Basic Mathematics for College Students

Introduction to basic mathematics for college students:

There are certain math topics are appeared in college which we already read in school period.Those topics are very basics for college students.The following mathematics topics are basics for college students. They are, algebra problems, geometry measures, probability problems and linear equations. By referring those topics, the college students recall the concepts of the topics , so that they can do themselves. Let we see some basic problems to basic mathematics for college students.


Example problems to basic mathematics for college students:


The following problems are examples for basic mathematics for college students.

Basic mathematics - problem1) Solve the linear equation, 3a+6b-14 = 16, a+3b- 6 = 12.

Solution:

Here we need to find the value of a and b,

The given equations are 3a+6b-14 =16 and a - 6b-6 =12

3a+6b-14 = 16

a  - 6b - 6  = 12.

We can write the above equation as

3a + 6b = 16+14

a - 6b  = 12+6

From the above equation, we get

3a + 6b = 30------------------------->1

a  - 6b  = 18------------------------->2

4a        = 48

4a = 48

Divide by 4 on both sides, we get

`(4a)/4` = `(48)/4`

a  = 12.

Apply the 'a' value in equation (1),we get

3a+6b= 30

3(12)+6b  = 30

36 + 6b    = 30.

Add -36 on both sides, we get

36-36 +6b = 30-36

6b = -6

Divide by 6 on both sides, we get

`(6b)/ 6` = -`6/ 6`

b  = -1.

The values of a and b are , 12 and -1.

Problem2) Find the value of  D , 2d+5e+6f = 80-4d+20, where e=2 and f=3

Solution:

Here we have the value of e and f as  2 and 3.

Apply the values in the given equation,we get

2d +5e+6f = 80-4d+20

2d + 5 ( 2) +6(3) = 80-4d+20

2d +10 +18       = 80-4d+20

2d +28            = 80+20-4a

2d +4d         = 100 - 28

6d    = 72

Divide by 6 on both sides, we get

6d /6  = 72 /6

d    = 12

The value of d =12.

Problem3) Solve the following linear equation, 3p+3q = 18, 2p+3q = 14

Solution:

Here we need to find the value for P and Q.

The given equations are

3p + 3q   =  18 -------->1

2p + 3q   =   14---------->2

By changing the sign of the second equation values, we can solve the above problem.

3p + 3q  =18

-2p - 3q  =14

p = 4

Apply the p value in equation (1) , we get

3 (4) + 3q  = 18

12 + 3q   = 18

Add -12 on both sides, we get

12-12 +3q = 18-12

3q = 6

Divide by 3 on both sides, we get

3q/3 = 6/3

q = 2

The values of m and q are , 4 and 2.

Problem 4:  If S and T values are 2 in the given equation , 10S +8T +6U = 78, find the value of U?

Solution:

Given equation          = 10S + 8T + 6U = 78

Known values  are , S = T =  2

Apply S and T values in the above equation,we get

10S + 8T + 6U      = 78.

10(2) + 8(2) + 6U  = 78.

20 + 16 + 6U     = 78

36 + 6U     = 78

Add -36 on both sides, we get

36 - 36 + 6U = 78 - 36

6U    =  42

Divide by 6 on both sides, we get

6U/6  = 42/6

U = 7

The value of U= 7.

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Practice problems to basic mathematics for college students:


Try to solve the following mathematics problems.

Problem1) 2a+2b+4c = 10, a+3b+c = 22, 2a+3b+2c=14

Answer: a=-11,b=10,c=3.

Problem2) 4x + 6y = 28, 2x+4y=18

Answer : x = 1,y=4

ABC of Mathematics

Introduction to ABC of maths:

Mathematics plays a very important role in humans life. Mathematical thinking is very important for everyone. In every work of a human we need mathematics.


Humans use the mathematics in doing any work. For example consider:
Arithmetic:  in counting and sorting,
Geometry: in spacing and distancing,
Statics: in balancing and weighting,
Probability: in guessing and judging


ABC OF MATHS means the fundamentals and basics of mathematics.

Brief explanation of ABC of maths:


NUMBER:
Number in mathematics is just a word or a symbol which is used for counting and also to say where something comes in a series.
Generally numbers are used for counting and calculating.


There are different kinds of numbers which includes: natural numbers, integers or whole numbers, cardinal numbers, ordinal numbers, rational numbers, real numbers, complex numbers etc.

NATURAL NUMBERS:
Natural numbers start from the 1, 2………infinity. They form the endless chain.

INTEGERS:
Integers also form the endless chain and it includes negatives and zero also.
…….……..-3, -2, -1, 0, 1, 2, 3……………..

Integers are also known as whole numbers.


CARDINAL NUMBERS:
Cardinal number shows how many there are. They are used for counting.


ORDINAL NUMBERS:
Ordinal numbers tells about the order of the object being counted. Places in a sequence are shown by ordinal numbers.


RATIONAL NUMBERS:
Rational numbers are the numbers which does not includes like an square root of 2. It includes the integers and fractions.


REAL NUMBERS:
Real numbers are the numbers which also includes the numbers like square root of 2.


COMPLEX NUMBERS:
Complex numbers includes the imaginary numbers also like square root of -1.


There are also some special numbers included in ABC of mathematics. They include zero and Arabic number system. Zero is the number which was discovered by Indian and it is a great discovery without zero there is no value or reason of mathematics.
There are different kinds of scales which are used to classify and quantify and also to measure things. The different types of scales includes Binary scale, Nominal scale, Ordinal scale, Interval scale, Ratio scale
We also use roman numbers. FORMULA in mathematics is nothing but a rule.

Everyday Mathematics 5th Grade

Introduction to everyday mathematics 5th grade:

For 5th grade students mathematics is divided into different categories. In grade 5 students can learn Simplification using BODMAS, word problems, unitary methods, Multiples and factors, HCF and LCM, Multiplication and Division of fractional numbers and decimals. They also learn about money, percentage, profit and loss, Simple interest etc.They learn measurement of length, mass and capacity. In geometry they can learn about the types of angles, triangles, element of circle, relation between diameter and radius etc.


everyday mathematics 5th grade - Some Examples


In 5th grade we can learn everyday common mathematics as follows:

(1) Unitary Method: The simple rule of unitary method that if we are given the cost of one object, we can find the cost of many objects by multiplying the cost of one object with the number of objects and if we are given the cost of several objects, we an find the cost of one object by dividing the cost of several objects by the number of objects.

Example:  The weight of 25 bags of rice is 650 kg. Find the weight of hundred bags of rice?

Solution: since the weight of 25 bags of rice is 650kg.

Then the weight of 1 bag of rice is 650 ÷ 25

26kg

Then weight of 100 bags of rice is 26 x 100

2600 kg

Answer: weight of 100 bags of rice is 2600 kg.

(2) HCF and LCM:  We define the HCF as the highest common factor.HCF of two given numbers is the highest number that divides the given numbers exactly without leaving any remainder.

Example: Find HCF of 136, 170 and 255 by division method:

Solutions: First we find the HCF of 136 and 170

136)170(1

136

34

HCF of 136 and 170 = 34

Now we find the HCF of 34 and 255

34)255(7

238

17)34(2

34

0

HCF of 34 and 255 =17

Hence HCF of 136,170 and 255 = 17

LCM: The lowest common multiple of two or more numbers is the smallest number which is a multiple of each of the numbers.

Example: Find the LCM of 20, 30 and 50 by division method:

2] 20, 30, 50

5] 10, 15, 25

2,   3,   5

LCM = 2x 5x 2x 3x 5 = 300


everyday mathematics 5th grade - more Examples


Decimal numbers: Conversion of a decimal fraction into fraction number:

Write the given number without a decimal point in the numerator.
Write 1 in the denominator followed by as many zeros as the decimal places.
Then write the resulting fraction in the lowest form.
Example:

(a)    1.5 = 15 / 10 =3 /2

(b)   54.972 = 54972 /1000

= 13743 / 250

Find the sum:

(a)    205.40, 80.75 and 1493.50

All decimal points should be in a column in the question and answer:

2 0 5.  40

80.  75

14 93 .50

1 7 7 9.65

Percent: Percent means for every hundred. To convert a % into fraction, place the given number over 100 and reduce it to its lowest term.

Example: 20% = 20 / 100

=  1/5

150% = 150/100

=3/2

To find the percent of a given number:

Example: (a)   Find 50% of 75

50/100 x 75

= 75/2 = 37 ½ Ans.

(b)   What % is 25 of 200

25/200 x 100

=    12 ½ Ans.

Perfect Square Root Numbers

Introduction to perfect square root numbers:

Let us study about perfect square root numbers. Square root is defined as the method to get the given values number square.
The perfect square root numbers are also similar to what the simple square root mean, that the number when a multiply with itself forms a perfect square root number.
In mathematics all numbers as both positive and negative can have their perfect squares in positive terms. Examples are below.

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Perfect square root numbers:


Some of the examples of perfect square root numbers are as follows:
1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, etc.


Example 1:

Find the perfect square root numbers for the numbers 12, 14, 16, 18 and 20


Solution:

The perfect square root number for the number 12 = 12^2 = 12 * 12 = 144.
The perfect square root number for the number 14 = 14^2 = 14 * 14 = 196.
The perfect square root number for the number 16 = 16^2 = 16 * 16 = 256.
The perfect square root number for the number 18 = 18^2 = 18 * 18 = 324.
The perfect square root number for the number 12 = 12^2 = 20 * 20 = 200.

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Example 2:

Find the numbers for the perfect square root numbers 441, 169, 361, 1089 and 625.


Solution:

The number which gives the perfect square root number 441 = `sqrt(441)` = `sqrt(21*21)` = 21.
The number which gives the perfect square root number 169 = `sqrt(169)` = `sqrt(13*13)` = 13.
The number which gives the perfect square root number 361 = `sqrt(361)` = `sqrt(19*19)` = 19.
The number which gives the perfect square root number 1089 = `sqrt(1089)` = `sqrt(33*33)` = 33.
The number which gives the perfect square root number 625 = `sqrt(625)` = `sqrt(25*25)` = 25.


Exercises:

Find the perfect square root numbers for the numbers 15, 17, 22, 38 and 42. (Answer: 225, 289, 484, 1444 and 1764)
Find the numbers for the perfect square root numbers 484, 121, 676, 1156 and 2116. (Answer: 22, 11, 26, 34 and 46)

What is Mathematics Education

Introduction to mathematics education:

Mathematics education means study of the quantity, properties, and dealings of quantities and sets, using numbers and shapes. The major parts of mathematics are algebra, analysis, probability, set theory, and statistics. In this article we will study the education of mathematics and solve some of mathematics concepts with examples. Now we will solve the examples in algebra, probability, set theory and statistics what is mathematics education.


Examples - what is mathematics education:

Let us we will solve the problem in algebra for what is mathematics education.

Algebra - Example Problem 1:

Solve given polynomial equations.

5x^2 + 9 + 15x + 2x^2 + 4x + 10 + 3x

Solution:

Step 1:

First we have to mingle terms x^2

5x^2 + 2x^2=7x^2

Step 2:

Now combine the terms x

15x + 4x + 3x = 22x

Step 3:

Then join the constants terms

9 + 10=19

Step 4:

Finally, combine all the terms

7x^2 + 22x +19

So, the final answer is 7x^2 + 22x +19

Probability – What is Mathematics education:

Example 2:

In a briefcase, there are 7 red color dresses and 3 blue color dresses and 6 white color dresses. Find the probability of choosing blue color and white color dresses?

Solution:

Given, Number of red color dresses = 7

Number of blue color dresses = 3

Number of white color dresses = 6

So, Number of total outcomes should be,

7 + 3 + 6 = 16.

Therefore, the probability of choosing blue color dresses = `3/16`

Choosing white color dresses = `6/16.`

These are algebra and probability examples for what is mathematics education.

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More Examples – What is Mathematics Education:


Set theory:

Given sets are, P = {17, 25, 36, 85} and R = {25, 52, 63, 85}

P∪R = {17, 25, 36, 85} ∪ {25, 52, 63, 85}

= {17, 25, 52, 63, 36, 85,}

P∩R = {17, 25, 36, 85} ∩ {25, 52, 63, 85}

= {25, 85}

Statistics – What is Mathematics Education:

Find the range for given values.

The given values are 52, 63, 70,21,52,10.

Solution:

52, 63, 70,21,52,10.

In statistics, the range is defined as different between smallest value and biggest value.

In this given value, the smallest value is 10.

Then, the highest value is 70.

So, the range is,

Range = Highest value – smallest value

= 70 – 10.

= 60.

Range = 60.

These are examples for set theory and statistics for what is mathematics education.

That’s all about what is mathematics education.

Mathematics Education Standards

Introduction to mathematics education standards:

The mathematics education include a different branches of unit conversions, algebra, subtraction, measurement, number sense, multiplication, functions, adding and subtraction of decimals, fractions & mixed numbers, division, algebra, geometry, median problems, algebra function, probability and statistics number using words decimals. This mathematics education supports all type of standards up to higher standards.


Example problems for Mathematics education standards:


Example 1:

Solve the quadratic equation `x^2 +5x + 6 =0`

Solution:

`X^2 +5x +6 =0`

`X^2 +2x +3x + 6 =0`

` x(x +2) +3 (x +2) = 0`

`(x +2)(x + 3) =0`

`x + 2 = 0 `                   `x` ` + 3 =0`

`X = -2 `                      `X =-3`

Example 2:

Solve the quadratic equation `x^2 +4x + 4 =0`

Solution:

` X^2 +4x +4 =0`

`X^2 +2x +2x + 4 =0`

` x(x +2) + 2(x +2) = 0`

`(x +2)(x + 2) =0`

`x + 2 = 0 `              ` x + 2 =0`

`X = -2 `                      `X =-2`

Example of polynomial exponent problems- Mathematics education standards:

Addition of polynomial exponent:

Two or more polynomials, adding the terms,

Suitable example adding polynomial exponent,

Example1:

` (2x^2+3x^3)+(x^2+7x^3)`

`=2x^2+x^2+3x^3+7x^3`

The variable and exponent must be same then we add the polynomial exponent,

`=3x^2+10x^3`

So the result is `=3x^2+10x^3`

Subtraction of polynomial exponent

Example2:

`(3x^2+3x^3)-(x^2+7x^3)`

`=3x^2-x^2+3x^3-7x^3`

The variable and exponent must be same then we subtract the polynomial exponent,

`=2x^2-4x^3 `

So the result is` =2x^2-4x^3 `

Adding polynomials- Mathematics education standards:

Example 1: Find the sum of `6x^2 + 7x + 16 and 1x - 3x^2 -4.`

Solution: By means properties of real numbers, we realize

`(6x^2 + 7x + 16) + (-3x^2 + 1x - 4) = 3x^2 + 7x + 1x + 16 - 4`

`= 3x^2 + 7x + 1x + 16 - 4`

`= 3x^2 + 8x + 12`

So the final result is `= 3x^2 + 8x + 12`

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Examples for finite difference problem- Mathematics education standards:


Example 1: calculate the values of Δ`y` and dy if `y = f(x) = x^3 + x^2 - 2x + 1`

Where x changes (i) from `1 to 1.05` and (ii) from` 1 to 1.01`

Solution:

(i) We have `f(1) = 1^3 + 1^2 - 2(1) + 1 = 1`

`f(1.05) = (1.05)^3 + (1.05)^2 - 2(1.05) + 1 = 1.15.`

and Δ`y = f(1.05)- f(1) = 0.15.`

in general `dy = f ^ 1(x) dx = (3x^2 + 2x - 2)dx`

When `x` ` = 1` , `dx = ` Δ`x =1 and dy = [(3(1)^2+2(1)-2] 1= 3`

(ii) `f(1.01) = (1.01)^3 - (1.01)^2 - 2(1.01) + 1 = -1.01`

∴ Δ`y = f(1.01) - f(1) = 1.99`

Mathematics Dealing With Functions

Introduction of mathematics dealing with functions:

The mathematics dealing with functions in the form of f(x) = 2x+3, we are assigning the value for variable x in the given function so that we can solve the functions  in math. Now we are given several values for variable x in the given function and finding the solution for each function. Example for s function is f(y)=19y+12,function f(2). Using the square equation solve a function rule  of f(x) = 2x2 +4x +28, function of f(2).

Example of mathematics dealing with functions:

f(2) = 2x+2 and  f(x) =` (x+2)/2` , here the variable of x is 2.

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Problems how to find mathematics dealing with functions in square equations


Problem 1 : Using  square equation find  the mathematics dealing with functions of f(9), when f(x) = `(x+2)/2` +2.

Solution : Here the variable is given as 9 find the function of f(9).

f(x) = `(x+2)/2` + 2 find the f(9)

The value of x is 9 is given

f(9) = ` (9+2)/2 ` + 2

f(9) =  `11/2 ` + 2

f(9) = 5.5+2

f(9) = 7.5

Problem 2 : Using  square equation find  the mathematics dealing with functions of f(8), when f(x) = `(x+3)/4` +12.

Solution :

Here the variable is given as 8 find the function notation of f(8).

f(x) =`(x+3)/4 ` +12 find the f(8)

The value of x is 8 is given

f(8) = `(8+3)/4 ` +12

f(8) = `(11)/4` +12

f(8) = 2.75 + 12

f(8) = 14.75

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Problems in mathematics dealing with functions


Problems1: using mathematics dealing with functions of f(3),  When f(x ) = `(2(x+3))/8`

Solution:

Using the function  f(3) in the constant function

f(x) =` (2x+6)/8`

f(3) = `(2xx3+6)/8 ` here substitute x value 3 in the given constant function

f(3) = `12/8.`

f(3) = 1.5

Problems 2: using mathematics dealing with functions of f(4). When f(x ) = `(2(x+6))/4`

Solution:

Using function f(4) in the constant function

f(x) =  `(2(x+6))/4`

f(4) = `(2(x+6))/4` here substitute x value 4 in the given constant function

f(4) =` (2xx4+12)/4.`

f(4) = `(8+12)/4`

f(4) = `20/4`

f(4) = 5