Diseño Electrónico: Página India atozmath de Piyush N Shah que muestra la solución paso a paso de Problemas matemáticos y de Métodos Numéricos

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Home > Numerical methods calculators

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Numerical Methods Calculators (examples)

1. Find a root an equation using
1. Bisection Method
2. False Position Method
3. Fixed Point Iteration Method
4. Newton Raphson Method
5. Secant Method
6. Muller Method
7. Halley's Method
8. Steffensen's Method
9. Birge-Vieta method (for nth degree polynomial equation)
10. Bairstow method

2. Find roots of non-linear equations using
Modified Newton Raphson method (Multivariate Newton Raphson method)

3. Numerical Interpolation using
0. Best suitable formula (among 2 to 10)
1. Newton's Forward Difference formula
2. Newton's Backward Difference formula
3. Newton's Divided Difference Interpolation formula
4. Lagrange's Interpolation formula
5. Lagrange's Inverse Interpolation formula
6. Gauss Forward formula
7. Gauss Backward formula
8. Stirling's formula
9. Bessel's formula
10. Everett's formula
11. Hermite's formula
12. Missing terms in interpolation table

4.1 Numerical Differentiation using
1. Best suitable formula (among 2 to 7)
2. Newton's Forward Difference formula
3. Newton's Backward Difference formula
4. Newton's Divided Difference formula
5. Lagrange's formula
6. Stirling's formula
7. Bessel's formula

4.2 Numerical Differentiation first order and second order using
1. 2 point Forward, Backward, Central difference formula
2. 3 point Forward, Backward, Central difference formula
3. 4 point Forward, Backward, Central difference formula
4. 5 point Forward, Central difference formula

4.3 Richardson extrapolation formula for differentiation

5. Numerical Integration using
1. Trapezoidal Rule
2. Simpson's 1/3 Rule
3. Simpson's 3/8 Rule
4. Boole's Rule
5. Weddle's Rule

6.1 Solve (1st order) numerical differential equation using
1. Euler method
2. Runge-Kutta 2 method
3. Runge-Kutta 3 method
4. Runge-Kutta 4 method
5. Improved Euler method
6. Modified Euler method
7. Taylor Series method
8. Adams bashforth predictor method
9. Milne's simpson predictor corrector method

6.2 Solve (2nd order) numerical differential equation using
1. Euler method
2. Runge-Kutta 2 method
3. Runge-Kutta 3 method
4. Runge-Kutta 4 method

7. Cubic spline interpolation

Numerical Methods with example

1. Bisection, False Position, Iteration, Newton Raphson, Secant Method

Find a real root an equation using
1. Bisection Method
2. False Position Method
3. Fixed Point Iteration Method
4. Newton Raphson Method
5. Secant Method
6. Muller Method
7. Birge-Vieta method (for nth degree polynomial equation)
8. Bairstow method

Enter an equation like...
f(x)=2x3-2x-5
f(x)=x3-x-1
f(x)=x3+2x2+x-1
f(x)=x3-2x-5
f(x)=x3-x+1
f(x)=cos(x)

2. Numerical Differentiation

Numerical Differentiation using Newton's Forward, Backward Method
1. From the following table of values of x and y, obtain dydx and d2ydx2 for x = 1.2 .

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
y	2.7183	3.3201	4.0552	4.9530	6.0496	7.3891	9.0250

2. From the following table of values of x and y, obtain dydx and d2ydx2 for x = 2.2 .

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
y	2.7183	3.3201	4.0552	4.9530	6.0496	7.3891	9.0250

3. Numerical Integration

Numerical Integration using Trapezoidal, Simpson's 1/3, Simpson's 3/8 Rule

1. From the following table, find the area bounded by the curve and x axis from x=7.47 to x=7.52 using trapezodial, simplson's 1/3, simplson's 3/8 rule.

x	7.47	7.48	7.49	7.50	7.51	7.52
f(x)	1.93	1.95	1.98	2.01	2.03	2.06

2. Evaluate I = ∫1011+xdx by using simpson's rule with h=0.25 and h=0.5

5. Solve numerical differential equation using Euler, Runge-kutta 2, Runge-kutta 3, Runge-kutta 4 methods

1. Find y(0.1) for y′=x-y2, y(0) = 1, with step length 0.1
2. Find y(0.5) for y′=-2x-y, y(0) = -1, with step length 0.1
3. Find y(2) for y′=x-y2, y(0) = 1, with step length 0.2
4. Find y(0.3) for y′=-(x⋅y2+y), y(0) = 1, with step length 0.1
5. Find y(0.2) for y′=-y, y(0) = 1, with step length 0.1

4. Numerical Interpolation

Numerical Interpolation using Forward, Backward Method

1. The population of a town in decimal census was as given below. Estimate population for the year 1895.

Year	1891	1901	1911	1921	1931
Population (in Thousand)	46	66	81	93	101

2. Let y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10. Find y(4) using newtons's forward difference formula.

3. In the table below the values of y are consecutive terms of a series of which the number 21.6 is the 6th term. Find the 1st and 10th terms of the series.

X	3	4	5	6	7	8	9
Y	2.7	6.4	12.5	21.6	34.3	51.2	72.9

4. The population of a town in decimal census was as given below. Estimate population for the year 1895.

X	0.10	0.15	0.20	0.25	0.30
tan(X)	0.1003	0.1511	0.2027	0.2553	0.3073

Find (1) tan 0.12 (2) tan 0.26

5. Certain values of x and log10x are (300,2.4771), (304,2.4829), (305,2.4843) and (307,2.4871). Find log10 301.

6. Find lagrange's Inerpolating polynomial of degree 2 approximating the function y = ln x defined by the following table of values. Hence find ln 2.7

X	2	2.5	3
ln(X)	0.69315	0.91629	1.09861

7. Using the following table find f(x) as polynomial in x

x	-1	0	3	6	7
f(x)	3	-6	39	822	1611

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Matrix & Vector Calculators (examples)

1.1 Matrix operations
1. Addition/Subtraction of two matrix
2. Multiplication of two matrix
3. Division of two matrix
4. Power of a matrix
5. Transpose of a matrix
6. Determinant of a matrix
7. Adjoint of a matrix
8. Inverse of a matrix
9. Prove that any two matrix expression is equal or not
10. Minor of a matrix
11. Cofactor of a matrix
12. Trace of a matrix

1.2 Matrix operations
1. Transforming matrix to Row Echelon Form
2. Transforming matrix to Reduced Row Echelon Form
3. Rank of matrix
4. Characteristic polynomial
5. Eigenvalues
6. Eigenvectors
7. Triangular Matrix
8. LU Decomposition using Gauss Elimination method
9. LU decomposition using Doolittle's method
10. LU decomposition using Crout's method
11. Diagonal Matrix
12. Cholesky Decomposition
13. QR Decomposition (Gram Schmidt Method)
14. QR Decomposition (Householder Method)
15. LQ Decomposition
16. Pivots of a Matrix
17. SVD - Singular Value Decomposition
18. Moore-Penrose Pseudoinverse of a Matrix
19. Power Method for dominant eigenvalue
20. Determinants using Sarrus Rule
21. Determinants using properties of determinants
22. Row Space
23. Column Space
24. Null Space

1.3 Matrix Structure
0. Auto Detect the matrix type
1. is Row Matrix
2. is Column Matrix
3. is Square Matrix
4. is Horizontal Matrix
5. is Vertical Matrix
6. is Diagonal Matrix
7. is Identity Matrix
8. is Scalar Matrix
9. is Null Matrix
10. is Lower Triangle Matrix
11. is Upper Triangle Matrix
12. is Orothogonal Matrix
13. is Singular Matrix
14. is Nonsingular Matrix
15. is Symmetric Matrix
16. is Skew Symmetric Matrix
17. is Nilpotent Matrix
18. is Involutary Matrix
19. is Idempotent Matrix
20. is Periodic Matrix
21. is Positive Definite Matrix
22. is Negative Definite Matrix
23. is Derogatory Matrix
24. is Diagonally Dominant Matrix
25. is Strictly Diagonally Dominant Matrix

2. Find inverse of a matrix using
1. Adjoint method
2. Gauss-Jordan Elimination method
3. Cayley Hamilton method method

3. Solve linear equation of any number of variables using
1. Inverse Matrix method
2. Cramer's Rule method
3. Gauss-Jordan Elimination method
4. Gauss Elimination Back Substitution method
5. Gauss Seidel method
6. Gauss Jacobi method
7. Elimination method
8. LU decomposition using Gauss Elimination method
9. LU decomposition using Doolittle's method
10. LU decomposition using Crout's method
11. Cholesky decomposition method
12. SOR (Successive over-relaxation) method
13. Relaxation method

4. Vector Algebra
1. Addition/Subtraction of two vectors
2. Scalar Multiplication of vectors
3. Dot Product of two vectors
4. Cross Product of two vectors
5. Magnitude(length) of a vector
6. Unit vector
7. Direction cosines of a vector
8. Component form of a vector given two points
9. Angle between two vectors
10. Vector projections of B onto A
11. Orthogonal vectors
12. Collinear vectors
13. Coplanar vectors
14. Scalar triple product
15. Area of triangle determined by two vectors
16. Area of parallelogram determined by two vectors
17. Volume of pyramid determined by vectors
18. Volume of Parallelepiped determined by vectors
19. Decomposition of vector in basis
20. Linearly dependent and linearly independent vectors

Home > Algebra calculators

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Algebra Calculators (examples)

1. Addition, subtraction, multiplication, division of two polynomials
1.1 Addition of two polynomials eg.(x3-4x2+4x-8)+(x-2)
1.2 Subtraction of two polynomials eg.(x3-4x2+4x-8)-(x-2)
1.3 Multiplication of two polynomials eg.(x3-4x2+4x-8)×(x-2)
1.4 Long division of two polynomials eg.(x3-4x2+4x-8)÷(x-2)
1.5 Synthetic division of two polynomials eg.(x3-4x2+4x-8)÷(x-2)
1.6 Remainder theorem eg.(x3-4x2+4x-8)÷(x-2)

2. Factoring Polynomials
2.1 Factor out GCF (Taking common) eg.x2+2x
2.2 Difference of squares eg.x2-9
2.3 Sum and Difference of cubes eg.x3-27,x3+27
2.4 Whole square of a binomial eg.4x2+12xy+9y2
2.5 Reverse FOIL method (Splitting the middle term) eg.x2+10x+24
2.6 Perfect square trinomial eg.4x2+y2+1+4xy+4x+2y
2.7 Factorization with the help of factor theorem eg.x3-3x2-6x+8
2.8 Cyclic Expressions eg.a2(b-c)+b2(c-a)+c2(a-b)
2.9 Factorization of fourth power polynomial eg.x4+x2+1

3. Expand and simplify polynomial
3.1 FOIL Method eg.(2x-1)(4x+5)
3.2 Expand Difference of Squares eg.(x-6)(x+6)
3.3 Expand Perfect Squares of binomial eg.(x-3)2
3.4 Expand Cubes eg.(x-3)3
3.5 Expand Trinomials eg.(a+b)(a2-ab+b2)
3.6 Expand Perfect Squares of trinomial eg.(2x+3y+4z)2

4. Complete square, Is perfect square, Find missing term
1. Completing the square for quadratic equation eg. 9x2+6x+1=9(x+13)2
2. Determining if the polynomial is a perfect square eg. (1) x2-4xy+4y2, (2) 3x2+5x+2
3. Find the missing term in a perfect square trinomial eg. (1) 9x2 - __ + 16, (2) __ + 12x2 + 9, (3) 49x2 + 56 xy + __

5.1 HCF(GCD)-LCM of Polynomials eg. Find GCD, LCM of (2x2-4x),(3x4-12x2),(2x5-2x4-4x3)
5.2 Find other polynomial when one polynomial its GCD and LCM are given

6. Rational Expression of Polynomials
1. Reduce rational expressions eg. 4x2-258x3-125
2. Adding, subtracting, multiplying, dividing of rational expressions polynomials eg. x-3x+1-x-6x

7. Simplifying Algebraic Expressions
eg. (1) 4x+13x-2=32, (2) 2x-13x+1+4x-16x=0

8.1 Quadratic Equation
1.1 Solving quadratic equations by factoring, eg. (1) 25x2-30x+9=0, (2) x2+10x-56=0
1.2 Solving quadratic equations using the quadratic formula, eg. (1) 25x2-30x+9=0, (2) x2+10x-56=0
1.3 Discriminant eg. (1) 25x2-30x+9=0, (2) x2+10x-56=0
1.4 Discriminant & Nature of Roots eg. (1) 25x2-30x+9=0, (2) x2+10x-56=0
2. Find the quadratic equation whose roots are α and β eg. (1) α=3,β=-4, (2) α=1+3√2,β=1-3√2
3. Roots for non-zero denominator eg. (1) 5x-18x+2=2x-6x-1, (2) xx+1+x+1x=52, (3) 4(4x+14x-1)2+4x+14x-1=3, (4) 4x+14x-1+4x-14x+1=3
4. Roots of non-quadratic equation eg. (1) 6(x2+1x2)-25(x-1x)+12=0, (2) (x2+1x2)-8(x+1x)+14=0

5. If α and β are roots of equation 2x2-3x-6=0, then find α2+β2
6. If α and β are roots of equation 2x2-3x-6=0, then find equation whose roots are α2 and β2
7. Find value of k for which 2x2+kx+2=0 has real roots

9. Solve linear equation in two variables by (eg. Solve 7y+2x-11=0 and 3x-y-5=0 using Substitution method)
1. Substitution method
2. Elimination method
3. Cross multiplication method
4. Addition-Subtraction method
5. Inverse matrix method
6. Cramer's Rule method
7. Graphical method

10. Solve linear equation of any number of variables (simultaneous equations) using
1. Inverse Matrix method
2. Cramer's Rule method
3. Gauss-Jordan Elimination method
4. Gauss Elimination Back Substitution method
5. Gauss Seidel method
6. Gauss Jacobi method
7. Elimination method
8. LU decomposition method / Crout's method
9. Cholesky decomposition method
10. SOR (Successive over-relaxation) method
11. Relaxation method

11.1 Find the value of h,k
1. Find the value of h,k for which the system of equations has a Unique solution
2. Find the value of h,k for which the system of equations has Infinite solution
3. Find the value of h,k for which the system of equations has No solution
4. Find the value of h,k for which the system of equations is consistent
5. Find the value of h,k for which the system of equations is inconsistent

11.2 Determine whether the system of linear equations
1. Determine whether the system of linear equations has a Unique solution
2. Determine whether the system of linear equations has Infinite solution
3. Determine whether the system of linear equations has No solution
4. Determine whether the system of linear equations is consistent
5. Determine whether the system of linear equations is inconsistent

12. Variation Equations
1. Find value of variation using given value
(1) x ∝ y and x=6 when y=3. Find y=? when x=18, (2) x ∝ yz. x=8 when y=4 and z=3. Find x=? when y=6 and z=4.
2. Prove results for given variation
(1) If x ∝ y then prove that x3+y3∝x2y-xy2, (2) If 3x-5y ∝ 5x+6y then prove that x ∝ y.

13. If x+1x=2 then find x-1x,x2-1x2,x3+1x3
1. If x-1x=6 then find (1) x2+1x2 (2) x+1x (3) x2-1x2
2. If x+y=5 and xy=6 then find x2+y2
3. If x2+y2+z2=29 and xy+yz+zx=-14 then find x+y+z
4. If x+y+z=1,xy+yz+zx=-1 and xyz=-1 then find x3+y3+z3

14. Interval notation and set builder notation eg. (1) 3≤x≤7, x is odd. (2) |x3-2|≤25, x in Z. (3) x∈[2,8)

15. Set Theory eg. A={x≤5;x∈N},B={2≤x≤8;x∈N},C={x3-3x2-4x=0}, Find
1. Union eg. A∪(B∪C)=(A∪B)∪C
2. Intersection eg. A∩(B∪C)=(A∩B)∪(A∩C)
3. Complement eg. (A∪B)′=A′∩B′
4. Power set (Proper Subset) eg. P(A)
5. Difference eg. (1)A-B, (2) A-(B∪C)=(A-B)∩(A-C)
6. Symmetric difference eg. (1)AΔB, (2) BΔC, (3) AΔC
7. Cross Product eg. A×B
8. Prove that any two expression is equal or not eg. A-(B∪C)=(A-B)∩(A-C)
9. Cardinality of a set eg. n(A)
10. is Belongs to a set eg. 2∈B ?
11. is Subset of a set eg. A⊂B ?
12. is two set Equal or not eg. A=B ?

16. Functions
1. Find Range of f:A→B eg. 1. f(x)=5x+2 where A={1≤x<5}, 2. f(x)=√x where A={1,4,16,36}
2. Composite functions and Evaluating functions
eg. 1. f(x)=2x+1, g(x)=x+5. Find fog(x), also evaluate at x=2
2. fog(x)=x+23x,f(x)=x-2. Find g(2).
3. gof(x)=1x2,f(x)=2+x2. Find g(x).
3. Find value eg. 1. f(x)=x(x+1)(2x+1). Find f(x)-f(x-1), 2. f(x)=x2-2x. Find f(2)-f(0)
4. Verifying if two functions are inverses of each other eg. 1. f(x)=x+3,g(x)=x-3, 2. f(x)=4x-3,g(x)=x+34, 3. f(x)=xx-1,g(x)=2x2x-1

17. Functions
1. Domain of a function
2. Range of a function
3. Inverse of a function
4. Properties of a function
5. Parabola Vertex form
6. Parabola Focus
7. axis symmetry of a parabola
8. Parabola Directrix
9. Intercept of a function
10. Parity of a function
11. Asymptotes of a function

18. Descartes' rule of signs eg. x5-x4+3x3+9x2-x+5

19. Ratio and Proportion
1. If a:b:c=2:3:5 then find value of a2+b2+c2ab+bc+ca
2. If a:b=2:3,b:c=4:5 then find a:b:c
3. If ab=cd=ef then prove that 2a+3c-4e2b+3d-4f=5a-4c+3e5b-4d+3f
4. If xy+z=yz+x=zx+y then prove the value of each ratio is 12 or -1
5. Geometric Mean
6. Ratios(duplicate, triplicate) and proportional(mean, third, fourth)
6.1 Duplicate Ratio
6.2 Triplicate Ratio
6.3 Sub-Duplicate ratio
6.4 Sub-Triplicate Ratio
6.5 Compounded Ratio
6.6 Mean proportional
6.7 Third proportional
6.8 Fourth proportional
6.9 Compare ratios

20. Partial Fraction decomposition eg. 5x-4x2-x-2

21. Logarithmic equations eg. log(20)+log(30)-12log(36)

22. Simple Interest
23. Compound Interest
24. Percentage

25. Arithmetic Progression
26. Geometric Progression

27. Polynomial
1. Polynomial in ascending order
2. Polynomial in descending order
3. Degree of a polynomial
4. Leading term of a polynomial
5. Leading coefficient of a polynomial
6. Determine expression is a polynomial or not
7. Zeros of a polynomial

Home > Statistical Methods calculators

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Problem Source	Your textbook, etc

Statistical Methods Calculators (examples)

1.1 Ungrouped data

85,96,76,108,85,80,100,85,70,95

1. Mean, Median and Mode for ungrouped data
2. Quartiles, Deciles and Percentiles for ungrouped data
3. Population Variance, Standard deviation and coefficient of variation for ungrouped data
4. Sample Variance, Standard deviation and coefficient of variation for ungrouped data
5. Population Skewness, Kurtosis for ungrouped data
6. Sample Skewness, Kurtosis for ungrouped data
7. Geometric mean, Harmonic mean for ungrouped data
8. Mean deviation, Quartile deviation, Decile deviation, Percentile deviation for ungrouped data
9. Five number summary for ungrouped data
10. Box and Whisker Plots for ungrouped data

1.2 Grouped data

Class	50-55	45-50	40-45	35-40	30-35	35-30	20-25
f	25	30	40	45	80	110	170

1. Mean, Median and Mode for grouped data
2. Quartiles, Deciles and Percentiles for grouped data
3. Population Variance, Standard deviation and coefficient of variation for grouped data
4. Sample Variance, Standard deviation and coefficient of variation for grouped data
5. Population Skewness, Kurtosis for grouped data
6. Sample Skewness, Kurtosis for grouped data
7. Geometric mean, Harmonic mean for grouped data
8. Mean deviation, Quartile deviation, Decile deviation, Percentile deviation for grouped data
9. Five number summary for grouped data
10. Box and Whisker Plots for grouped data

1.3 Mixed data

Class	1	2	5	6-10	10-20	20-30	30-50	50-70	70-100
f	3	4	10	23	20	20	15	3	2

1. Mean, Median and Mode for mixed data
2. Population Variance, Standard deviation and coefficient of variation for mixed data
3. Sample Variance, Standard deviation and coefficient of variation for mixed data

Home > Operation Research calculators

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Educational Level	Secondary school, High school and College
Program Purpose	Provide step by step solutions of your problems using online calculators (online solvers)
Problem Source	Your textbook, etc

Operation Research Calculators (examples)

1. Assignment problem
1.1 Assignment problem (Using Hungarian method-2)
1.2 Assignment problem (Using Hungarian method-1)
2.1 Travelling salesman problem using hungarian method
2.2 Travelling salesman problem using branch and bound (penalty) method
2.3 Travelling salesman problem using branch and bound method
2.4 Travelling salesman problem using nearest neighbor method
2.5 Travelling salesman problem using diagonal completion method
3. Crew assignment problem

2. Simplex method (Solve linear programming problem using)
1. Simplex method (BigM method)
2. Two-Phase method
3. Dual Simplex method
4. Integer Simplex method (Gomory's cutting plane method)
5. Graphical method
6. Primal to dual conversion
7. Branch and Bound method
8. 0-1 Integer programming problem
9. Revised Simplex method

3. Transportation Problem using
1. North-West corner method
2. Least cost method
3. Vogel's approximation method
4. Row minima method
5. Column minima method
6. Russell's approximation method
7. Heuristic method-1
8. Heuristic method-2
9. Optimal solution using MODI method
10. Optimal solution using stepping stone method

4. PERT and CPM
1. Network diagram
1. Activity, Predecessors
2. Activity i-j
3. Activity i-j, Name of Activity

2. Critical path, Total float, Free float, Independent float
1. Activity, Predecessors, Duration
2. Activity i-j, Duration
3. Activity i-j, Name of Activity, Duration

3. Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic)
1. Activity, Predecessors, to, tm, tp
2. Activity i-j, to, tm, tp
3. Activity i-j, Name of Activity, to, tm, tp

4. Project crashing to solve Time-Cost Trade-Off with fixed Indirect cost
1. Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and Indirect Cost
2. Activity i-j, Normal Time & Cost, Crash Time & Cost and Indirect Cost
3. Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and Indirect Cost

5. Project crashing to solve Time-Cost Trade-Off with varying Indirect cost
1. Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost
2. Activity i-j, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost
3. Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost

5. Sequencing Problems
1. Processing n Jobs Through 2 Machines Problem
2. Processing n Jobs Through 3 Machines Problem
3. Processing n Jobs Through m Machines Problem
4. Processing 2 Jobs Through m Machines Problem

6. Replacement and Maintenance Models
1. Model-1 : Replacement policy for items whose running cost increases with time and value of money remains constant during a period
1.1 Model-1.1
1.2 Model-1.2
1.3 Model-1.3
2. Model-2 : Replacement policy for items whose running cost increases with time but value of money changes constant rate during a period
3. Model-3 : Group replacement policy

7. Game Theory
1. Saddle Point
2. Dominance method
3. Algebraic method
4. Calculus method
5. Arithmetic method
6. Matrix method
7. 2Xn Games
8. Graphical method
9. LPP method
10. Bimatrix method

8. Data envelopment analysis (DEA method)

Operation Research with example

1. Assignment Problem

1.1 Balanced Assignment Problem (Using Hungarian method)

1. A department has five employess with five jobs to be permormed. The time (in hours) each men will take to perform each job is given in the effectiveness matrix.

		Employees
		I	II	III	IV	V
Jobs	A	10	5	113	15	16
	B	3	9	18	13	6
	C	10	7	2	2	2
	D	7	11	9	7	12
	E	7	9	10	4	12

How should the jobs be allocated, one per employee, so as to minimize the total man-hours?

1.2 Unbalanced Assignment Problem (Using Hungarian method)

2. In the modification of a plant layout of a factory four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The cost of locating a machine at a place (in hundred rupess) is as follows.

		Location
		A	B	C	D	E
Machine	M1	9	11	15	10	11
	M2	12	9	--	10	9
	M3	--	11	14	11	7
	M4	14	8	12	7	8

Find the optimal assignment schedule.

2. Travelling salesman problem
2.1 using hungarian method
2.2 using branch and bound (penalty) method
2.3 using branch and bound method
2.4 Travelling salesman problem using nearest neighbor method
2.5 Travelling salesman problem using diagonal completion method

1. A travelling salesman has to visit five cities. He wishes to start from a particular city, visit each city only once and then return to his starting point. The travelling cost of each city from a particular city is given below.

		To city
		A	B	C	D	E
From city	A	x	2	5	7	1
	B	6	x	3	8	2
	C	8	7	x	4	7
	D	12	4	6	x	5
	E	1	3	2	8	x

How should the jobs be allocated, one per employee, so as to minimize the total man-hours?

3 Crew assignment problem

1. Best-ride airlines that operates seven days a week has the following time-table.

Delhi - Mumbai

Mumbai - Delhi

Flight No	Departure	Arrival
1	7.00	8.00
2	8.00	9.00
3	13.00	14.00
4	18.00	19.00

Flight No	Departure	Arrival
101	8.00	9.00
102	9.00	10.00
103	12.00	13.00
104	17.00	18.00

Crews must have a minimum layover of 5 hours between flights. Obtain the pairing of flights that minimizes layover time away from home. For any given pairing, the crew will be based at the city that results in the smaller layover. For each pair also mention the city where crew should be based.

2. Simplex method
Solve the following LP problem by using
1. Simplex method (BigM method)
2. Two-Phase method
3. Graphical method
4. Primal to dual conversion
5. Dual Simplex method
6. Integer Simplex method (Gomory's cutting plane method)
7. Branch and Bound method
8. 0-1 Integer programming problem
9. Revised Simplex method

2.1 Simplex method

1. Use the simplex method to solve the following LP problem.
Maximize Z = 3x1 + 5x2 + 4x3
subject to the constraints
2x1 + 3x2 ≤ 8
2x2 + 5x3 ≤ 10
3x1 + 2x2 + 4x3 ≤ 15
and x1, x2, x3 ≥ 0

2. Use the simplex method to solve the following LP problem.
Maximize Z = 4x1 + 3x2
subject to the constraints
2x1 + x2 ≤ 1000
x1 + x2 ≤ 800
x1 ≤ 400
x2 ≤ 700
and x1, x2 ≥ 0

2.2 BigM method

1. Use the penalty (Big - M) method to solve the following LP problem.
Minimize Z = 5x1 + 3x2
subject to the constraints
2x1 + 4x2 ≤ 12
2x1 + 2x2 = 10
5x1 + 2x2 ≥ 10
and x1, x2 ≥ 0

2. Use the penalty (Big - M) method to solve the following LP problem.
Minimize Z = x1 + 2x2 + 3x3 - x4
subject to the constraints
x1 + 2x2 + 3x3 = 15
2x1 + x2 + 5x3 = 20
x1 + 2x2 + x3 + x4 = 10
and x1, x2, x3, x4 ≥ 0

2.3 Two-Phase method

1. Solve the following LP problem by using the Two-Phase method.
Minimize Z = x1 + x2
subject to the constraints
2x1 + 4x2 ≥ 4
x1 + 7x2 ≥ 7
and x1, x2 ≥ 0

2. Solve the following LP problem by using the Two-Phase method.
Minimize Z = x1 - 2x2 - 3x3
subject to the constraints
-2x1 + 3x2 + 3x3 = 2
2x1 + 3x2 + 4x3 = 1
and x1, x2, x3 ≥ 0

2.4 Dual Simplex method

2.5 Gomorys Integer Cutting method

1. Solve the following integer programming problem using Gomory's cutting plane algorithm.
Maximize Z = x1 + x2
subject to the constraints
3x1 + 2x2 ≤ 5
x2 ≤ 2
and x1, x2 ≥ 0 and are integers.

2. Solve the following integer programming problem using Gomory's cutting plane algorithm.
Maximize Z = 2x1 + 20x2 - 10x3
subject to the constraints
2x1 + 20x2 + 4x3 ≤ 15
6x1 + 20x2 + 4x3 ≤ 20
and x1, x2, x3 ≥ 0 and are integers.

2.6 Graphical method

1. Use graphical method to solve following LP problem.
Maximize Z = x1 + x2
subject to the constraints
3x1 + 2x2 ≤ 5
x2 ≤ 2
and x1, x2 ≥ 0

2. Use graphical method to solve following LP problem.
Maximize Z = 2x1 + x2
subject to the constraints
x1 + 2x2 ≤ 10
x1 + x2 ≤ 6
x1 - x2 ≤ 2
x1 - 2x2 ≤ 1
and x1, x2 ≥ 0

2.7 Primal to dual conversion

1. Write the dual to the following LP problem.
Maximize Z = x1 - x2 + 3x3
subject to the constraints
x1 + x2 + x3 ≤ 10
2x1 - x2 - x3 ≤ 2
2x1 - 2x2 - 3x3 ≤ 6
and x1, x2, x3 ≥ 0

2. Write the dual to the following LP problem.
Minimize Z = 3x1 - 2x2 + 4x3
subject to the constraints
3x1 + 5x2 + 4x3 ≥ 7
6x1 + x2 + 3x3 ≥ 4
7x1 - 2x2 - x3 ≤ 10
x1 - 2x2 + 5x3 ≥ 3
4x1 + 7x2 - 2x3 ≥ 2
and x1, x2, x3 ≥ 0

2.8 Branch and Bound method

1. Solve the following LP problem by using Branch and Bound method
Max Z = 7x1 + 9x2
subject to
-x1 + 3x2 ≤ 6
7x1 + x2 ≤ 35
x2 ≤ 7
and x1,x2 ≥ 0

2. Solve the following LP problem by using Branch and Bound method
Max Z = 3x1 + 5x2
subject to
2x1 + 4x2 ≤ 25
x1 ≤ 8
2x2 ≤ 10
and x1,x2 ≥ 0

2.9 0-1 Integer programming problem

1. Solve LP using zero-one Integer programming problem method
Max Z = 300x1 + 90x2 + 400x3 + 150x4
subject to
35000x1 + 10000x2 + 25000x3 + 90000x4 ≤ 120000
4x1 + 2x2 + 7x3 + 3x4 ≤ 12
x1 + x2 ≤ 1
and x1,x2,x3,x4 ≥ 0

2. Solve LP using 0-1 Integer programming problem method
MAX Z = 650x1 + 700x2 + 225x3 + 250x4
subject to
700x1 + 850x2 + 300x3 + 350x4 ≤ 1200
550x1 + 550x2 + 150x3 + 200x4 ≤ 700
400x1 + 350x2 + 100x3 ≤ 400
x1 + x2 ≥ 1
-x3 + x4 ≤ 1
and x1,x2,x3,x4 ≥ 0

2.9 Revised Simplex method

1. Solve the following LP problem by using Revised Simplex method
MAX Z = 3x1 + 5x2
subject to
x1 ≤ 4
x2 ≤ 6
3x1 + 2x2 ≤ 18
and x1,x2 ≥ 0

2. Solve the following LP problem by using Revised Simplex method
MAX Z = 2x1 + x2
subject to
3x1 + 4x2 ≤ 6
6x1 + x2 ≤ 3
and x1,x2 ≥ 0

3. Transportation Problem using
1. North-West corner method
2. Least cost method
3. Vogel's approximation method
4. Row minima method
5. Column minima method
6. Russell's approximation method
7. Heuristic method-1
8. Heuristic method-1
9. optimal solution by MODI method
10. Optimal solution using stepping stone method

1. A Company has 3 production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100's) per week of a product, respectively. These units are tobe shipped to 4 warehouses D1, D2, D3 and D4 with requirement of 5,6,7 and 14 units (in 100's) per week, respectively. The transportation costs (in rupees) per unit between factories to warehouses are given in the table below.

	D₁	D₂	D₃	D₄	Capacity
S₁	19	30	50	10	7
S₂	70	30	40	60	9
S₃	40	8	70	20	18
Demand	5	8	7	14	34

Find initial basic feasible solution for given problem by using
(a) North-West corner method
(b) Least cost method
(c) Vogel's approximation method
(d) obtain an optimal solution by MODI method
if the object is to minimize the total transportation cost.

2. Find an initial basic feasible solution for given transportation problem by using
(a) North-West corner method
(b) Least cost method
(c) Vogel's approximation method

	D₁	D₂	D₃	D₄	Supply
S₁	11	13	17	14	250
S₂	16	18	14	10	300
S₃	21	24	13	10	400
Demand	200	225	275	250

3. A company has factories at F1, F2 and F3 which supply to warehouses at W1, W2 and W3. Weekly factory capacities are 200, 160 and 90 units, respectively. Weekly warehouse requiremnet are 180, 120 and 150 units, respectively. Unit shipping costs (in rupess) are as follows:

	W₁	W₂	W₃	Supply
F₁	16	20	12	200
F₂	14	8	18	160
F₃	26	24	16	90
Demand	180	120	150	450

Determine the optimal distribution for this company to minimize total shipping cost.

4. Find an initial basic feasible solution for given transportation problem by using
(a) North-West corner method
(b) Least cost method
(c) Vogel's approximation method

	P	Q	R	S	Supply
A	6	3	5	4	22
B	5	9	2	7	15
C	5	7	8	6	8
Demand	7	12	17	9	45

4. PERT and CPM

1. An assembly is to be made from two parts X and Y. Both parts must be turned on a lathe Y must be polished where as X need not be polished. The sequence of acitivities, together with their predecessors, is given below

Activity	Description	Predecessor Activity
A	Open work order	-
B	Get material for X	A
C	Get material for Y	A
D	Turn X on lathe	B
E	Turn Y on lathe	B,C
F	Polish Y	E
G	Assemble X and Y	D,F
H	Pack	G

Draw a network diagram of activities for the project.

2. An established company has decided to add a new product to its line. It will buy the product from a manufacturing concern, package it, and sell it to a number of distributors that have been selected on a geographical basis. Market research has already indicated the volume expected and the size of sales force required. The steps shown in the following table are to be planned.

Activity	Description	Predecessor Activity	Duration (days)
A	Organize sales office	-	6
B	Hire salesman	A	4
C	Train salesman	B	7
D	Select advertising agency	A	2
E	Plan advertising campaign	D	4
F	Conduct advertising campaign	E	10
G	Design package	-	2
H	Setup packaging campaign	G	10
I	Package initial stocks	J,H	6
J	Order stock from manufacturer	-	13
K	Select distributors	A	9
L	Sell to distributors	C,K	3
M	Ship stocks to distributors	I,L	5

(a) Draw an arrow diagram for the project.
(b) Indicate the criticla path.
(c) For each non-critical activity, find the total and free float.

5. Sequencing Problems

1. There are seven jobs, each of which has to go through the machines A and B in the order AB. Processing times in hours are as follows.

Job	1	2	3	4	5	6	7
Machine A	3	12	15	6	10	11	9
Machine B	8	10	10	6	12	1	3

Decide a sequence of these jobs that will minimize the total elapsed time T. Also find T and idle time for machines A and B.

2. Find the sequence that minimizes the total time required in performing the following job on three machines in the order ABC. Processing times (in hours) are given in the following table.

Job	1	2	3	4	5
Machine A	8	10	6	7	11
Machine B	5	6	2	3	4
Machine C	4	9	8	6	5

6. Replacement and Maintenance Models

Model-1.1

1. A firm is considering the replacement of a machine, whose cost price is Rs 12,200 and its scrap value is Rs 200. From experience the running (maintenance and operating) costs are found to be as follows:

Year	1	2	3	4	5	6	7	8
Running Cost	200	500	800	1,200	1,800	2,500	3,200	4,000

When should the machine be replaced?

Model-1.2

1. The data collected in running a machine, the cost of which is Rs 60,000 are given below:

Year	1	2	3	4	5
Resale Value	42,000	30,000	20,400	14,400	9,650
Cost of spares	4,000	4,270	4,880	5,700	6,800
Cost of labour	14,000	16,000	18,000	21,000	25,000

Determine the optimum period for replacement of the machine.

Model-1.3

1. Machine A costs Rs 45,000 and its operating costs are estimated to be Rs 1,000 for the first year increasing by Rs 10,000 per year in the second and subsequent years. Machine B costs Rs 50,000 and operating costs are Rs 2,000 for the first year, increasing by Rs 4,000 in the second and subsequent years. If at present we have a machine of type A, should we replace it with B? if so when? Assume that both machines have no resale value and their future costs are not discounted.

Model-2
Replacement policy for items whose running cost increases with time but value of money changes constant rate during a period

1. An engineering company is offered a material handling equipment A. It is priced at Rs 60,000 includeing cost of installation. The costs for operation and maintenance are estimated to be Rs 10,000 for each of the first five years, increasing every year by Rs 3,000 in the sixth and subsequent years. The company expects a return of 10 percent on all its investment. What is the optimal replacement period?

Year	1	2	3	4	5	6	7
Running Cost	10,000	10,000	10,000	10,000	10,000	13,000	16,000

2. A company is buying mini computers. It costs Rs 5 lakh, and its running and maintenance costs are Rs 60,000 for each of the first five years, increasing by Rs 20,000 per year in the sixth and subsequent years. If the money is worth 10 percent per year, What is the optimal replacement period?

Model-3
Group replacement policy

1. A computer contains 10,000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same time, the cost per resistor would be reduced to 35 paise. The percentage of surviving resistors say S(t) at the end of month t and the probability of failure P(t) during the month t are as follows:

t	0	1	2	3	4	5	6
P(t)	0	0.03	0.07	0.20	0.40	0.15	0.15

What is the optimal replacement plan?

2. The following mortality rates have been observed for a certain type of fuse:

t	0	1	2	3	4	5
P(t)	0	0.05	0.10	0.20	0.40	0.25

There are 1,000 fuses in use and it costs Rs 5 to replace an individual fuse. If all fuses were replaced simultaneously it would cost Rs 1.25 per fuse. It is proposed to replace all fuses at fixed intervals of time, whether or not they have burnt out, and to contiune replacing burnt out fuses as they fail. At what time intervals should the group replacement be made? Also prove that this optimal policy is superior to the straight forward policy of replacing each fuse only when it fails.

7. Game Theory
1. Saddle Point
2. Dominance method
3. Algebraic method
4. Calculus method
5. Arithmetic method
6. Matrix method
7. 2Xn Games
8. Graphical method
9. LPP method
10. Bimatrix method

7.1 Saddle Point

1. For the game with payoff matrix

		Player B
		B1	B2	B3
Player A	A1	-1	2	-2
Player A	A2	6	4	-6

determine the best strategies for players A and B. Also determine the value of game. Is this game saddle point?

7.2 Dominance method

1. Dominance Example

		Player B
		B1	B2	B3	B4
Player A	A1	3	5	4	2
	A2	5	6	2	4
	A3	2	1	4	0
	A4	3	3	5	2

7.3 Algebraic method

1. Find the solution of game using algebraic method for the following pay-off matrix

		Player B
		B1	B2
Player A	A1	1	7
Player A	A2	6	2

7.4 Calculus method

1. Find the solution of game using calculus method for the following pay-off matrix

		Player B
		B1	B2
Player A	A1	1	3
Player A	A2	5	2

7.5 Arithmetic method

1. Find the solution of game using arithmetic method for the following pay-off matrix

		Player B
		B1	B2	B3
Player A	A1	10	5	-2
	A2	13	12	15
	A3	16	14	10

7.6 Matrix method

1. Find the solution of game using matrix method for the following pay-off matrix

		Player B
		B1	B2	B3
Player A	A1	1	7	2
	A2	6	2	7
	A3	5	1	6

7.7 2Xn Games

1. Find the solution of game using 2Xn Games method for the following pay-off matrix

		Player B
		B1	B2
Player A	A1	-3	4
	A2	-1	1
	A3	7	-2

7.8 Graphical method

1. Find the solution of game using graphical method method for the following pay-off matrix

		Player B
		B1	B2
Player A	A1	1	-3
	A2	3	5
	A3	-1	6
	A4	4	1
	A5	2	2
	A6	-5	0

7.9 LPP method

1. Find the solution of game using linear programming method for the following pay-off matrix

		Player B
		B1	B2	B3
Player A	A1	3	-4	2
	A2	1	-7	-3
	A3	-2	4	7

7.10 Bimatrix method

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Word Problems Calculators (examples)

1. Arithmetic Progression
2. Geometric Progression
3. Simple Interest
4. Compound Interest
5. HCF-LCM Word Problem
6. Installment
7. Percentage
8. Profit Loss Discount
9. Statistics Word Problem
10. Problems on Ages
11. Unitary Method (Chain Rule)

Word Problems with example

1. Arithmetic Progression

1. For given arithemetic progression series 1,4,7,10,13 ,... find 10 th term and addition of first 10 th terms.
Here first term a = 1,
d = 4 - 1 = 3
We know that, f(n) = a + (n - 1)d
f(10) = 1 + (10 - 1)(3)
= 1 + (27)
= 28
We know that, S_n = ⁿ/₂ [2a + (n - 1)d]
∴ S_10 = 10/2 [2(1) + (10 - 1)(3)]
= 5 [2 + (27)]
= 5 [29]
= 145
Hence, 10th Term Of The Given Series is 28 And Sum of First 10th Term is 145

2. For arithemetic progression addition of 3 terms is 27 and their multiplication is 648, then that nos.

3. Find the sum of all natural nos between 100 to 200 and which are not divisible by 4.

4. For arithemetic progression, addition of three terms is 51 and multiplication of end terms is 273, then find that nos.

5. For arithemetic progression, addition of 4 terms is 4 and addition of multiplication of end terms and multiplication of middle terms is -38, then find that nos.

2. Geometric Progression

1. For given geometric progression series 3,6,12,24,48,... find 10 th term and addition of first 10 th terms.
Here a = 3,
r = 6 / 3 = 2
We know that, a_n = a * r^(n-1)
a_10 = 3 * (2)^(10 - 1)
= 3 * (512)
= 1536
We know that, S_n = a (r^n - 1)/(r - 1)
∴ S_10 = 3 * ((2)^10 - 1) / (2 - 1)
=> S_10 = 3 * (1024 - 1) / 1
=> S_10 = 3 * 1023 / 1
=> S_10 = 3069
Hence, 10th term of the given series is 1536 and sum of first 10th term is 3069

2. For geometric progression addition of 3 terms is 26 and their multiplication is 216, then that nos.

3. For geometric progression multiplication of 5 terms is 1 and 5th term is 81 times then the 1st term.

4. Arithmetic mean of two no is 13 and geometric mean is 12, then find that nos.

5. Find 6 geometric mean between 1 and 256.

6. Find 1² + 2² + 3² + ... + 10²

7. Find 1³ + 2³ + 3³ + ... 10 terms

3. Simple Interest

1. Find the simple intereset on Rs. 730 for 184 days at 25/4 % per annum.
Here P = Rs. 730 , R = 25/4 % and Time = 184 days = 184 / 365 years
S.I. = P*R*N/100 = ( 730 * 25/4 * 184 /365) / 100 = 23
Simple Interest is Rs. 23 .

2. The interest on a certain amount of money at 8% per year for a period of 4 years is Rs 512. Find the sum of money.

3. A sum of money lent at simple interest amounts to Rs 1596 in 3/2 years and to 1860 in 5/2 years. Find the sum & the rate of interest.

4. A sum was put at simple interest at a certain rate for 2 years. Had it been put at 3 % highere rate, it whould have fetched Rs. 300 more. Find the sum.

5. A shopkeeper borrowed Rs. 20000 from two money lenders. For one loan he paid 12% and for the other 14% per annum. After one year, he paid Rs. 2560 as interest. How much did he borrow at each rate ?

6. At what rate percent per annum will sum of money double in 8 years?

7. Rajeev deposited money in the post office which is doubled in 20 years at a simple rate of interest. In how many years will the original sum triple itself?

4. Compound Interest

1. Calculate Compound Interest and amount on Rs 4500 at 10 % per annum in 3 years.

Here P = Rs. 4500 , R = 10 %, N = 3 yrs.
A = P (1 + R/100)^N
= 4500 * ( 1 + 10 / 100 ) ^ 3
= 4500 * ( 110 / 100 ) ^ 3
= 5989.5

Therefore, C.I. = Rs. ( 5989.5 - 4500 ) = Rs. 1489.5 .

2. What sum of money becomes Rs. 9261 in 3 years at 5% per annum, compounded annually ?

3. A sum of money amounts to Rs 6690 after 3 years and to Rs 10035 after 6 years on compound interest. Calculate the sum of money.

4. The difference between the compound interest and the simple interest on a certain sum at 10 % per annum for 2 years is Rs. 52 . Find the sum.

5. If the compound interest on a certain sum for 3 years at 10 % per annum be Rs. 331, what would be the simple interest ?

6. A sum of money 2 times itself at compound interest in 15 years. In how many years, it will become 8 times of itself ?

5. HCF-LCM Word Problem

1. The HCF of two nos is 14 and their LCM is 11592 . If one of the nos is 504 , find the other?

Other no = ( HCF * LCM ) / Given No = ( 14 * 11592 ) / 504 = 322 .

2. Find the largest no which can exactly divide 513 , 783 , 1107

3. Find the smallest no exactly divisible by 12 , 15 , 20 and 27 .

4. Find the least no which when divided by 6 , 7 , 8 , 9 , 12 leaves the same remainder 2 in each case.

5. Find the largest no which divides 77 , 147 , 252 to leave the same remainder in each case.

6. The greatest no that will divide 290 , 460 , 552 leaving respectively 4 , 5 , 6 as remainder.

7. LCM of two nos is 14 times their HCF. The sum of LCM and HCF is 600 . If one no is 280 , then find the other no ?

6. Installment

1. A briefcase is available for Rs 440 cash or for Rs 200 cash down payment and Rs 244 to be paid after 1 months. Find the rate of intereset charged under the installment plan.

Cash Price = 440
Down Payemnt = 200
Remaining Balance = 440 - 200 = 240
The installment to be paid at the end of 1 months = 244
Therefore the interest charged on Rs 240, for a period of 1 months = Rs 244 - Rs 240 = 4
If R % is the rate of interest per annum, then
(240 × R × 1) / (100 × 12) = 4
R = 20
Thus, the rate of interest charged under the installment plan is = 20 per annum

2. A washing machine is available at Rs 6400 cash or for Rs 1400 cash down payment and 5 monthly installments of Rs 1030 each. Calculate the rate of interest charged under the instalment plan.

3. A computer is sold by a company for Rs 19200 cash or for Rs 4800 cash down payment together with 5 equal monthly installments. If the rate of interest charged by the company is 12% per annum, find each installment.

4. A man borrows money from a finance company and has to pay it back in 2 equal half yearly installments of Rs 4945 each. If the interest is charged by the finance company at the rate of 15 % per annum compounded as installment plan, find the principal and the total interest paid.

5. Ram borrowed a sum of money and returned it in 3 equal quarterly installments of Rs 17576 each. Find the sum borrowed, if the rate of interest charged was 16 % per annum compounded as installment plan. Find also the total interest charged.

7. Percentage

1. A number is 20 % of 80 .

20 % of 80 = ( 20 / 100 × 80 ) = 16 .

2. A's annual income is increased from Rs 60000 to Rs 75000 . Find the percentage of increase in A's income.

3. In a school of 225 boys, 15 were absent then what percent were present ?

4. A earns 25 % more than B. By what percent does B earn less then A.

5. A reduction of 20 % in the price of basmati rice would enable a man to buy 2 kg of rice more for Rs 250. Find the reduced price per kg.

6. Find the selling price of an item, of which the printed price is Rs 25000 if the successive discounts given are 10 %, 8 % and 4 %.

7. The successive discount of 10 % and 5 % are given on the purchased Computer. If the final price of the Computer is Rs 10260, then find the printed price of the Computer.

8. Profit Loss Discount

1.Find selling price when cost price = Rs. 50 , Loss/Gain = 10 % .

C.P. = Rs. 50 , Gain = 10 %
S.P. = C.P. × ^{( 100 + Gain% )} / ₁₀₀ = 50 × ^{( 100 + 10 )} / ₁₀₀= 55 Rs.

C.P. = Rs. 50 , Loss = 10 %
S.P. = C.P. × ^{( 100 - Loss% )} / ₁₀₀ = 50 × ^{( 100 - 10 )} / ₁₀₀= 45 Rs.

2. Find cost price when selling price = 55 , Loss/Gain = 10 %

3. Find Loss/Gain % when cost price = 50 and selling price = 55

4. By selling an article for Rs 110 a man loses 12 %. For how much should he sell it to gain 8 %.

5. A man sells two houses for Rs 536850 each. On one he gains and on the other he loses 5 %. Find his gain or loss % on the whole transaction.

6. If selling price of 10 articles is the same as the cost price of 12 articles, find the gain %.

7. If the marked price of an article is Rs. 380 and a discount of 5 % is given on it, what is the selling price ?

8. A cycle dealer marks his goods 25 % above his cost price and allows a discount of 8 % on it. Find his gain percent.

9. Successive discounts of 20 % and 10 % equivalent to a single discount of how many percent?

9. Statistics Word Problem

1. The mean of 10 observations is 12.5 . While calculating the mean one observation was by mistake taken as (-8) instead of (+8) . Find the correct mean.
Here Mean X = 12.5 and n=10
Σ(X) = X × n = 12.5 × 10 = 125
∴ the correct sum Σ(X) = 125 - (Wrong Observation) + (Correct Observation)
= 125 - (-8) + (8) = 141 .
∴ correct mean = ^{correct sum} / _n = ¹⁴¹ / ₁₀ = 14.1

2. The sum of 15 observation is 343 . If we remove two observation 18 and 26 , then find out the mean of remaining observations.

Here Σ(X) = 343 and n= 15
If we remove two observation then 13 observation left.
Sum of 13 observation Σ(X') = 343 - ( ( 18 ) + ( 26 ) ) = 299
Then the mean of remaining 13 observations = ^Σ(X') / _n= ²⁹⁹/ ₁₃= 23. and many more...

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Calculus Calculators (examples)

1. Trigonometry
1. Simplifying trigonometric equations, proving identities, eg. (1) sin(30)⋅cos(60)+sin(30)⋅cos(60), (2) tan(x)+cot(x), (3) tan2(x)-sin2(x), (4) csc(x)cos(x)-cos(x)sin(x)
2.1 If sin(x)=513 then Find the value of all the other five trigonometric functions
2.2 If sin(x)=35 then solve expression cos(x)csc(x)+tan(x)sec(x)
3 For P(3,4), find the value of all six trigonometric functions

2. Derivative
1. Find Derivative, eg. (1) ddx(2x3-2x2-5x+4), (2) ddx(log(x)ex), (3) ddx(3cos(sin(x)))
2. Find maximum and minimum value of y=x3+6x2-15x+7

Home > Geometry calculators

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Geometry Calculators (examples)

1. Coordinate Geometry

2. Graphs
1. Graphs - Plotting of mathematical functions f(x) or f(y)

2. Graph - Using Points and Slope
2.1 Points : (3,-5),(5,3)
2.2 Points on X-axis : -3,5,12,-7,0
2.3 Points on Y-axis : -3,5,12,-7,0
2.4 Lines : x=-3y+5; 2x+y=1; x+2y<=5; x+y>=15
2.5 Line using Slope & point : Slope=7 and Point=(4,6)
2.6 Line using Slope & Y-Intercept : Slope=2 and Y-Intercept=-4
2.7 Line passing through two points : Point1=(3,-5) and Point2=(5,3)

3. Circle
4. Ellipse
5. Parabola
6. Hyperbola
7. Polar Graph

3. Area
1. Area of a Circle
2. Area of a Semi-Circle
3. Area of a Regular Hexagon
4. Area of a Square
5. Area of a Rectangle
6. Area of a Parallelogram
7. Area of a Rhombus
8. Area of a Trapezium
9. Area of a Scalene Triangle
10. Area of a Rightangle Triangle
11. Area of a Equilateral Triangle
12. Area of a Isosceles Triangle
13. Area of a Sector Segment

4. Volume
1. Volume of a Cuboid
2. Volume of a Cube
3. Volume of a Cylinder
4. Volume of a Cone
5. Volume of a Sphere
6. Volume of a Hemi-Sphere

5. Pythagorean Theorem

Geometry with example

2. Graph

1. General Graph

1. y=x
2. y≤sin(x)
3. y≥cos(x)
4. y=√x
5. y≤√x3-3x
6. y≥|x|
7. y=x3-x
8. y≤ex-e-x2

1. x=sin(y)
2. x≤cos(y)
3. x≥√y
4. x=√y3-3y
5. x≤|y|
6. x≥sin(y)3+cos(y)3
7. x=y+sin(y)
8. x≤(sin(9y)+sin(10y))⋅sin(0.1y)

2. Graph - Using Points and Slope

1. Points
like (3,-5),(5,3)

2. Points on X-axis
like -3,5,12,-7,0

3. Points on Y-axis
like -3,5,12,-7,0

4. Lines
like x=-3y+5; 2x+y=1; x+2y<=5; x+y>=15

5. Line using Slope & point
like Slope=7 and Point=(4,6)

6. Line using Slope & Y-Intercept
like Slope=2 and Y-Intercept=-4

7. Line passing through two points
like Point1=(3,-5) and Point2=(5,3)

3. Circle

1. Circle-1
X2+Y2=9

2. Circle-2
(X+1)2+Y2=12

3. Circle-3
X2+(Y-2)2=15

4. Circle-4
(X+1)2+(Y-2)2=9

4. Ellipse

1. Ellipse-1
X24+Y29=9

2. Ellipse-2
(X+1)24+Y29=12

3. Ellipse-3
X24+(Y-2)29=15

4. Ellipse-4
(X+1)24+(Y-2)29=9

5. Parabola

1. Parabola-1
Y=3X2

2. Parabola-2
Y=3X2+1

3. Parabola-3
Y=3(X+1)2

4. Parabola-4
Y=3(X+1)2+1

5. Parabola-5
X=3Y2

6. Parabola-6
X=3Y2+1

7. Parabola-7
X=3(Y+1)2

8. Parabola-8
X=3(Y+1)2+1

6. Hyperbola

1. Hyperbola-1
X24-Y29=9

2. Hyperbola-2
2.(X+1)24-Y29=12

3. Hyperbola-3
X24-(Y-2)29=15

4. Hyperbola-4
(X+1)24-(Y-2)29=9

5. Hyperbola-5
Y24-X29=9

6. Hyperbola-6
(Y+1)24-X29=12

7. Hyperbola-7
Y24-(X-2)29=15

8. Hyperbola-8
(Y+1)24-(X-2)29=9

7. Polar Graph

1. R=4⋅cos(2⋅t)
2. R=4⋅sin(2⋅t)
3. R=2-4⋅sin(2⋅t)
4. R=2-4⋅cos(2⋅t)
5. R=2+4⋅cos(2⋅t)
6. R=2+4⋅sin(2⋅t)

8. Statistics Graph

1. Histogram
2. Frequency Polygon
3. Frequency Curve
4. Less than type cumulative frequency curve
5. More than type cumulative frequency curve

1. The frequency distribution of the marks obtained by 100 students in a test of Mathematics carrying 50 marks is given below.
Draw Histogram, Frequency Polygon, Frequency Curve, Less than type cumulative frequency curve and More than type cumulative frequency curve of the data.

Marks obtained	0 - 9	10 - 19	20 - 29	30 - 39	40 - 49
number of students	8	15	20	45	12

1. Histogram	2. Frequency Polygon

3. Frequency Curve	4. Less than type cumulative frequency curve

5. More than type cumulative frequency curve

3. Area

1. Circle

Area (A)=πr2
Circumference (C)=2πr=πd
Diameter (d)=2r

I know that for a circle Radius = 10 . From this find out Area of the circle.

Here radius(r)=10(Given)

Diameter(d)=2⋅r

=2⋅10

=20

Perimeter=2πr

=2⋅227⋅10

=4407

Area=πr2

=227⋅(10)2

=22007

2. Semi-Circle

Area (A)=12πr2
Circumference (C)=πr=πd2
Perimeter (P)=πr+2r
Diameter (d)=2r

I know that for a Semi-Circle Radius = 10 . From this find out Area of the Semi-Circle.

Here radius(r)=10(Given)

Diameter(d)=2r

=2⋅10

=20

Circumference=πr

=227⋅10

=2207

Perimeter=πr+2r

=227⋅10+2⋅10

=2207+20

=3607

Area=πr22

=227⋅(10)22

=11007

3. Regular Hexagon

Perimeter (P)=6a
Area (A)=√34×6×a2

I know that for a Regular Hexagon Side = 10 . From this find out Area of the Regular Hexagon.

Here Side(a)=10(Given)

Perimeter=6a

=6⋅10

=60

Area=√34⋅6⋅a2

=√34⋅6⋅102

=259.8076

4. Square

Diagonal (d)=√2a
Perimeter (P)=4a
Area (A)=a2=d22

I know that for a square Side(a) = 10 . From this find out Area of the square.

Here a=10(Given)

Diagonal=√2⋅a

=√2⋅10

=14.1421

Perimeter=4⋅a

=4⋅10

=40

Area=a2

=102

=100

5. Rectangle

Diagonal (d)=√l2+b2
Perimeter (P)=2(l+b)
Area (A)=lb

I know that for a rectangle Length = 5 and Breadth = 12 . From this find out Area of the rectangle.

Here one-Side(l)=5and other-Side(b)=12(Given)

Diagonal=√l2+b2

=√52+122

=√25+144

=√169

=13

Perimeter=2⋅(sum of Sides)

=2⋅(5+12)

=34

Area = Product of Sides

=5⋅12

=60

6. Parallelogram

Area (A)=ah
Perimeter (P)=2a+2b

I know that for a parallelogram a = 9 , b = 22 and h = 14 . From this find out Area of the parallelogram.

Herea=9,b=22,h=14(Given)

Perimeter=2⋅(a+b)

=2⋅(9+22)

=62

Area=a⋅h

=9⋅14

=126

7. Rhombus

Radius (r1)=d12
Radius (r2)=d22
Side (a)=√r21+r22
Perimeter (P)=4a
Area (SA)=d1d22

I know that for a rhombus d1 = 10 and d2 = 24 . From this find out Area of the rhombus.

Here, we haved1=10andd2=24(Given)

a2=(d12)2+(d22)2

a2=(102)2+(242)2

a2=(5)2+(12)2

a2=169

a=13

Perimeter=4⋅a

=4⋅13

=52

Area=12(Product of diagonals)

=12⋅d1⋅d2

=12⋅10⋅24

=120

8. Trapezium

Area (A)=h2(a+b)
Perimeter (P)=a+b+c+d

I know that for a trapezium a = 22 , b = 18 , c = 16 , d = 16 and h = 4 . From this find out Area of the trapezium.

Herea=22,b=18,c=16,d=16,h=4(Given)

Perimeter=a+b+c+d

=22+18+16+16

=72

Area=(a+b)⋅h2

=(22+18)⋅42

=80

Area >> Triangle

9. Scalene Triangle

Perimeter (P)=a+b+c
S=P2=a+b+c2
Area (A)=√S(S-a)(S-b)(S-c)

I know that for a scalene Triangles a = 3 , b = 4 and c = 5 . From this find out Area of the scalene Triangles.

Herea=3,b=4,c=5(Given)

We know that,

Perimeter=a+b+c

=3+4+5

=12

Semi-Perimeter=s=a+b+c2

=122

=6

Herea=3,b=4,c=5and semi-Perimeter=6

We know that,

Area=√s(s-a)(s-b)(s-c)

=√6(6-3)(6-4)(6-5)

=6

10. Right angle Triangle

Diagonal (d)=√a2+b2
Perimeter (P)=a+b+c
Area (A)=12(ab)

I know that for a right angle Triangles AB = 5 and BC = 12 . From this find out Area of the right angle Triangles.

Here one Side=5and other Side=12(Given)

We know that,

In triangle ABC, by Pythagoras' theorem

AC2=AB2+BC2

AC2=52+122

AC2=25+144

AC2=169

AC=13

Perimeter=AB+BC+AC

=5+12+13

=30

Here base=5and height =12

We know that,

Area=12⋅AB⋅BC

=12⋅5⋅12

=30

11. Equilateral Triangle

Perimeter (P)=3a
Area (A)=√34a2

I know that for a equilateral Triangles Side = 6 . From this find out Area of the equilateral Triangles.

Herea=6(Given)

We know that,

Perimeter=3⋅a

=3⋅6

=18

We know that,

Area=√34⋅a2

=1.7324⋅6⋅6

=15.5885

12. Isosceles Triangle

Height (h)=√a2-b24
Perimeter (P)=2a+b
Area (A)=bh2

I know that for a isosceles Triangles a = 5 and b = 6 . From this find out Area of the isoceles Triangles.

Here base(b)=6and equal side(a)=5(Given)

We know that,

Perimeter=(2⋅equal Side)+third Side

=(2⋅a)+b

=(2⋅5)+6

=16

We know that,

Area=12⋅base⋅height

=12⋅b⋅√a2-b24

=12⋅6⋅√52-624

=12⋅6⋅4

=12

13. Sector Segment

Length of the arc =l=πrθ180
Area of a minor sector =πr2θ360

I know that for a sector & segment Radius = 10 and angle of measure = 180 . From this find out length of arc of the sector & segment.

Herer=10andθ=180(Given)

Length of the arc=l=πrθ180

=227⋅10⋅180180

=31.4286

Area of a minor sector=πr2θ360

=227⋅102⋅180360

=157.1429

4. Volume

1. Cuboid

Diagonal (d)=√l2+b2+h2
Surface Area (SA)=2(lb+bh+hl)
Volume (V)=lbh

I know that for a cuboid Length = 3 , Breadth = 4 , and Height = 5 . From this find out Volume of the cuboid.

Here, we havel=3,b=4,h=5(Given)

Diagonal2=l2+b2+h2

Diagonal2=32+42+52

Diagonal2=50

Diagonal=7.0711

Here, we havel=3,b=4,h=5(Given)

Volume=l⋅b⋅h

=3⋅4⋅5

=60

Here, we havel=3,b=4,h=5(Given)

Total Surface Area=2(lb+bh+lh)

=2⋅(3⋅4+4⋅5+3⋅5)

=2⋅(47)

=94

Here, we havel=3,b=4,h=5(Given)

Curved Surface Area=2h(l+b)

=2⋅5(3+4)

=70

2. Cube

diameter (d)=√2l
Diagonal =√3l
Surface Area (SA)=6l2
Volume (V)=l3

I know that for a cube Length = 3 . From this find out Volume of the cube.

Here, we havel=3

Diagonal=√3l

=√3⋅3

=5.1962

Here, we havel=3

Volume=l3

=33

=27

Here, we havel=3

Total Surface Area=6l2

=6⋅32

=6⋅9

=54

Here, we havel=3

Curved Surface Area=4l2

=4⋅32

=4⋅9

=36

3. Cylinder

Curved Surface Area (CSA)=2πrh
Total Surface Area (TSA)=2πr(r+h)
Volume (V)=πr2h

I know that for a cylinder Radius = 3 and Height = 10 . From this find out Curved Surface Area of the cylinder.

Here, we have Radius(r)=3and Height(h)=10(Given)

Volume=πr2h

=π⋅32⋅10

=282.7433

Total Surface Area=2πr(r+h)

=2⋅π⋅3⋅(3+10)

=245.0442

Curved Surface Area=2πrh

=2⋅π⋅3⋅10

=188.4956

4. Cone

Height (h)=√l2-r2
Curved Surface Area (CSA)=πrl
Total Surface Area (TSA)=πr(l+r)
Volume (V)=πr2h3

I know that for a cone Radius = 3 and Length = 5 . From this find out Volume of the cone.

Here, we have Radius(r)=3and Slant Height(l)=5(Given)

l2=r2+h2

h2=l2-r2

h2=52-32

h2=16

h=4

Volume=πr2h3

=π⋅32⋅43

=37.6991

Total Surface Area=πr(l+r)

=π⋅3(5+3)

=75.3982

Curved Surface Area=πrl

=π⋅3⋅5

=47.1239

5. Sphere

Surface Area (SA)=4πr2
Volume (V)=43πr3
diameter (d)=2r

I know that for a sphere Radius = 3 . From this find out Volume of the sphere.

Here, we have Radius(r)=3(Given)

Volume=43πr3

=43⋅π⋅33

=84.823

Total Surface Area=4πr2

=4⋅π⋅32

=113.0973

Curved Surface Area=4πr2

=4⋅π⋅32

=113.0973

6. Hemi-Sphere

Curved Surface Area (CSA)=2πr2
Total Surface Area (TSA)=3πr2
Volume (V)=23πr3
diameter (d)=2r

I know that for a Hemi-Sphere Radius = 3 . From this find out Volume of the Hemi-Sphere.

Here, we have Radius(r)=3(Given)

Volume=23πr3

=23⋅π⋅33

=56.5487

Total Surface Area=3πr2

=3⋅π⋅32

=84.823

Curved Surface Area=2πr2

=2⋅π⋅32

=56.5487

5. Pythagoras Theorem

Pythagorean Theorem : In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the remaining sides.

i.e. AC2=AB2+BC2

Using Pythagoras Theorem, Find out AC when AB = 5 and BC = 12
Here AB=5 and BC=12 (Given)
We know that,
In triangle ABC, by Pythagoras' theorem
AC2=AB2+BC2
AC2=52+122
AC2=25+144
AC2=169
AC=13

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Pre-Algebra Calculators