Introduction to 4th grade geometry problems

In general, 4th grade geometry problems includes types of polygons, area and perimeter of the polygons. Here we are going to see about  4th grade geometry problems on area and perimeter.  Area (A) is a two-dimensional measure. The area is measured in terms of square units such as square inches, square feet and square centimeters. Hectares and acres are also considered in some special cases. The entire boundary of a figure or the distance around the figure is called as the perimeter. For circle it is called as the circumference, such lengths are measured with inches, feet, and centimeters.

4th Grade Geometry Problems in Square:

Formula to find area and perimeter:

Area of square = side x side square unit.

Area of square (A) = a^2 square units,   (a is the side length of the square)

Perimeter of the square = 4 x side length.

Perimeter of the square (P)=4 x a.

1. Find the area and perimeter of the square, whose side length is 5 meters.

Sol:

Area of square =a^2

= 5 x 5

Area of square =25m^2

Perimeter of the square = 4 x a

=4 x 5

Perimeter of the square  = 20 meters

2. Find the area and perimeter of the square, whose side length is 12 feet.

Sol:

Area of square   =a^2

= 12 x 12

Area of square  =144ft^2

Perimeter of the square    = 4 x a

=4 x 12

Perimeter of the square = 48 feet ft.

4th grade geometry problems in Rectangle:

Formula to find the Area and perimeter of the rectangle:

Area of the rectangle (A) = length x width

Area (A) = l x  w (l is the length and w is the width of the rectangle )

perimeter of the rectangle = 2(length + width)

perimeter (p) =2(l x w)

1. Find the area and perimeter of rectangle, whose length and width are 12meter and 6 meter respectively.

Sol:

Area of rectangle = l x w  square unit.

Given:    Length= 12 meters, Width =6 meters

=12x6

Area of rectangle  = 72 m^2                

Perimeter of the rectangle   = 2(l + w)

=2(12 + 6)

= 2 (18)

Perimeter of the rectangle  = 36 meter

2. Find the area and perimeter of rectangle, whose length and width are 8.5meter and 3 meter respectively.

Sol:

Given: Length= 8.5 meters, Width =3 meters

Area of rectangle   = l x w square unit.

=8.5x3

Area of rectangle = 25.5m^2                

Perimeter of the rectangle    = 2(l + w)

=2(8.5 + 3)

= 2 (11.5)

Perimeter of the rectangle = 23 meter

4th Grage Geometry Problems in Circle:

Formula to find the area and circumference of the circle:

Area of the circle = πr^2

( r is the radius of the circle)

Circumference of the circle = 2πr


1. The radius(r) of a circle is 5 inches. Find the area and circumference of that circle?

Sol:

Given: r = 5inches

Area of the circle= π x r^2

π = 3.14

A = 3.14 x (5)^2

=3.14 x 25 inch^2.

Area  = 78.5 in^2

Circumference of the circle = 2πr.

π=3.14, r = 5 inches

Circumference of the circle = 2 x 3.14 x 5

= 31.4 inches


2. The radius(r) of a circle is 7 inches. Find the area and circumference of that circle?

Sol:

Area of the circle= π x r^2

Given:  r= 7inches,            

value of π=3.14

Area = 3.14 x (7)^2

= 3.14 x 49 inch^2.

Area  =  153.86 in^2

Circumference of the circle = 2πr.

π=3.14 , Given r = 7 inches

=2 x 3.14 x 7

Circumference of the circle = 43.96inches

Introduction to Word Problems based on Ratio and Proportion

A ratio is the comparison of two quantities by division. It is a relation that one quantity bears to another with respect to magnitude. If A and B are two numbers, than the ratio of A to B is A/B and is denoted by A:B. The ratio does not have any unit.

The equality of two ratios is called Proportion. If (A/B) = (C/D) , then A,B,C,D are said to be in proportion and can be written as

A : B :: C:D

The Ratio and Proportion world problems are a kind of problems in which , the relation between different quantity has to be determined and then using the definition of ratio and proportion, the unknown has to be calculated.

Here are some of the ratio and proportion word problems:

Ratio and Proportion: Word Problems.

Problem: Find the value of k that must be added to 7, 16, 43, 79 so that they are in proportion. (Answer: 5)

Problem: Find the fourth proportional to the numbers 60, 48, 30. (Answer: 24)

Problem: The Income of Alex and Bob are in the ratio of 3:2 and their expenditure in the ratio of 5:3. Find the income of Alex if each saves dollars 1000. (Answer:  $6000)

Problem: A mixture contains alcohol and water in the ratio of 12:5. On adding 14 litres of water, the ratio of alcohol to water becomes 1:1. Find the quantity of alcohol in the mixture. (Answer: 24 litres)

Ratio and Proportion: Multiple Choice Word Problems:

Problem 1: If the ratio of ages of Alex and Bob is 6:5 at present and fifteen years from now, the ratio will get changed to 9:8, then find Alex's age.

(A). 24 years

(B) 30 years

(C) 18 years

(D) 33 years

(Answer: (B) 30 years)

Problem 2: If dollars 58 is divided among 150 children such that each girl and each boy gets 25 dollars and 50 dollars respectively. Then how many girls are?

(A) 52

(B) 54

(C) 68

(D) 62

(Answer: (C) 68)

Problem 3: The number that must be added to each of the numbers 8, 21, 13 and 31 to make the ratio of first two numbers equal to the ratio of last two numbers is

(A) 5

(B) 7

(C) 9

(D) None of these.

(Answer: (A) 5)

Unit conversion chart

Introduction: Units may be written in full or using the agreed symbols, but no other abbreviation may be used. The letter ‘s’ is never added to symbols to indicate the plural form. A full stop is not written after symbols for units unless it occurs at the end of a sentence. When unit symbols are combined as a quotient

e.g. metre per second, it is recommended that they may be written as m/s or better still as ms^-1. Three decimal signs are used commonly internationally and they are the full point, followed by the mid-point and lastly the comma. The full point sometimes is used as a sign of multiplication and the comma for spacing the digits in large numbers; we shall be on the safe side if we use the mid-point for decimals.

Basic units of measurement: the unit of measure for length is metre and it is denoted by the letter ‘m’. The unit of measure for mass is kilogram and it is denoted by the symbol ‘kg’. The unit of measure for time is second and it is denoted by the symbol‘s’.

Unit conversion chart: 
1. Units of weight: SI unit for weight is kg and let us understand unit weight through weight conversions  chart. Following are the units of weight and weight  conversions chart
(i) 10 milligrams = 1 centigram 10 mg = 1 cg
(ii) 100 centigram = 1 gram 100 cg = 1 g
(iii) 100 grams = 1 kilogram 1000 g = 1 kg
(iv) 100 kilograms = 1 quintal
(v) 10 quintals = 1 metric tonne
2. Units for volume: The SI unit of volume is cubic metres (m^3). Litre is also sometimes used as the unit of volume. Following are the units of volume and volume conversions:
(i) 1 cm^3 = 1 ml = 1 cm × 1 cm × 1 cm = 10 mm × 10 mm × 10 mm = 1000 mm^3
(ii) 1 litre = 1000 ml = 1000 cm^3
(iii) 1 m^3 = 1 m × 1 m × 1 m = 100 cm × 100 cm × 100 cm = 10^6 cm^3 = 1000 litre = 1 kilolitre
(iv) 1 dm^3 = 1000 cm^3
(v) 1 m^3 = 1000 dm^3
(vi) 1 km^3 = 10^9 m^3
3. Units of length: The SI unit of length is metre.Following are the unit of length and length conversions:
(i) 10 millimetres = 1 centimetre 10 mm = 1 cm
(ii) 100 centimetres = 1 metres 100 cm = 1 m
(iii) 1000 metres = 1 kilometre 1000 m = 1 km
4. Square and cubic units
1 cm^2 = 100 mm^2 1 cm^3 = 1000 mm^3
1 m^2 = 10000 cm^2 1 m^3 = 1000000 cm^3
1 litre = 1000 cm^3
1 m^3 = 1000 litres
1 hectare = 10000 m^2.

Introduction to vectors

Introduction to vectors: In mathematics we have many systems which are employed to handle problems in Geometry, Mechanics and other branches of Applied Mathematics. Vectors constitute one of these systems. Vectors facilitate analytic study of the previous type of physical objects which have direction in addition to magnitude. No doubt, the set of real numbers provides an analytical tool for study of various types of physical problems for which vectors are useful.

The use of vectors in these problems is more natural. We have to associate a physical entity involving direction to the set of real numbers for we have to split up the entity into components and associate a number with each. The use of vectors avoids this splitting up. What is a vector? Quantities that have magnitude as well as direction are called vectors.

Definition of a vector: A directed line segment is called a vector. A vector from P to Q is denoted by  . P and Q are called respectively initial and terminal points of the vector  . Vector Example: Such quantities are called vectors Displacement, velocity, acceleration, momentum, weight, force etc.

Types of vectors are Zero or null vector: A vector whose initial and terminal points are coincident is called the zero or the null vector. Vectors other than the null vector are called proper vectors.

Unit vector: A vector whose modulus is unity is called the unit vector.

Like and unlike vectors: Vectors are said to be like when they have the same sense of direction and unlike when they have opposite directions.

Collinear or parallel vectors: Vectors having the same or parallel supports are called collinear vectors.

Co-initial vectors: Vectors having the same initial point are called co-initial vectors.

Scalars quantities: Quantities that have only magnitude but no direction are called scalars. For example, time, mass, volume, population, temperature, energy, power are all scalars. We need a unit and a real number to specify a scalar. For example, 3 hours,  2.5 kg, 8 cubic centimetres, 23°F etc.

Vector and scalar quantities

Physical quantities are divided into two categories- Vector and scalar. Those quantities which have only magnitude and which are not related to any fixed direction in space are called scalar quantities or scalars. Example of scalars are mass, volume, density, work, temperature etc. Second kinds of quantities are those which have both magnitude and direction.


Co-planar vectors: A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Two vectors are always coplanar.

Percentile (Statistics)

Define percentile:

Instead of dividing the total frequency into 4 parts by quartiles, we may divide it into 100 parts by what are called percentiles. Or we may divide into 10 parts by decile. The theory of percentiles is precisely analogous to that of the quartiles. Like the quartiles, there may, for instance, be certain indeterminacies in their exact percentile definition which are removed by supplementary conventions. Percentiles can be obtained by arithmetical or graphical interpolation. Percentiles have obvious meanings.

These quantities such as quartiles, deciles and percentiles, which divide the total frequency into a number of parts, are called quantiles or grades, and when we speak of the grade of an individual we mean thereby the proportion of the total frequency which lies below it. For example, when we say that a student has scored 75 percentile, we mean that of the total number of students who have appeared for the test, 75% of them have scored below this student.

Finding the percentile:

The percentile can be conveniently found by a graphical method which is an extension  of the graphical method of finding the median. Against the variate value as abscissa we graph as ordinate the cumulated frequency up to and including the corresponding variate value. This is called the distribution curve. By reading off the ordinate corresponding to a given variate we can find the number of members of the population bearing that or lower value. Similarly, by reading off the variate corresponding to the given ordinate we can find the percentiles.

A somewhat similar form of graph (with the percentiles as abscissa and the variate a ordinate) was formerly in use and was known as Galton’s ogive. The curve was not, however, always shaped like an ogive. The distribution curve appears to provide a more natural method of representation and a better name. We recognize it as the graph of the integral of the frequency curve.

Percentile uses:

In statistics percentile has been found to be very useful when dealing with non measurable characters. For example, the capacity of different boys in a class as regards some school subject cannot be directly measured, but it may not be very difficult for the teacher to arrange them in order of merit as regards this particular character. If the boys are then numbered up in that order, the number of each boy, or his rank, becomes more or less his representative of his percentile. So the third ranker in a group of 50 boys would be the 94th percentile and so on.

Skew Symmetric Matrix

What is Skew Symmetric Matrix?
A square matrix is said to be skew symmetric matrix, if its transpose is its negative. Symbolically it should satisfy the equation Z = -ZT. Also, a matrix will be a skew symmetric matrix if and only if it satisfies the condition xTAy =  -yTAx for all the vectors of x and y.   For all x, this is equivalent to xT Ax=0.  The skew symmetric matrix is also termed as anti-symmetric matrix or anti-metric matrix.

Difference between Skew Symmetric Matrices and Symmetric Matrices
The skew symmetric matrix is similar to symmetric matrix except the following property: 1+1 not equal to 0 in skew symmetric matrix assuming that the core field doesn’t have the characteristic 2. Here one represents multiplicative identity and zero denotes additive identity.

The following equation helps to understand the difference between skew symmetric matrices and symmetric matrices better:
K = (1/2) (K+KT) + (1/2) (K-KT) = L+M, where K is an n x n matrix. L and M are two resultant matrices.   In these two matrices, LT = L and so L is symmetric matrix and MT = -M and thus M is the skew symmetric matrix.

Skew Symmetric Matrix Properties
Listed below are the properties of Skew Symmetric Matrix:
1. Addition of two skew symmetric matrices will result in a skew symmetric matrix.
2. The scalar multiplication of two skew symmetric matrices will result in a skew symmetric matrix. The skew symmetric matrices are of dimension n (n-1)/2 and they form vector space.
3. The dimension m (m-1) / 2 scalars determine the skew symmetric matrix. These scalars represent the count of entries located on top of main diagonal of the matrix.
4. In a skew symmetric matrix, all the entries in main diagonal are zero. Therefore, trace value is also zero.
5. Cross products can be represented as matrix multiplications using skew symmetric matrices of dimension 3 x 3.
6. All the Eigen values are zero in value or they can be considered as purely imaginary.

Does a Skew Symmetric Matrix have an Inverse?
The skew symmetric matrix does not have an inverse, however the symmetric matrix has inverse. The symmetric matrix inverse for a diagonal matrix is calculated by replacing the diagonal elements with their reciprocal. The elements other than the diagonal elements will have the value zero in the diagonal matrix. In case of m x m matrix, the inverse of symmetric matrix is calculated with the help of determinant.

List of Irrational numbers

All Real numbers can be classified as rational and irrational numbers. Irrational numbers are . Irrational numbers are non –terminating non recurring decimals that cannot be expressed as fraction .

List of irrational numbers are as follow :

Irrational numbers

1.  All  non perfect square roots are irrational numbers.Some example of irrational squareroots are √2 , √3 , √5  √7, √11, √13, √17 , √19 , √21…………….etc. as  √2=1.4141.4142135623…………………, √3=1.73205 08075 68877………….and so on ….All these squarroots have non-terminating  non-recurring decimals which can never be expressed as fractions.so they are irrational.
2.  Like squareroots many cuberoots are also irrational . 3√ 3 , 3√ 5 , 3√ 7 , 3√ 14 etc  
3.  Some natural constants  like ∏=3.14159 26535 89793…………………..and e=2.71828 18284 59045…………………………….., etc are irrationals
4.  Sum of rational and irrationals are irrational .Let us take an example of 3 + √2  …which is irrational.
5.  Difference of rational and irrationals are irrational . For example 2 - √5 .. is irrational.
6.  Product of rational and irrational are irrational.For example 2 √7. are irrational
7.  Quoitent of rational and irrational numbers are irrational √7/√5 are irrational
8.  Negative of an irrational number is irrational. - √5 , - √7 , - √11… all are irrational number.
9.  The product of non-zero rational number and an irrational number is an irrational number. Let us take an example of irrational number . List  two irrational number between . √2 and √7.In order to find  irrational number  we will first square of . √2 and . √7. (. √2) ² = 2 and (√7.) ²= 7 , As 2< 3< 5<7  it follows that √2< √3 < √5 < √7  , therefore √3 and √5 lies between √2 and √7 . Hence two irrational number between √2 and √7 are √3 and √5.