Understanding the Concept of Inequalities

In mathematics there is system of linear equations. The equations are formed when there is an ‘equal to’ symbol present in the given expression. If the ‘equal to’ symbol is replaced by a ‘greater than’ or ‘lesser than’ symbol the equation becomes an inequality. When a set of equations are given it forms a system of equations. The equations can be linear in nature and also quadratic in nature. Equations of higher degrees are also present. The higher the degree of the equation the more complex the equation becomes. It is difficult and also time consuming to solve the equations of higher degrees. But the linear equations can be easily solved. The simple concept of transposing is used in solving them. There are both variables and constants present in an equation. The value of constants does not change throughout the equation but the value of the variables changes. The variables can assume various values.

Just like the system of equations there is also the system of inequalities in mathematics. The method of solving them is similar to the system of equations. But the ‘greater than’ or ‘lesser than’ inequality must be kept in mind before solving them, otherwise the whole answer can go wrong. The inequality can also be treated as a constraint. The final solution must satisfy this inequality or constraint. If the solution does not do so then it is not the correct answer. The solution must be found out again and must be checked for its feasibility. The systems of inequalities can be easily solved just like the system of equations.

The art of solving systems of inequalities can be easily learnt if one is thorough with the concept of solving the system of linear equations. The difference between the two is only in the symbols used. The graphical method can also be used to solve the same. The graphical method is a pictorial method of finding the solution to this method. The equation or the inequality given in the question is plotted on the graph and the solution is found accordingly. It is an easy method of understanding the concept. Once the solution is obtained through this method it is very easy to check its feasibility. The feasibility of the solution can be understood by looking at the graph itself. Once the solutions that are obtained are found to be feasible they must be accepted as the solutions.

Graph Logarithmic Functions

A logarithmic function relates the output corresponding to the logarithms of the variables. In many applications in real life, it is easier to study the output to the logarithmic values of the variable rather than the values of the variables. This is because; in certain cases the output increases very rapidly with even nominal increases of values of the variables.

Since the logarithm picks up only the increase in the exponent size, the increase rate is controlled to smaller figures. For example if a function is described as f(x) = x, for every1 increase of the variable, the function rises by 1. On the other hand if we define a function g(x) = log [(f(x)] = log (x), then the function rises by 1 only for every 1 increase of exponent of 10.(Here as per normal conventions we referred ‘log’ to mean logarithm to base 10).

In other words, a logarithmic-function has a low increase for large inputs. But however much we explain, the concept can be more clearly understood with a logarithmic function graph. Always a visual presentation is much more effective than verbal algebraic explanation. Thus graphing logarithmic functions have taken a very important place in algebra especially in the study of functions.
Because of the large variations of scales of the input and output in such functions and also because of the domain restrictions one must adopt the correct technique to graph logarithmic functions.

The following may be used as some tips to do the graphing for functions of logarithm.
1) First find the domain of the given functions. Remember that logarithms are defined only for positive expressions. So accordingly determine the domain and start graphing only from the minimum domain value. This will give an idea about the scaling the variable axis.
2) According to the domain of the function, determine the range. You can determine a practical range so that graph is properly sized. This will help in selecting proper scale for the output or function axis.
3) Next find the x-intercept and y-intercept. In some cases one or even both may not exist at least in the selected domain and range of the graph. Plot these points on the set of coordinate axes.
4) Make a table of values for a few points (practically it will be only 2 or 3) selecting compatible values of the variables. For example, if f(x) = log (x), make the table for values of powers of 10. Plot the table values on the same set of axes.
5) Draw a smooth curve connecting all points that are plotted. The ends of the curve should have arrow heads in the appropriate directions to indicate that the function is continuous.

Quadratic functions

Let us first see how quadratic functions are defined. A function is a relation between two or more variables which is one to one and onto. If the independent variable is single with degree 2 and the dependent variable is of degree 1, then the function is said to be quadratic. That is, the output of the function is related to the square of the input.

The general form of a quadratic function is y = f(x) = ax2 + bx + c, where a, b and c are all constants. Obviously, the constant a cannot be 0, in which case the degree of the variable will be no longer 2 and hence the function will not qualify as quadratic function.

Quadratic-functions are the most prominent functions in algebra as many practical situations can be cited as quadratic functions examples. Thus the knowledge of solving quadratic functions is extremely important. The solving can mean in two types of solving. One is to find y for a given value of x. But this is very simple task to do and has no great practical implications.

On the other hand, for a given value of y, to solve forx is extremely important. This is what generally referred as solving quadratic functions. The very first exercise is to solve for solve for x,when y = 0. The values of x under such conditions are called as the zeroes of the function or the x-intercepts of the function. In case of linear functions y-intercepts are more prominent but in case of functions of quadratic form, the x-intercepts are more prominent.

In certain cases, you may find the solutions to be imaginary and at a later stage we will explain the implication of such cases.

A quadratic function can be transformed to a quadratic equation for a given value of y. That is, the constant term is adjusted accordingly and is equated to 0. There are many methods of solving for the variable. The easiest method, if possible, is to factor the trinomial as the product to two binomials and apply the zero product property. In cases where factorization is not possible, one can use the quadratic formula and find the solutions.

We cannot predict for sure, the shape of a graph for a given function except in two cases. In case of a linear function, the graph is a straight line. The shapes of graphs of quadratic functions are invariably vertical parabolas. The points of intersections of the parabolas with x-axis are the zeroes of the function. If there are no such intercepts, the zeroes of the functions are imaginary.

The Art of Simplifying Rational Expressions

The concept of simplifying rational expressions can be understood only after the studying the basics. Once the basics are clear, it becomes very easy to solve these. The concept of solving algebraic equations can also be learnt from these basics.

Once this is understood simplifying algebraic expressions can be very simple. Basic algebraic expressions will contain the arithmetic operations of addition, subtraction, multiplication and division. They also contain constants and variables. The value of constants does not change throughout and the value of the variables can change. This is the basic difference between the two.

To simplify rational expressions one must understand what the definition of these expressions is. These can be written with the fractions. The denominator in the fraction must not be zero; otherwise it is difficult to define the fraction.

The degree of the expression plays a very important role in deciding the method to solve the expression. The question how to simplify rational expressions can be understood only if the basics are understood. It is important to learn the concept of polynomials to understand this concept. Ratio of the polynomials can be treated as an expression representing the concept. The concept of ratio is very clear and simple to understand.

There can be various terms in the polynomial. These terms can be rational or need not be rational in nature. But this will be true only for the real numbers. The complex numbers do not come into the picture here. The complex numbers consist of a real part and an imaginary part in them.

So, this cannot become a part of the polynomials that appear in the ratio. The polynomial that appears in the denominator must not be reducible to zero otherwise the whole expression is valid.This concept is also useful in geometry. The expression can be represented graphically. Simplifying these fractions can be learnt from the simplification of simpler fractions. The process in simplification is very simple.

The common terms that are present in both the numerator and the denominator are cancelled, so that the common terms do not appear twice. This is the basic idea of simplification. The same process is followed in simplifying these expressions as well. The polynomials must be reduced into simpler terms. If there are any common terms in the numerator and the denominator they must be cancelled. This makes the term simpler and easy to understand.

Mixture Problems as Words

We are solving the equations in mathematics always having direct problems but when we are solving the problems in the word or description format then it is known to be as mixture problems. Formation of the question or getting the problems in the form of word problems is the approach to check your analytical skills.

It is very useful in the future profits as if you go further in the studies then you have to face the Operational Research type of subject or the optimization techniques used in the process of various functional aspect of the organization.

When the problem comes in the word format then you have a better experience to do that it means that you have to do the lot of problems before attempt the problems given in your papers. As it is must to know the formats in the word problems. Algebra mixture problems are creating a mix of two or more than two things to determine something related to quantity of the mixture problems.

It is the conversion of qualitative analysis into quantitative analysis. Say for example of mixture problems in algebra like if the trains are moving in a same direction and both the speeds of the train is 7km/s then tell us the relative speed of both the trains?

This is an example of the problems which is word problem. Solving this kind of the equation you have to focus on what the question want and then proceed further to reach to the final equation.

The other part of solving the equation is form the tabular form as when the polynomial type of the equation is given and then solves it as per the logical reasoning. Mixture problems in algebra is to mix up the qualitative type of problems in a quantify manner to check the data interpretation of yours while solving this kind of problems.

As if you understand these kinds of problems then you are able to solve all the problems in the mathematics. Operation research is a big scope for good data interpreter and it works in your competitive advantage also. So try to better solve more and more number of problems based on word problems to increase your IQ level.

Few tips we want to share is use go through the problem and in rough do the analysis part what is given in the problem and then to start the solution. Procedure wise solve the question to reach the final goal what you want to achieve and make it as a sequential wise approach.

Concept of Simplification of Radicals

Various concepts are used in mathematics. Radicals are also one among them. They can also be represented with the help of a symbol. Basically they are used to represent the root of a number. The number can from the various number systems in mathematics. There are various number systems in mathematics.

The natural number system, whole number system, integer number system and real number system are some of them. There can also be irrational numbers and also complex numbers. The complex numbers have a real part and an imaginary part attached to them.

Do they are said to be complex numbers. The operations that are performed on real numbers like addition and subtraction can also be performed on the complex numbers as well. But to perform these operations one must be clear with the concept of the complex numbers.

The next question that arises is how do you simplify radicals in mathematics. The radicals have to be simplified in order to arrive at the final answer. The process to simplify radicals is quite simple. The root that has been applied to a number can be a square root, cube root and so on.

Sometimes fourth root can also be used. The number inside the square root symbol must be present two times then it can be taken out of the square root symbol. In case of cube root symbol it should be three times and in the case of the fourth root it should four times and so on.

In simplifying radicals practice is required; otherwise it becomes very tough to solve the problems. In mathematics practice is very important otherwise it will be very difficult to arrive at the final answer. Many mistakes can also be committed en-route to the answer. This can be avoided by proper practice.

There can questions to simplify radicals with variables like the simplification of normal radicals. The process is quite similar to the process of simplification of the radicals of the normal nature. Only proper practice can help one to reach at the right answer. There can be signs that can be used before the radical symbol.

The signs can be positive or negative in nature. Depending on the sign preceding the radical symbol the answer also changes. The positive denotes a positive number and a negative sign denotes a negative number. The magnitude also plays a very important role.

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Expert guidance helps students in solving Math problems
                                                                                                                         
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Math Problems and its instant solutions
Most of the students face difficulties while solving Math problems and they start searching the right solutions for these. In that matter, students can take assistance from online tutors who are available round the clock. With this service, they don’t need to travel to attend any learning session anywhere. They can schedule the required numbers of learning sessions at their preferred time from home. Moreover, they can opt for repeated sessions on a same topic and thus, they can achieve a good command over any topic. Additionally, the free worksheets make students familiar with the question pattern.

Online tutors guide students in solving Math homework and assignments

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Role of Private tutors

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Positive Aspects of having a Private Tutor

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BODMAS

BODMAS is one of the most important concepts in mathematics. Taught in the middle school, BODMAS plays a major role throughout every mathematical calculation. BODMAS is an abbreviation of rules of operations in mathematics. It is used to make the rules easy to remember and perform. BODMAS is referred by various other names. BODMAS is also known as BEDMAS where E means Exponents. It is also known as PEMDAS where P means Parenthesis. Let’s have an in depth look at the full-forms of BODMAS and its meaning in this post.

B – Brackets First
O – Orders such as powers, square roots etc.
DM – Division and Multiplication
AS – Addition and Subtraction
• BODMAS defines the rules to be followed in orders of operations. According to the rule, the operation within brackets should always be calculated first.
Example: (2 toys for infants x 3 kids) + (2 crib toys for baby girls x 2 girls) + 5 toy action figure for kids
6 toys for infants + 4 crib toys for baby girls + 5 toy action figure for kids
15 toys for kids
• According to the BODMAS rule, exponents such as power and roots should be calculated prior to multiplication, division, addition and subtraction.
Example: 5 infant toys x 22
= 5 infant toys x 4
= 20
• According to the BODMAS rule, multiplication and division should be performed prior to addition and subtraction.
Example 1: 2 + 5 x 3
= 2 + 15
= 17
Example 2: 45 – 16 / 4
= 45 – 4
= 41
• According to the BODMAS rule, addition and subtraction comes at last when there are other operations to be performed.
Example 1: 2 + 3 x 6
= 2 + 18
= 20
Example 2: 7 – 2 x 3
= 7 – 6
= 1
These are the basics about BODMAS.

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Factoring polynomials

When factoring out polynomials,  we find   the  polynomial  that divide out evenly from the original polynomial  .   How to factor  Polynomials:  For this we have to find all the terms that if multiplied together we get the  original polynomial.  This is continued to all the terms until this cannot be simplified any more.  If the polynomial cannot be factored any more then the polynomial is said to be completely factored.


A factor of a polynomial is any polynomial which divides evenly into   the given polynomial   For example, x + 2 is a factor of the polynomial x^2 – 4.
The factorization of a polynomial is its representation as a product its factors. For example, the factorization of x^2 – 4 is (x – 2)(x + 2).

In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials. If you choose, you could then multiply these factors together, and you should get the original polynomial.
This example shows how to factoring polynomials.    Take 3x^2 – 12x + 9.   The common term in these is 3.  So take 3 out and divide each term by 3.  We get   3 (x 2-4x + 3).

Step by step explanation as how to Factor out Polynomials is given below
Factor 4a^2 + 20a – 3a - 15


The first two terms have a common factor in 4a.  The last two terms have a   common factor in 3.
We need to factor those terms out.

4a ( a + 5)  -3(a+5)

Now you have a binomial.  Each term     has a factor of (a + 5).

(4a -3) (a+5) is the factored terms.


Another example is given below on factoring of polynomials
X2 -8x + 15

start by looking at the factor pairs of 15.  We are looking for a pair of factors which add up to equal -8.  Look for the factor pairs of  +15 so that they add up to -8    The negative factor pairs of 15 are:
-15  and  -1
 -5 and  -3
Since -5 + -3 = -8 this is the pair we are looking for and we can factor the
original expression into:  (x-5)(x-3)

Matrix types: Diagonal matrix

The term diagonal matrx refers to the topic of linear algebra which is a branch of mathematics. In general a matrx would look like as follows:
[a11 a12 a13 … … a1m]
[a21 a22 a23 … … a2m]
[a31 a32 a33 … … a3m]
[… … … … … …    ]
[… … … … … …    ]
[an1 an2 an3 … … anm]

The above matrx has n rows and m columns. The diagonal of such a matrx consists of all the entries where in the row number = column number. Therefore if r = row number and j = column number. Then the entries of the type a(ij) where i=j are the diagonal entries. Therefore in the above matrx the diagonal would be the highlighted entries as shown below:

[a11 a12 a13 … … a1m]
[a21 a22 a23 … … a2m]
[a31 a32 a33 … … a3m]
[… … … … … …    ]
[… … … … … …    ]
[an1 an2 an3 … … anm]

Now if we have a matrx where in all the entries except the diagonal entries are zero, then such a matrx would be called a diagonal matrix. In general a diagonal-matrx would look like this:

[a11 0 0 … … 0]
[0 a22 0 … … 0]
[0 0 a33 … … 0]
[0 0 0 … … 0]
[… … … … .. 0]
[0 0 0 … … ann]

Note that a diagonal-matrx has to essentially be a square matrx. An example of a 3x3 diagonal-matrx is shown below:

[2 0 0]
[0 -1 0]
[0 0 5]

Scalar matrix:

A diagonal-matrx in which all the entries of the diagonal are equal is called a scalar-matrx. The general form of a scalar matrx would be like this:
[a 0 0 … … 0]
[0 a 0 … … 0]
[0 0 a … … 0]
[0 0 0 … … 0]
[… … … … .. 0]
[0 0 0 … … a]

Determinant of diagonal matrix:
Let us try to understand how to calculate the determinant of a diagonal matrix using an example.

Example: Calculate the determinant of the following diagonal-matrx:
[2 0 0]
[0 -1 0]
[0 0 5]
Solution:
D = 2*(-1*5 – 0*0) – 0*(5*0-0*0) + 0*(0*0-(-1)*0))
= 2*(-1)*(5) – 0 + 0
= 2*(-1)*5 = -10
So we see that the value of determinant of a diagonal-matrx is the product of the terms on the diagonal (also called the principal diagonal)

Discount and Discount Percentage

Discount and market price are two of the most important concepts in mathematics. These two concepts are also studied in finance and economics and play an important role in the real life market scenario. Let’s try to understand both the concepts in this post.

Market price is the price that is written in the price tag of a product. It is the amount that one needs to pay to buy a product. At times, customers are allowed to buy the product at a lesser price than the market price. For example: The market price of cot mobile for kids was Rs. 50. Maria was allowed to buy the same in Rs.45. Therefore, Rs. 5 is the discount offered to Maria while buying cot mobile for kids. The mathematical formula of finding discount price is Discount = Market Price – Selling Price (MP – SP).

Example 1: The market price of action figure toy is Rs.199. It is available in online shops for Rs.190. What is the discount offered?
Discount = Market Price – Selling Price
Discount = 199 – 190
= Rs. 9 is the discount offered on action figure toy in online shops.

Example 2: Mary bought kids’ toy guns at Rs. 50 and sold it for Rs. 40. What is the discount offered?
Discount = Market Price – Selling Price
Discount = 50 – 40
= Rs. 10 is the discount offered on kids toy guns by Mary.
Discount Percentage
The discount percentage is calculated on market price. The mathematical formula to find out discount percentage is [Discount / MP] X 100.

Example: Arun bought mobile phone for Rs.10000 and sold it out at Rs.9500. Find the discount and discount percentage:
Discount = Market Price – Selling Price
Discount = 10000 – 9500 = 500
Discount Percentage = [Discount / MP] X 100
= [500 / 10000] x 100
= 5%
The discount offered by Arun is 5%.
These are the basics on discount and discount percentages.

Subset

So what is the basic difference between a set and a sub-set? The answer is simple. A set has all the elements that are present in  a subset but a subset. does not contain all the elements of a set, it contains only a few elements. But it crucial to note that a subset. of a particular set will have no element that is other then what is present in that set.

So a subset. is a smaller or condensed form of a set. Let us consider a set of integers less than 5 and greater than 1. A = {  2,3,4 } now B which is a subset. of A can be any one of these: B = { 2,3} or B = {2} or B = { 3,4} etc. but not that B has no element other than that of A.

Even a null set is a sub-set of all the set may it be any set. A set of numbers can be either a set of real numbers or a set of integers or a set of fractions or a set of floating point number etc.  A set of numbers hence can be considered as a superset of all the other sets. A super set is which that contains all the sets elements of the given sub-sets. In the above example A can be called a superset of B and B can be called a S. S of A or just Subsets of Numbers.

The sum of subsets of a particular set will make up that set. It should be noted that null set belongs to every set.
To explain the Subsets of integers we can take up another example. The subsets can be A (sub-set of I) : { -1, -2 , 0 ,1, 2, 3} etc. All subsets are the part of a universal set which contains all the numbers. For a S. s of a integer we can have as many numbers as there are integers and then we can add a null set to it to get the original set.

The Set and Subset are the essentials for analyzing particular type of data and hence should be understood very clearly. The proper set and the improper set are yet another types of set, in a proper set we also have null element or entity. Similarly we can define s. s. for the same set.
R denotes the set of real numbers. Say ‘r’ is a s. s. of real numbers we write it as r = {0}

Percentage Increase

When we get a discount at a shop, we say that we got this much % off.  Now if the same shop offers 5 % more off, then the price of a product will be decreased by a certain amount. The increase and decrease in anything is usually calculated in terms of %. This is actually the measure of perc change which helps us to know the extent to which a thing has gained or lost its value over time.

For example if Sam works at shop with the pay of $ 10 per hour and if his pay gets increased to 4 15 per hour then we can Find Percentage Increase in his pay by making use of Formula for Percentage Increase which is difference of old value and new value divided by old value multiplied by hundred which can be written as (new value – old value) / old value X 100. Some more example of this would be change in temperature in two days, change in height or weight gain in children, share prices of a company or stock market or the prices of any product in a specific year compared to the last year.

How do you Calculate Percentage Increase- Finding Percentage Increase is done by following the series of steps given below: -

1. In the first step, we find out the difference between the two numbers.
2. In the second step, we take the difference and divide it by the original number.
3. In the third step, we multiply the number obtained in second step by 100.
4. In the last step, we put a % sign with the number or figure obtained in the third step.

Let us take an example to understand this concept more clearly. If the staff of a company is increased from 20 to 30 in one year then what is the Increase Percentage? To solve this, we follow the same steps given above. First we find the difference between two numbers which is 30 - 20 = 10. Then in the next step, we divide 10 by 20, which is the original number. That gives 10/20. Now in the next step, we multiply it by 100 which give 50. And in the last step, we put a 5 sign with 50 which makes the solution as 50 %.

Less than symbol

Less than symbol
A less than symbol looks like this “ <”  which is commonly used in mathematics. A less than symbol shows the relationship between in two values. The value that is on the left hand side is less than the value that is on the right hand is a less than symbol.

What is less than Symbol? In a way we can think about this is symbol with, two lines when they meet it becomes a point and that is where we put smaller value. But the Right hand side it is open wide and that side we put larger value. So for example if we are putting number 3 and number 5 like this, 3 less than(<) 5 Here we will put 3 at the left side and number 5 at right side, so we can say 3 is less than 5, this is less than sign example.

Symbols less than can be useful when we do not know the value of something, but we can compare to a value, we know that it is less than a certain value. So for instance I can never remember how old is my cousin sister is. But I know that she is younger to me. Let us say I am 20 years old, but I do know my cousin sister’s age is less than 20 years old. So we can write like this, age of my cousin < 20 years. Here we can say by looking this less than symbol, that age of my cousin sister is less than 20 years.

Another example in math symbols less than we shall understand it by, let us say that I have the expression 12 times 6 less than 8000,that is 12*6 < 8000 Maybe I cannot do that very quickly in my head but I am pretty confident that whatever I get 12 times 6 is going to be less than 8000. So there are times, in which we know rules of relationship between two numbers and which we can demonstrate it with the less than symbol.

The less than symbol does not tell us, how much less one value than the other is. It just puts them in that order. Which indicates one portion is smaller than the other. So let us use few more examples, where we can use less than symbols. Let us say, 7 and 56 can we use less than symbol here? Yes, very much as we know on the left side 7 and on the right side 56, so as per the symbol indicates <   56 is larger than 7. So we can use less than symbol here. Thus we can write 7 < 56.

Word Problems on Subtraction

Word problem is a very important concept of basic mathematics. A Word problem is nothing but a textual representation of a mathematical operation. Word problems are usually used to solve a problem easily. Word problems can be on all four operations of mathematics including addition, subtraction, multiplication and division. Let’s have a look at word problems on subtraction.
Word problem on subtraction is nothing but a text representation of subtraction. For example: Rohan bought 6 toys from Melissa & Doug brand. He gave 2 Melissa & Doug toys to Sam’s child and kept the others for his son. How many toys is he left with? This word problem is a text representation of mathematical problem (6 toys – 2 toys =?).

Solving word problems on subtraction:

Solving word problems on subtraction includes certain steps.  Firstly, both the numbers to be subtracted needs to be identified. Secondly, the word problem needs to be converted to a mathematical expression and finally, the number needs to be subtracted. For example: Mohan gifts three maternity wear from Mother Care India brand to his wife. His wife gives one of the Mother Care India maternity wear to her friend. How many maternity wear does Mohan’s wife now? Identifying the numbers, we get 3 maternity wear and 1 maternity wear. Converting the word problem to a mathematical expression and then subtracting, (3-1 = 2). Thus, the answer is 2 maternity wear.

Examples of Word Problems on Subtraction:

1. Philip has bought five fun toys from baby store India online collection. He gave two toys to Rohan’s son and kept one for his own son. How many toys from baby store India online does his son get?
Answer: 5 toys – 2 toys
(5-2) = 3 toys.
2. There are 18 Angry Bird toys in a basket. 7 Angry Bird toys were given to Mohan and 2 toys to Divya. How many Angry Bird toys were there in the basket after giving toys to Mohan and Divya?
Answer: 18 Angry Bird toys – 7 Angry Bird toys – 2 toys
(18-7-2) = 9 toys.
These are some basic examples on word problems on subtraction.