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The Secrets of Speed Math: How to Perform Lightning-Fast Calculations

There are different tips and tricks to improve your mathematical calculation skills and get the right answer with your calculation. But to start you need more practice. Without practice, you can’t able to improve your math calculation accurate and fast.

Author: Amelia Margaret


online calculators


There are few points that help you to improve calculation as given below:

Understand the basics: 

Before you can excel at more complex mathematical calculations, it's important to have a solid understanding of the basics. This includes things like basic arithmetic, algebra, and geometry. Make sure you have a good grasp of these concepts before moving on to more advanced calculations.


Practice regularly: 

One of the best ways to improve your mathematical calculation skills is to practice regularly. Set aside time each day or each week to work on mathematical problems. This could be anything from simple arithmetic to more advanced calculus. The more you practice, the more comfortable you will become with different types of calculations.


Use different resources: 

In order to improve your mathematical calculation skills, it's important to use a variety of resources. This can include textbooks, online resources, and even apps. Each resource will offer a different approach to mathematical concepts, which can help you to understand the material better.


Take online courses: 

Online courses can be a great way to improve your mathematical calculation skills. They offer a structured approach to learning and provide you with the opportunity to work at your own pace. There are many free and paid online courses available, so you're sure to find one that fits your needs.


Study with a tutor or mentor: 

Working with a tutor or mentor is another great way to improve your mathematical calculation skills. They can provide you with personalized instruction and help you to understand the material better. They can also answer any questions you might have and provide you with feedback on your progress.


Use visualization techniques:

Visualization techniques can be a great way to improve your mathematical calculation skills. This can include things like drawing diagrams, using manipulative, or even using virtual reality. These techniques can help you to better understand mathematical concepts and make it easier to solve problems.


Challenge yourself: 

Challenging yourself is key to improve your mathematical calculation skills. As you become more comfortable with different types of calculations, try tackling more difficult problems. This will help you to continue growing and improving your skills.


Practice mental math: 

Practicing mental math can also be a great way to improve your mathematical calculation skills. It helps to improve your memory, concentration and speed of calculation. Try to work on simple arithmetic problems in your head and then check your answers.


Keep a notebook:

Keep a notebook or journal to track your progress. You can use it to write down formulas and equations, take notes during your studies, or even jot down questions you might have. This can be a helpful tool when you're reviewing material later on.


Learn from mistakes:

Lastly, learn from your mistakes. Don't get discouraged if you can't solve a problem or if you make a mistake. Instead, try to understand where you went wrong and use that knowledge to improve your skills.


Conclusion:

To improving your mathematical calculation skills takes time and effort, but it's definitely worth it. By following these tips and putting in the work, you'll be able to master a wide range of mathematical calculations and become more confident in your abilities.


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The restructured syllabus of SSC-2021 examination

Higher Math

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Roman Numerals

 Ancient Romans used a special method of showing numbers. Roman numerals are a number system developed in ancient Rome where letters represent numbers. The modern use of Roman numerals involves the letters I, V, X, L, C, D, and M.


01. The Roman Symbols

Romans Numerals are based on the following symbols:
  • Roman Numerals : Number (in English)
  • I : 1 - One
  • V : 5 - Five
  • X : 10 - Ten
  • L : 50 - Fifty
  • C : 100 - One Hundred
  • D : 500 - Five Hundred
  • M : 1000 - One Thousand

02. Basic Roman Numerals Chart

  • Roman Numerals : Number (in English)
  • I : 1 - One
  • II : 2 -  Two
  • III : 3 - Three
  • IV : 4 - Four
  • V : 5 - Five
  • VI : 6 - Six
  • VII : 7 - Seven
  • VIII : 8 - Eight
  • IX : 9 -  Nine
  • X : 10 - Ten
Roman Numerals

  • X : 10 - Ten
  • XX : 20 - Twenty
  • XXX : 30 - Thirty
  • XL : 40 - Fourty
  • L : 50 - Fifty
  • LX : 60 - Sixty
  • LXX : 70 - Seventy
  • LXXX : 80 - Eighty
  • XC : 90 - Ninety

03. Numerals Chart (1-100 : I-C)

  • Numeric Numbers <=> Roman Numerals
  • 1 <=> I
  • 2 <=> II
  • 3 <=> III
  • 4 <=> IV
  • 5 <=> V
  • 6 <=> VI
  • 7 <=> VII
  • 8 <=> VIII
  • 9 <=> IX
  • 10 <=> X
  • 11 <=> XI
  • 12 <=> XII
  • 13 <=> XIII
  • 14 <=> XIV
  • 15 <=> XV
  • 16 <=> XVI
  • 17 <=> XVII
  • 18 <=> XVIII
  • 19 <=> XIX
  • 20 <=> XX
  • 21 <=> XXI
  • 22 <=> XXII
  • 23 <=> XXIII
  • 24 <=> XXIV
  • 25 <=> XXV
  • 26 <=> XXVI
  • 27 <=> XXVII
  • 28 <=> XXVIII
  • 29 <=> XXIX
  • 30 <=> XXX
  • 31 <=> XXXI
  • 32 <=> XXXII
  • 33 <=> XXXIII
  • 34 <=> XXXIV
  • 35 <=> XXXV
  • 36 <=> XXXVI
  • 37 <=> XXXVII
  • 38 <=> XXXVIII
  • 39 <=> XXXIX
  • 40 <=> XL
  • 41 <=> XLI
  • 42 <=> XLII
  • 43 <=> XLIII
  • 44 <=> XLIV
  • 45 <=> XLV
  • 46 <=> XLVI
  • 47 <=> XLVII
  • 48 <=> XLVIII
  • 49 <=> XLIX
  • 50 <=> L
  • 51 <=> LI
  • 52 <=> LII
  • 53 <=> LIII
  • 54 <=> LIV
  • 55 <=> LV
  • 56 <=> LVI
  • 57 <=> LVII
  • 58 <=> LVIII
  • 59 <=> LIX
  • 60 <=> LX
  • 61 <=> LXI
  • 62 <=> LXII
  • 63 <=> LXIII
  • 64 <=> LXIV
  • 65 <=> LXV
  • 66 <=> LXVI
  • 67 <=> LXVII
  • 68 <=> LXVIII
  • 69 <=> LXIX
  • 70 <=> LXX
  • 71 <=> LXXI
  • 72 <=> LXXII
  • 73 <=> LXXIII
  • 74 <=> LXXIV
  • 75 <=> LXXV
  • 76 <=> LXXVI
  • 77 <=> LXXVII
  • 78 <=> LXXVIII
  • 79 <=> LXXIX
  • 80 <=> LXXX
  • 81 <=> LXXXI
  • 82 <=> LXXXII
  • 83 <=> LXXXIII
  • 84 <=> LXXXIV
  • 85 <=> LXXXV
  • 86 <=> LXXXVI
  • 87 <=> LXXXVII
  • 88 <=> LXXXVIII
  • 89 <=> LXXXIX
  • 90 <=> XC
  • 91 <=> XCI
  • 92 <=> XCII
  • 93 <=> XCIII
  • 94 <=> XCIV
  • 95 <=> XCV
  • 96 <=> XCVI
  • 97 <=> XCVII
  • 98 <=> XCVIII
  • 99 <=> XCIX
  • 100 <=> C

04. Large Roman Numerals

  • V  <=> 5,000
  • X  <=> 10,000
  • L  <=> 50,000
  • C  <=> 100,000
  • D  <=> 500,000
  • M  <=> 1,000,000

05. Years in roman numerals

  • Year <=> Roman Numerals
  • 1000 <=> M
  • 1100 <=> MC
  • 1200 <=> MCC
  • 1300 <=> MCCC
  • 1400 <=> MCD
  • 1500 <=> MD
  • 1600 <=> MDC
  • 1700 <=> MDCC
  • 1800 <=> MDCCC
  • 1900 <=> MCM
  • 1990 <=> MCMXC
  • 1991 <=> MCMXCI
  • 1992 <=> MCMXCII
  • 1993 <=> MCMXCIII
  • 1994 <=> MCMXCIV
  • 1995 <=> MCMXCV
  • 1996 <=> MCMXCVI
  • 1997 <=> MCMXCVII
  • 1998 <=> MCMXCVIII
  • 1999 <=> MCMXCIX
  • 2000 <=> MM
  • 2001 <=> MMI
  • 2002 <=> MMII
  • 2003 <=> MMIII
  • 2004 <=> MMIV
  • 2005 <=> MMV
  • 2006 <=> MMVI
  • 2007 <=> MMVII
  • 2008 <=> MMVIII
  • 2009 <=> MMIX
  • 2010 <=> MMX
  • 2011 <=> MMXI
  • 2012 <=> MMXII
  • 2013 <=> MMXIII
  • 2014 <=> MMXIV
  • 2015 <=> MMXV
  • 2016 <=> MMXVI
  • 2017 <=> MMXVII
  • 2018 <=> MMXVIII
  • 2019 <=> MMXIX
  • 2020 <=> MMXX
  • 2021 <=> MMXXI
  • 2022 <=> MMXXII
  • 2023 <=> MMXXIII
  • 2024 <=> MMXXIV
  • 2025 <=> MMXXV
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Solution of Equation in One Variable

Introduction:
The equation of the form f(x)=0 is called an equation in one variable. It may be algebraic or transcendental or a combination of both. When f(x) is a polynomial in x, that is f(x)=a0xn+a1xn-1++a2xn-2+ ..... +an then f(x)=0 is called algebraic equation, Again if f(x) contains some other functions such as logarithmic, exponential, trigonometric etc, then f(x)=0 is called transcendental equation.

Example:
A few one variable equation:
(i) x3-2x-5=0
(ii) x3+x2-1=0
(iii) x2-3=0 etc.

A few one variable equation:
(i) e-x=sinx
(ii) ex tanx=1
(iii) 2x=log10x+7 etc.
If  f(x)=0 is an equation and f(a)=0 then x=a is called a root of the equation. Again if the graph of y=f(x) crosses the x-axis at x=a then x=a is a root of the equation f(x)=0. Sometimes root of equation is called zero or solution of the equation.

Mathematical Example:
f(x)=x3-6x2+11x-6=0 is an equation and f(1)=1-6+11-6=0, So x=1 is a root of the equation. Similarly x=2 and x=3 are roots of the equation.

Graph: f(x)=x3-6x2+11x-6=0

The graph of f(x)=x3-6x2+11x-6 crosses the x-axis at (-2, 0) and (3,0). So x=-2 and x=3 are the roots of the equation f(x)=0.

Graphical Method

Discuss the Graphical method to find a real root of an equation f(x)=0 

Solution: If we take a set of rectangular coordinate axes and plot the graph of y=f(x), then the abscissas of the point where the graph crosses the x-axis are the real roots of the equations f(x)=0.

In most cases the approximate values of the real roots of f(x)=0 are most easily found by writing the equation in the form f1(x)=f2(x) and then plotting the two equations y=f1(x) and y=f2(x) on the same axes. The abscissas of the point of intersection of these two functions y=f1(x) and y=f2(x) are the real roots of the equation f(x)=0.

Problem: Find the approximate value of a root of the equation cosx=3x-1 usnig graphical method.

Solution:
Given equation, cosx=3x-1
Let, y=cosx ............................ (i)
and y=3x-1 ............................ (ii)

Now we plot (i) and (ii) separately on the same set of axes as shown on the following diagram.

Graph: cosx=3x-1

The abscissa of the point of intersection not the graphs of theses equations is seen to be about 0.6. hence the approximate value of the root is 0.6.
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The word 'Simplification' is a noun. Simplifying makes a algebraic expressions easily understandable and solvable.

What is Simplification?

According to Cambridge Dictionary, The process of making something less complicated and therefore easier to do or understand, or the thing that results from this process:
  • The organization advises on the simplification of trade procedures.
  • This is a gross simplification of what actually happened.

Mathematical Concept:
To reduce a fraction to its lowest terms by canceling to the lowest common factor for both numerator and denominator or to condense an algebraic expressions by grouping and combining similar terms. Simplifying makes a algebraic expressions easily understandable and solvable. (from: www.splashlearn.com)

Learn how to simplify with BODMAS rule:

'BODMAS', We've named the simplification process easy to remember. By this rule (BODMAS), we can simplify mathematical expressions easily.

BODMAS Stands for letters:
B ↠ Bracket : ( ) { } [ ]

O ↠ Of or Orders

D ↠ Division

M ↠ Multiplication

A ↠ Addition

S ↠ Subtraction

According to BODMAS rule, we need to follow the following steps corresponding:
  • At first, We need to solve  brackets corresponding:  (), {}, []
  • Next: Of or Order
  • Next: Division or Multiplication
  • Finally: Addition  or Subtraction
Learn about: A Model

To know about BODMAS, follow the following photo attentively:

Rule: BODMAS

By following a few algebraic expressions, we can understand the BODMAS rule clearly. Please try to understand the below examples:

01. Example (Bracket related algebraic expressions):
[ 4 + { 3 + ( 7 + 2 ) - 5 } ]
= [ 4 + { 3 + 9 } - 5 ] (by completing First backer, according to BODMAS rule)
= [ 4 + 12 - 5 ] (by completing Second backer, according to BODMAS rule)
= [ 16 -5 ] ( Addition or Subtraction, as you with)
= 11 (Results)

02. Example:
20  ÷4 - 1 + 10✕2 - 8
= 5 - 1 + 20 - 8  (by completing Division & Multiplication)
= 5 + 20 - 1 - 8 (Rearranged: Positive and Negative values)
=  25 - 9 (by completing Addition)
= 16 (Results)

03. Example:
20 - [ { 10 + ( 12 - 6 ) - 4 } ]
= 20 - [  { 10 + 6 - 4 } ]
= 20 - [ { 16 -4 } ]
=  20 - [ 12]
= 20- 12
= 8

We have tried to show you basics form of simplification algebraic expressions, We believe that by following BODMAS rule, learner will be solved complex algebraic expressions easily. If you have a great idea, let's share to comments.

Thanks in Advance.
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Trigonometry Unit Circle




ASTC Diagram



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Sets

A set is the representation of a collection of objects; distinct objects with one or more common properties. Grouping up the objects in a set is an act of distinguishing those objects from the members of another set. We can use the terms – elements or members of a set instead of the term objects.


How to denote Sets?

If ‘A’ is a set and ‘a’ one of its elements then: ‘a ∈ A’ denotes that element ‘a’ belongs to ‘A’ whereas, ‘a ∉ A’ denotes that ‘a’ is not an element of A. Alternatively, we can say that ‘A’ contains ‘a’. A set is usually represented by capital letters and an element of the set by the small letter.

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Derivatives and Anti-derivatives

The derivative of a function is one of the basic concepts of mathematics. Together with the integral, derivative occupies a central place in calculus. The process of finding the derivative is called differentiation. The inverse operation for differentiation is called integration and you can find integration using integral calculator.

The derivative of a function at some point characterizes the rate of change of the function at this point. We can estimate the rate of change by calculating the ratio of change of the function Δy to the change of the independent variable Δx. In the definition of derivative, this ratio is considered in the limit as Δx → 0. Let us turn to a more rigorous formulation.


The chart above shows to differentiate and integrate the most common functions. There are, Integration reverses differentiation, returning the function to its original state, up to a constant C.

Credit: math24
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ABC =>> Uppercase

abc =>> Lowercase

~ => Tilde

` => Acute/ Back quote/ Grave/ Grave accent/ Left quote/ Open quote/ A push

! =>> Exclamation Mark/ Exclamation Point/ Bang

@ =>> At Sign (Each)/ At/ Ampersat/ Arobase/ Asperand


# =>> Hash/ Octothorpe/ Number/ Pound/ Sharp

$ =>> Dollar Sign/ Generic currency

% =>> Percent

^ =>> Caret/ Circumflex

& =>> Ampersand/ Epershand/ And

* =>> Asterisk/ Sometimes referred to as star

( =>> Open parenthesis

) =>> Close parenthesis

() =>> Parenthesis

- =>> Hyphen/ Minus/ Dash

- ==>> Underscore

+ =>> Plus

=  =>> Equals

{ =>> Curly Bracket/ Open Brace/ Squiggly brackets

} =>> Curly Bracket/ Close Brace/ Squiggly brackets

{} =>> Curly Brackets

[ =>> Open Bracket

] =>> Close Bracket

[] =>> Square Brackets

: =>> Colon

; =>> Semicolon

" =>> Quotation mark/ Quote/ Inverted commas

"" =>> Double Quotes

' =>> Apostrophe Mark/ Single Quote

/ =>> Forward slash/ Solidus/Virgule/ Whack

| =>> Bar/ Pipe

, =>> Comma

. =>> Period/ Decimal Point/ Dot/ Full stop

\ =>> Back Slash/ Reverse Solidus

? =>> Question Mark

> =>> Greater than/ Angle brackets

< =>> Less than/ Angle brackets
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Basic Rules of Integration



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Trigonometrical ratios of associated and compound angles

01. (i) sin (A+B) = sinA cosB + cosA sinB
(ii) sin (A-B) = sinA cosB - cosA sinB

02. (i) cos (A+B) = cosA cosB - sinA sinB
(ii) cos (A-B) = cosA cosB + sinA sinB

03. (i) tan (A+B) = (tanA + tanB) / (1 - tanA tanB)
(ii) tan (A-B) = (tanA - tanB) / (1 + tanA tanB)

04. (i) cot (A+B) = (cotA cotB - 1) / (cotB + cotA)
(ii) cot (A-B) = (cotA cotB + 1) / (cotB - cotA)

05. 2 sinA cosB = sin (A+B) + sin (A-B)
06. 2 cosA sinB = sin (A+B) - sin (A-B)
07. 2 sinA sinB = cos (A-B) - cos (A+B)
08. 2 cosA cosB = cos (A+B) + cos (A-B)

09. sinC + sinD = 2 sin(C+D)/2 cos (C-D)/2
09. sinC - sinD = 2 cos(C+D)/2 sin (C-D)/2


Quick navigation: Basic Rules of Trigonometry


Trigonometric Ratios



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Definition of Parabola

: A parabola is a curve that looks like the one on the right. Its open end can point up, down, left or right. A curve of this shape is called 'parabolic', meaning 'like a parabola'.

Quick navigation: Geometric Figure

: A u-shaped curve with certain specific properties. Formally, a parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix. [Adopted from MathWords]

Note: For a parabolic mirror, all rays of light emitting from the focus reflect off the parabola and travel parallel to each other (parallel to the axis of symmetry as well).


: A parabola is the set of all points in a plane that are equidistant from the focus and the directrix of the parabola. [Adopted from iCoachMath]

Focus:

Focus of a parabola lies on the axis of symmetry.

Directrix: 

Directrix is a line that is perpendicular to the axis of symmetry of a parabola.

Geometric figure and Formulas:


Large image below:


Short Note:

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Secondary School Certificate (SSC) Exam - 2016
Suggestion: Higher Mathematics (For All Education Board in Bangladesh)

Algebra





Geometry



Trigonometry & Perimetry


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Secondary School Certificate (SSC) Exam - 2016
Suggestion: Mathematics (For All Education Board in Bangladesh)

Algebra






Geometry

Trigonometry & Perimetry






Statistics


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Problem-02: A survey conducted over the last 20 years in Hamburg., Germany indicated that in 8 of them the winter was mild, in 7 of them it was cold, and in the remaining 5 it was very cold. A company sells 1,000 heavy coats in a mild year., 1,300 in a year, and 2,000 in a very cold year.
                   Find the yearly expected profit of the company if a coat costs 85 deutsche marks (DM) and is sold to stores for 123 DM.

Solution:
According to the question,
* States of nature: (i) Mild, (ii) Cold, (iii) Very cold
* Alternative courses of Action: Selling of heavy coats.
Probabilities of states of natures:
(i) Mild = 8/20 = 0.4
(ii) Cold = 7/20 = 0.35
(iii) Very cold = 5/20 = 0.25

You can also read: Decision Analysis » Problem-01 Solution | Operations Research

Table for the calculation of Expected Value

Probabilities 0.4 0.35 0.25 1
States of Nature Mild Cold Very cold Expected Value
Courses of Action
Selling of heavy coats 1,000 1,300 2,000 1,355

Expected profit of coat = 123 - 85 = 38
∴ Expected total profit = 38 × 1,355 = 51,490

N.B:

How to calculate Expected value:
EV = (1,000 × 0.4) + (1,300 × 0.35) + (2,000 × 0.25) = 1,355
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Problem-01: A marketing agent frequently flies from Montreal to Boston. She can ride the airport bus from her hotel to the airport, which costs $3; but if she takes it, there is a .08 chance that she will miss the flight. A hotel limousine costs $7, with a .96 chance of being on time for the flight. For $15, she can take a taxi that will make 99 of 100 flights. Each time she catches the plane, she will conclude a business transaction that will produce a profit of $1,000; otherwise, she will lose the deal. Which mode of transportation should the marketing agent use in order to maximize her profits?

Solution:
According to given data, we can draw following decision tree:
Figure: Decision Tree, according to given data.

Net Profit Calculation from different mode of transportation:

For Airport Bus :

EMV=$1,000 (.92) + $0 (.08) = $920
Net Profit: $920 - $3 = $917

For Hotel Limousine :

EMV=$1,000 (.96) + $0 (.04) = $960
Net Profit: $960 - $7 = $953

For Taxi :

EMV=$1,000 (.99) + $0 (.01) = $990
Net Profit: $990 - $15 = $975


Figure: Decision Tree for making final decision.
Here, Taxi, represents the highest profit from others. Since, the marketing agent should use a taxi in order to maximize her profits.
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Quantitative Methods: Duality in Linear Programming

Associated with every linear programming problem, there is another intimately related LPP, called the dual problem of the original LPP. The original LPP is called the Primal Problem. According to the duality theorem:
“For every maximization (or minimization) problem in linear programming, there is a unique similar problem of minimization (or maximization) involving the same data which describes the original problem.”


The rules for constructing the Dual from the Primal (or Primal from the Dual) are:
i) If the objective of one problem is to be maximized, the objective of the other is to be minimized.
ii) The maximization problem should have all ≤ constraints and the minimization problem has all ≥ constraints.
iii) All primal and dual variables must be non-negative (> 0).
iv) The element of the right hand side of the constraints in one problem are the respective coefficient of the objective functions in the other problem.
v) The matrix of constraints coefficients for one problem is the transpose of the matrix of constraint coefficients for the either problem.
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