LazyMaths.com http://www.lazymaths.com The Home of Speed Math and Smart Math Thu, 02 Feb 2012 21:44:43 +0000 en hourly 1 http://wordpress.org/?v=3.3.1 [Speed Math] CUBING NUMBERS NEAR 10-5 http://www.lazymaths.com/speed-math/cubing-numbers-near-10-5/ http://www.lazymaths.com/speed-math/cubing-numbers-near-10-5/#comments Thu, 02 Feb 2012 21:44:43 +0000 LazyMaths.com http://www.lazymaths.com/?p=4133

Here’s an example of a SPEED MATH shortcut for CUBING NUMBERS NEAR 10 : (Cub) Nr10 from the CUBING AND CUBE ROOTS category.

Cubing Numbers near 10

When can I use this method?

For cubing numbers near 10.

The number can be either < or > 10.

One can use this method to cube numbers away from 10 as long as it is comfortable to cube the difference portions.

pdf Download Practice sheet for CUBING NUMBERS NEAR 10

Notes –

  1. This method uses the concept (10 + a)3 = [(10 + 3a) 102] + [(3a2) 10] + [a3], where ‘(10 + a)’ is the number near 10 and ‘a’ is the difference.
  2. Whenever the number is more than 10, the difference is written as positive (+ve).
  3. Whenever the number is less than 10, the difference is written as negative (-ve).

Related Shortcuts –

Cubing Numbers near 100: (Cub) Nr100
Cubing Numbers near 1000: (Cub) Nr1000
Cubing 2-digit Numbers near a Base: (Cub) NrB2
Cubing 3-digit Numbers near a Base: (Cub) NrB3
Cubing 4-digit Numbers near a Base: (Cub) NrB4

We hope this helps you in getting to the answer faster. You can apply this shortcut in the exams and get to the answer before anyone else can.

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]]> http://www.lazymaths.com/speed-math/cubing-numbers-near-10-5/feed/ 0 [Speed Math] CUBING 2-DIGIT NUMBER-5 http://www.lazymaths.com/speed-math/cubing-2-digit-number-5/ http://www.lazymaths.com/speed-math/cubing-2-digit-number-5/#comments Thu, 26 Jan 2012 22:00:31 +0000 LazyMaths.com http://www.lazymaths.com/?p=4141

Here’s an example of a SPEED MATH shortcut for CUBING 2-DIGIT NUMBER : (Cub) 2D from the CUBING AND CUBE ROOTS category.

Cubing 2-digit Number

When can I use this method?

For cubing any 2 digit number.

pdf Download Practice sheet for CUBING 2-DIGIT NUMBER

Notes –

  1. Notice the way the individual values are added.
  2. This method uses the concept of (a + b)3 = a3 + 3a2b + 3ab2 + b3

Related Shortcuts –

None

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]]> http://www.lazymaths.com/speed-math/cubing-2-digit-number-5/feed/ 0 [Speed Math] MULTIPLYING NUMBERS WITH FIRST DIGITS SUM 1000-4 http://www.lazymaths.com/speed-math/multiplying-numbers-with-first-digits-sum-1000-4/ http://www.lazymaths.com/speed-math/multiplying-numbers-with-first-digits-sum-1000-4/#comments Tue, 24 Jan 2012 17:51:22 +0000 LazyMaths.com http://www.lazymaths.com/?p=3503

Here’s an example of a SPEED MATH shortcut for MULTIPLYING NUMBERS WITH FIRST DIGITS SUM 1000 : (Mul) F1000dx from the MULTIPLICATION category.

Multiplying Numbers with first digits sum 1000

When can I use this method?

For multiplying any 4-digit number with another 4-digit number such that the sum of the initial digits of the multiplier and multiplicand = 1000 and the remaining digits of multiplier are same as that of the multiplicand.

For multiplying any 5-digit number with another 5-digit number such that the sum of the initial digits of the multiplier and multiplicand = 1000 and the remaining digits of multiplier are same as that of the multiplicand.

You cannot use this method to multiply numbers with unequal number of digits, i.e. multiplying a 4-digit number with a 5-digit number.

pdf Download Practice sheet for MULTIPLYING NUMBERS WITH FIRST DIGITS SUM 1000

Notes –

  1. Notice the way the common digits are added.
  2. When multiplying 4-digit numbers with common last digit, remember to write the RHS of the answer in 2-digit form by prefixing a zero, if the value is single digit, e.g. write ‘9’ as ‘09’.
  3. When multiplying 5-digit numbers with last 2 digits common, remember to write the RHS of the answer in 4-digit form by prefixing zero(s), if the value is less than 4-digit, e.g. write ‘9’ as ‘0009’ or ‘21’ as ‘0021’ or ‘121’ as ‘0121’.

Related Shortcuts –

Multiplying Numbers with first digits sum 10: (Mul) F10dx
Multiplying Numbers with first digits sum 100: (Mul) F100dx


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]]> http://www.lazymaths.com/speed-math/multiplying-numbers-with-first-digits-sum-1000-4/feed/ 0 [Speed Math] CHECKING SUBTRACTION-5 http://www.lazymaths.com/lazymaths/checking-subtraction-5/ http://www.lazymaths.com/lazymaths/checking-subtraction-5/#comments Fri, 20 Jan 2012 01:54:48 +0000 LazyMaths.com http://www.lazymaths.com/?p=2489

Here’s an example of a SPEED MATH shortcut for CHECKING SUBTRACTION: (Check) Sub from the CHECKING ANSWERS category.

Make sure to check out SEED NUMBER CONCEPT to understand how Seed numbers are calculated.

Checking Subtraction

When can I use this method?

For checking any Subtraction.

This method can also be used to check problems with a combination of additions and subtractions also.

pdf Download Practice sheet for CHECKING SUBTRACTION

Notes –

  1. Make sure to check out SEED NUMBER CONCEPT to understand how Seed numbers are calculated.
  2. Any time you come across a seed number values greater than 1-digit, reduce it to a single digit value using the seed number concept.
  3. Any time you come across a negative (-ve) seed number; add 9 to it to get is positive (+ve) value. Do the same if the seed number is = 0.
  4. It is the fastest and easiest way to verify your answers.

Related Shortcuts –

Checking Division : (Check) Div

Checking Multiplication : (Check) Mul

Checking Addition : (Check) Add


We hope this helps you in getting to the answer faster. You can apply this shortcut in the exams and get to the answer before anyone else can.

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]]> http://www.lazymaths.com/lazymaths/checking-subtraction-5/feed/ 0 [Speed Math] SQUARING 2-DIGIT NUMBERS NEAR A BASE-4 http://www.lazymaths.com/speed-math/squaring-2-digit-numbers-near-a-base-4-2/ http://www.lazymaths.com/speed-math/squaring-2-digit-numbers-near-a-base-4-2/#comments Tue, 17 Jan 2012 21:40:56 +0000 LazyMaths.com http://www.lazymaths.com/?p=2670

Here’s an example of a SPEED MATH shortcut for SQUARING 2-DIGIT NUMBERS NEAR A BASE : (Sq) NrB2 from the SQUARING AND SQUARE ROOTS category.

Squaring 2-digit Numbers near a Base

When can I use this method?

For squaring any 2-digit numbers.

The number can be either < or > a Base number.

pdf Download Practice sheet for SQUARING 2-DIGIT NUMBERS NEAR A BASE

Notes –

  1. When selecting a Base number, prefer a number that is near the number to be squared as well as it should be easy to multiply with (prefer round numbers).
  2. This method uses the concept (B + a)2 = [(B + a) + a] + a2, where ‘(B + a)’ is the number to be squared, ‘B’ is the Base number and ‘a’ is the difference.
  3. Whenever the number is more than the Base number, the difference is written as positive (+ve).
  4. Whenever the number is less than Base Number, the difference is written as negative (-ve).
  5. This method is a corollary to the Multiplication method NrB2

Related Shortcuts –

Squaring Numbers near 100 : (Sq) Nr100

Squaring Numbers near 1000 : (Sq) Nr1000

Squaring Numbers near 50 : (Sq) Nr50

Squaring Numbers near 500 : (Sq) Nr500

Squaring 3-digit Numbers near a Base : (Sq) NrB3


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]]> http://www.lazymaths.com/speed-math/squaring-2-digit-numbers-near-a-base-4-2/feed/ 0 [Speed Math] CUBING NUMBERS NEAR 100-5 http://www.lazymaths.com/speed-math/cubing-numbers-near-100/ http://www.lazymaths.com/speed-math/cubing-numbers-near-100/#comments Thu, 12 Jan 2012 21:28:38 +0000 LazyMaths.com http://www.lazymaths.com/?p=4125

Here’s an example of a SPEED MATH shortcut for CUBING NUMBERS NEAR 100 : (Cub) Nr100 from the CUBING AND CUBE ROOTS category.

Cubing Numbers near 100

When can I use this method?

For cubing numbers near 100.

The number can be either < or > 100.

One can use this method to cube numbers away from 100 as long as it is comfortable to cube the difference portions.

pdf Download Practice sheet for CUBING NUMBERS NEAR 100

Notes –

  1. This method uses the concept (100 + a)3 = [(100 + 3a) 1002] + [(3a2) 100] + [a3], where ‘(100 + a)’ is the number near 100 and ‘a’ is the difference.
  2. Whenever the number is more than 100, the difference is written as positive (+ve).
  3. Whenever the number is less than 100, the difference is written as negative (-ve).

Related Shortcuts –

Cubing Numbers near 10: (Cub) Nr10
Cubing Numbers near 1000: (Cub) Nr1000
Cubing 2-digit Numbers near a Base: (Cub) NrB2
Cubing 3-digit Numbers near a Base: (Cub) NrB3
Cubing 4-digit Numbers near a Base: (Cub) NrB4

We hope this helps you in getting to the answer faster. You can apply this shortcut in the exams and get to the answer before anyone else can.

If you like this
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- Share it with your friends

Want more of these? Click SPEED MATH

Wanna learn from other resources? Head to our MORE MATH page

You can also join OUR CLASSES and learn hundreds of such techniques. Learn to apply them in your exams and even in everyday life! We offer classroom training for SAT, GRE, GMAT and many more exams.

Is this a helpful shortcut? Don’t forget to rate this shortcut

Tell us what you think. How can we improve? Feel free to comment or ask questions. We can be reached at math(at)lazymaths(dot)com

]]> http://www.lazymaths.com/speed-math/cubing-numbers-near-100/feed/ 0 [Speed Math] SQUARING NUMBERS NEAR 500-4 http://www.lazymaths.com/speed-math/squaring-numbers-near-500-4/ http://www.lazymaths.com/speed-math/squaring-numbers-near-500-4/#comments Tue, 10 Jan 2012 21:18:44 +0000 LazyMaths.com http://www.lazymaths.com/?p=2652

Here’s an example of a SPEED MATH shortcut for SQUARING NUMBERS NEAR 500 : (Sq) Nr500 from the SQUARING AND SQUARE ROOTS category.

Squaring Numbers near 500

When can I use this method?

For squaring numbers near 500.

The number can be either < or > 500.

One can use this method to square numbers away from 500 as long as it is comfortable to square the difference portions.

pdf Download Practice sheet for SQUARING NUMBERS NEAR 500

Notes –

  1. This method uses the concept (500 + a)2 = [(250 + a) 100] + a2, where ‘(500 + a)’ is the number near 500 and ‘a’ is difference.
  2. Whenever the number is more than 500, the difference is written as positive (+ve).
  3. Whenever the number is less than 500, the difference is written as negative (-ve).

Related Shortcuts –

Squaring Numbers near 100 : (Sq) Nr100

Squaring Numbers near 1000 : (Sq) Nr1000

Squaring Numbers near 50 : (Sq) Nr50

Squaring 2-digit Numbers near a Base : (Sq) NrB2

Squaring 3-digit Numbers near a Base : (Sq) NrB3


We hope this helps you in getting to the answer faster. You can apply this shortcut in the exams and get to the answer before anyone else can.

If you like this
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Want more of these? Click SPEED MATH

Wanna learn from other resources? Head to our MORE MATH page

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Is this a helpful shortcut? Don’t forget to rate this shortcut

Tell us what you think. How can we improve? Feel free to comment or ask questions. We can be reached at math(at)lazymaths(dot)com

]]> http://www.lazymaths.com/speed-math/squaring-numbers-near-500-4/feed/ 0 [Smart Math] ARITHMETIC PROBLEM 27 http://www.lazymaths.com/smart-math/arithmetic-problem-27/ http://www.lazymaths.com/smart-math/arithmetic-problem-27/#comments Fri, 06 Jan 2012 21:27:27 +0000 LazyMaths.com http://www.lazymaths.com/?p=2902

Here’s and example of a SMART MATH problem for ARITHMETIC.

Arithmetic

Problem

How many integer solutions exists for the equation \left| y-\sqrt{(y+2)^{2}-9} \right|=5?

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

The Usual Method

\left| y-\sqrt{(y+2)^{2}-9} \right|=5 can be written as:

y-\sqrt{(y+2)^{2}-9}=\pm 5

\therefore -\sqrt{(y+2)^{2}-9}=\pm 5-y

\therefore \sqrt{(y+2)^{2}-9}=(y\pm 5)

Squaring both sides, we get:

\therefore (y+2)^{2}-9=(y\pm 5)^{2}

\therefore (y+2)^{2}-9=(y\pm 5)^{2}

\therefore (y+2)^{2}-9=(y\pm 10y+25)

\therefore (y+2)^{2}=(y\pm 10y+34)

\therefore y^{2}+4y+4=y^{2}+34\pm 10y

\therefore \pm 10y+4y=30

\therefore 14y=30      or         6y=30

\therefore y=\frac{30}{14}=\frac{15}{7} or y=\frac{30}{6}=5

Although the equation gives two answers, but only 5 is an integer, there is only one integer solution for the equation.

(Ans: 2)

Estimated Time to arrive at the answer = 100 seconds.

Using Technique

For \left| y-\sqrt{(y+2)^{2}-9} \right|=5, \sqrt{(y+2)^{2}-9} should be a perfect square of = 0.

Equating \sqrt{(y+2)^{2}-9} to 0, we get:

\sqrt{(y+2)^{2}-9} = 0

(y+2)^{2}-9 = 0

\therefore (y+2)^{2}=9

\therefore (y+2)=\pm 3

\therefore y= 1or –5

Substituting y = 1 in the original equation, we get:

\left| 1-\sqrt{0} \right|=5, but this cannot be true, as 1\ne 5, hence, 1 is not the correct solution to the equation.

Substituting y = –5 in the original equation, we get:

\left| -5-\sqrt{0} \right|=5, this is true, as the equation is satisfied. Hence, –5 is the correct solution to the equation. Thus there is only a single solution to the equation.

Note that we are not checking by equating with perfect squares, as:

\sqrt{(y+2)^{2}-9} = 4 (a perfect square)

(y+2)^{2}-9 = 16

\therefore (y+2)^{2}=25

\therefore (y+2)=\pm 5

\therefore y= 3 or –7

Substituting y = 3 in the original equation, we get:

\left| 3-\sqrt{4} \right|=1, but this cannot be true, as 1\ne 5, hence, 3 is not the correct solution to the equation.

Substituting y = –7 in the original equation, we get:

\left| -7-\sqrt{4} \right|=9, but this cannot be true, as 9\ne 5, hence, –7 is not the correct solution to the equation.

(Ans: 2)

Estimated Time to arrive at the answer = 15 seconds.

We hope this helps you in getting to the answer faster. You can apply this technique in the exams and get to the answer before anyone else can.

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Is this technique really smart? Don’t forget to rate this technique

Tell us what you think. How can we improve? Feel free to comment or ask questions. We can be reached at math(at)lazymaths(dot)com

]]> http://www.lazymaths.com/smart-math/arithmetic-problem-27/feed/ 0 [Speed Math] CUBING NUMBERS NEAR 10-4 http://www.lazymaths.com/speed-math/cubing-numbers-near-10-4/ http://www.lazymaths.com/speed-math/cubing-numbers-near-10-4/#comments Thu, 05 Jan 2012 21:41:21 +0000 LazyMaths.com http://www.lazymaths.com/?p=4131

Here’s an example of a SPEED MATH shortcut for CUBING NUMBERS NEAR 10 : (Cub) Nr10 from the CUBING AND CUBE ROOTS category.

Cubing Numbers near 10

When can I use this method?

For cubing numbers near 10.

The number can be either < or > 10.

One can use this method to cube numbers away from 10 as long as it is comfortable to cube the difference portions.

pdf Download Practice sheet for CUBING NUMBERS NEAR 10

Notes –

  1. This method uses the concept (10 + a)3 = [(10 + 3a) 102] + [(3a2) 10] + [a3], where ‘(10 + a)’ is the number near 10 and ‘a’ is the difference.
  2. Whenever the number is more than 10, the difference is written as positive (+ve).
  3. Whenever the number is less than 10, the difference is written as negative (-ve).

Related Shortcuts –

Cubing Numbers near 100: (Cub) Nr100
Cubing Numbers near 1000: (Cub) Nr1000
Cubing 2-digit Numbers near a Base: (Cub) NrB2
Cubing 3-digit Numbers near a Base: (Cub) NrB3
Cubing 4-digit Numbers near a Base: (Cub) NrB4

We hope this helps you in getting to the answer faster. You can apply this shortcut in the exams and get to the answer before anyone else can.

If you like this
- Support us by donating
- Share it with your friends

Want more of these? Click SPEED MATH

Wanna learn from other resources? Head to our MORE MATH page

You can also join OUR CLASSES and learn hundreds of such techniques. Learn to apply them in your exams and even in everyday life! We offer classroom training for SAT, GRE, GMAT and many more exams.

Is this a helpful shortcut? Don’t forget to rate this shortcut

Tell us what you think. How can we improve? Feel free to comment or ask questions. We can be reached at math(at)lazymaths(dot)com

]]> http://www.lazymaths.com/speed-math/cubing-numbers-near-10-4/feed/ 0 [Smart Math] ARITHMETIC PROBLEM 26 http://www.lazymaths.com/smart-math/arithmetic-problem-26/ http://www.lazymaths.com/smart-math/arithmetic-problem-26/#comments Wed, 04 Jan 2012 21:26:00 +0000 LazyMaths.com http://www.lazymaths.com/?p=2898

Here’s and example of a SMART MATH problem for ARITHMETIC.

Arithmetic

Problem

Simplify \frac{\sqrt{11+4\sqrt{7}}+\sqrt{8+\sqrt{64}}}{\sqrt{11-4\sqrt{6}}}

  1. \frac{2\sqrt{14}+12\sqrt{2}+\sqrt{21}-6\sqrt{3}}{5}
  2. \frac{2\sqrt{14}+12\sqrt{2}-\sqrt{21}+6\sqrt{3}}{5}
  3. \frac{2\sqrt{14}-12\sqrt{2}-\sqrt{21}+6\sqrt{3}}{5}
  4. \frac{2\sqrt{14}-12\sqrt{2}+\sqrt{21}-6\sqrt{3}}{5}
  5. \frac{2\sqrt{14}+12\sqrt{2}+\sqrt{21}+6\sqrt{3}}{5}

The Usual Method

Simplifying the numerator of the equation, we get:

\sqrt{11+4\sqrt{7}} = \sqrt{7+4+2\sqrt{7\times 4}}=\sqrt{7}+\sqrt{4}

and,

\sqrt{8+\sqrt{64}} = \sqrt{8+8}=\sqrt{16}=4

Hence numerator is \sqrt{7}+\sqrt{4}+4

Similarly, simplifying the denominator, we get:

\sqrt{11-4\sqrt{6}} = \sqrt{8+3-2\sqrt{6\times 4}}=\sqrt{8+3-2\sqrt{8\times 3}}=\sqrt{8}-\sqrt{3}

Thus the equation now becomes:

\frac{\sqrt{7}+\sqrt{4}+4}{\sqrt{8}-\sqrt{3}}

This can be written as:

\frac{\sqrt{7}+\sqrt{4}+4}{\sqrt{8}-\sqrt{3}} = \frac{\sqrt{7}+\sqrt{4}+4}{\sqrt{8}-\sqrt{3}}\times \frac{\sqrt{8}+\sqrt{3}}{\sqrt{8}+\sqrt{3}}

= \frac{\sqrt{56}+6\sqrt{8}+\sqrt{21}+6\sqrt{3}}{8-3}

= \frac{2\sqrt{14}+12\sqrt{2}+\sqrt{21}+6\sqrt{3}}{5}

(Ans: 5)

Estimated Time to arrive at the answer = 150 seconds.

Using Technique

Simply observe that since all the terms in the numerator are positive, simplifying the numerator will also give all positive terms.

The denominator has a negative sign. However, the process of rationalization would require its conjugate to be multiplied to result in a rational number in the denominator. This conjugate multiplier will have positive terms. Hence, when it multiplies with the all positive term numerator, we will get a product in the numerator with all positive terms. The only option that has all positive terms in the numerator is option ‘5’.

(Ans: 5)

Estimated Time to arrive at the answer = 15 seconds.

We hope this helps you in getting to the answer faster. You can apply this technique in the exams and get to the answer before anyone else can.

If you like this
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Want more of these? Click SMART MATH

Wanna learn from other resources? Head to our MORE MATH page

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Is this technique really smart? Don’t forget to rate this technique

Tell us what you think. How can we improve? Feel free to comment or ask questions. We can be reached at math(at)lazymaths(dot)com

]]> http://www.lazymaths.com/smart-math/arithmetic-problem-26/feed/ 0