digits of numbers which end in 3,7 and 9 c. Last two digits of numbers which end in 2 d. Last two digits of numbers which end in 4,6 and 8. Last two digits of a number ending with 1.Let's start with an example. What is the last two digits of 31^786? Note :: multiply the tens digit of the number with the last digit of the exponent to get the tens digit. The units digit is equal to one. So, last two digits of 31^786 is 81. What is the last two digits of 41^2789? Ans. (4*9=36), Therefore 6 will be the tens digit and one will be the units digit. What is the last two digits of 71^56747 Ans. (7*7=49), Therefore 9 will be the tens digit and one will be the units digit. Find the last two digits of 51^456 * 61^567. The last two digits of 51^456 will be 01 and the last two digits of 61^567 will be 21. Therefore, the last two digits of 51^456*61^567 will be the last two digits of 01*21=21 Last two digits of numbers ending in 3,7 or 9. Find the last two digits of 19^266. 19^266 = (19^2)133. Now, 19^2 ends in 61(19^2=361) therefore, we need to find the last two digits of (61)^133. Once the number is ending in 1 we can straight away get the last two digits the help of the previous method. The last two digits are 81(6*3 = 18, so the tens digit will be 8 and last digit will be 1) Find the last two digits of 33^288. 33^288 = (33^4)72, Now 33^4 ends in 21(33^4 = 33^2 * 33^2 = 1089 * 1089 = xxxxx21) therefore, we need to find the last two digits of 21^72. By the previous method, the last two digits of 21^72 = 41(tens digit = 2*2 = 4, unit digit =1) So here's the rule for finding the last two digits of numbers ending in 3,7 and 9. Convert the number till the number gives 1 as the last digit and then find the last two digits according to the previous method. Find the last two digit of 87^474. Ans. 87^474 = 87^472 * 87^2 = (69*69)^118 * 69 (The last two digits of 87^2 are 69) = 61^118*69 = 81 * 69 = 89 Last two digits of numbers ending in 2,4,6 or 8. There is only one even two-digit number which always ends in itself(last two digits) - 76 i.e. 76 raised to any power gives the last two digits as 76. Therefore, our purpose is to get 76 as last two digits for even numbers. We know that 24^2 ends in 76 and 2^10 ends in 24. Also, 24 raised to an even power always ends with 76 and 24 raised to an odd power always ends with 24. Therefore, 24^34 will end in 76 and 24^53 will end in 24. Find the last two digits of 2^543. Ans. 2^543 = (2^10)^54 * 2^3 = (24)^54 (24 raised to an even power) * 2 ^3 = 76 * 8 = 08. Find the last two digits of 64^236. Ans. (2^6)^236 =2^1416=(2^10)^141*2^6 = 24^141 (24 raised to odd power) * 64 = 24*64 =36. Now those numbers which are not in the form of 2n can be broken down into the form 2n*odd number. We can find the last two digits of both the parts separetely. Find the last two digits of 62^586. Ans. (2*31)^586 = 2^586 * 3^586 = (2^10)^58*2^6*31^586 = 76 * 64* 81 = 84 Find the last two digit of 54^380 Ans. (2*3^3)^380 = 2^380 * 3^1140 = (2^10)^38*(3^4)^285=76*81^285=76*01=76 Find the last two digit of 56^283 Ans. 56^283 = (2^3*7)^283=2^849*7^283=(2^10)^84*2^9*(7^4)*70*7^3 = 76*12*(01)^70*43=16 Find the last two digit of 78^379 Ans. 78^379 = (2*39)^379=2^379*(39)^379=(2^10)^37*2^9*(39^2)^189*39 = 24*12*81*39 = 92 Capsule :: "You must have long term goals to keep you from being frustrated by short term failures." -- Chalres C. Noble ________________________________________________________ * Divisibility Rules * A number is divisible by 2, if the last two digit is even (0,2,4,6,8) E.g. 66,68,12,14 are divisible by 2, while 129,39,49 are not. A number is divisible by 3,if the sum of the digits is divisible by 3. E.g. 216,213 are divisible by 3, while 215,217,107 are not divisible by 3. A number is divisible by 4, if last two digits are divisible by 4. E.g. 1312 is divisible by 4, while 7019 is not. A number is divisible by 5, if last digit is 0 or 5. E.g. 120,125,110 divisible by 5, while 809 is not. A number is divisible by 6, if the number is divisible by 2 and 3. E.g. 216,312 divisible by 6, while 308 is not divisible by 6. A number is divisible by 7, if you double the last digit and substract it from the rest of the number and the answer is 0 or divisible by 7. E.g. 2835 Here in first step we will take last digit, double it and subtract it from rest of the number. Step -1 2835 = 283 - (5*2) = 283 - 10 = 273 Step-2 273 = 27 - (3*2) = 27 - 6 = 21 , and 21 is divisible by 7. 805 is divisible by 7, while 905 is not divisible by 7. A number is divisible by 8, if the last three digits are divisible by 8. E.g. 109816, here last three digit 816 is divisible by 8 = (816/8 = 102) while 216302 is not divisible by 8. A number is divisible by 9, if the sum of total digit is divisible by 9. E.g. 1629 (1+6+2+9=18, and again, 1+8=9), so 1629 is divisible by 9. while 2013 is not divisible by 9. A number is divisible by 10, if the number ends in 0. E.g. 110,100,120 etc. , while 111,121,107 are not divisible by 10. A number is divisible by 11, when we substract sum of odd digit from sum of even digit and resulting number is either 0 or divisible by 11. E.g. 1364 = (3+4) - (1+6) = 7 -7 =0 ; 121 = (1+1) - (2+0) = 2 -2 = 0 ; 561 = (5+1)-(6+0)=0 So, 1364,121,561 all are divisible by 11. A number is divisible by 12, if it's divisible by both 3 and 4. E.g. 648, here last two digit of 648 is 48 and it's divisible by 4. Also (6+4+8 = 18, which is divisible by 3. Finally 648 is divisible by 12. Capsule :: Success is not final, failure is not fatal:it is the courage to continue that counts. _______________________________________________________
We find the remainder when y is divided by 4. Let y=4q+r where r is the remainder when y is divided by 4, and 0 < r < 4.
We observe the following conditions:
Even numbers 2,4,6,8 when raised to powers which are multiple of 4 give the units digit as 6.
Odd numbers 3,7 and 9 when raised to powers which are multiple of 4 give the units digit as 1.
Find the unit digit of 7^15^16^17.
0 comments
Post a Comment