Algebra Word Problems

In the field of mathematics where many forms are present algebra is one such form that deals with the basic concepts of Mathematics. We all know that Maths is a game of mathematical operations, so to deal with addition, subtraction, multiplication, division with the known and unknown values of the variables is known as algebra. In a simple way to say is that when we play computer games then we deal with jumping, running or finding secret things and in Maths we play with letters, numbers and symbols to extract the secret things. Now let’s see what is included in algebra.

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WHAT IS THERE IN ALGEBRA?
1. Algebraic terms – Algebraic terms are those terms which are infused with the values of variables. For example 3a.
In this example 3 is the numerical coefficient and a is the variable. The main work of numerical coefficient is also to deal with + 3 or – 3 of the variable.

2. Algebraic expressions – Algebraic expressions deal with a logical collection of numbers, variables, positive or negative numbers in the mathematical operations. For example we can write an expression – 3a + 2b.

3. Algebraic equations – Algebraic equations refer to the equivalency of expressions considering both the left hand side and the right hand side. For example an equation is x – 35 = 56k2 + 3. Over here both left hand side and right hand side are dealing with expression. If both the expressions are equal then that is said to be an equation.

ALGEBRA WORD PROBLEMS

The main part in Algebra is to solve the word problems, which we also did in Pre Algebra Word Problems.  It can be harder if you don’t think and do it but it can be easy if you think and do it. If we say in English how are you and the simple answer to the question is I am fine which can be simply understood but to convert the English language in to the language of maths requires certain techniques. You need to practice a lot with proper understanding of the translations. Let’s go step by step.

1. First of all read the entire word problem carefully. Make it a point that we should not start solving the problem by just reading half of a sentence. Read it properly so that you can gather the information that what you have and what you still need to have.

2.  Then, pick the information so that it can be changed into certain variables. Some might be known and some might be unknown. If required then draw pictures and symbols in the rough work so that you can deeply understand the problem without leaving any loopholes.

3. Note the key words as for different mathematical operations there are different key words –

a) Addition – increased by, total of , sum, more than, plus
b) Subtraction – decreased by, minus, less than, fewer, difference between
c) Multiplication – of, times. Multiplied by, product of, increased or decreased by the factor of
d) Division – per, out of, quotient
e) Equals -  is, gives, yields, will be


4. After going though these expressions carefully you can solve various word problems logically. Limit Problems are part of Calculus.

Rational Inequalities

Rational Inequalities : - The rational expressions are written in the P(x)/q(x) form where p(x) and q (x) are the polynomials i.e p(x), q(x)∈z(x), and q(x)≠0 are known as rational functions. the set of rational function is denoted by Q (x).

Therefore Q(x) = (p(x))/(q(x)) (where p(x), q(x) ∈z(x) and q(x) ≠0. for, if any a(x)∈z(x), then we can write a(x) as  ( a(x))/1, so that a(x)∈z(x)  and therefore,(a(x))/1  ∈Q(x);i.e  a(x)∈Q(x). The rational function example is given below.   

(2x+4)/(x²+5x+8)>0
(ax+b)/(cx+d)<0 p="">
Rational inequalities means the left hand side is not equal to the right hand side of the equation. The inequalities are associated with the linear programming.  The step by step procedure to solve the inequalities is given here. 

The very first step to solve the inequality problems is to write the equation in the correct form. There are two sides in the equation, left hand side and the right hand side. All the variables are written in the left hand side and zero on the right hand side. The second step is to find the critical values or key. To find this we have to keep the denominator and the numerator equal to zero.

The sign analysis chart is prepared by using the critical values. In this step the number line is divided into the sections. The sign analysis is carried out by assuming left hand side values of x and plugs them in the equation and marks the signs. Now using the sign analysis chart find the section which satisfies the inequality equations.

To write the answer the interval notation is used. Let us solve an example (x + 2) / (x² – 9) < 0. Now put the numerator equal to zero, we get x = -2. By putting the denominator equal to zero we get x = +3 and x = -3.  Mark these points on the number line. We get minus three, minus two and plus three on the number line.
Now try minus four, plug the value of x equal to minus four we get – 0.286.

Plug x equal to minus 2.5 we get + 0.1818. Now plug 2 in the equation we get minus zero point eight.
Now we try x equal to plus four which is at the right hand side of the number line. We get plus 0.86. The sign is minus, plus and plus sign. The equation is less than zero. To get less than zero we need all the negative values of the equation.

For x equals to plus three and minus three the equation is undefined. For x equals to minus two the equation is zero. The equation is satisfied by the value of x which is less than minus three. Therefore the answer which is the solution of the equation is minus infinite to minus three.

Rational Expressions Applications : - The equations which do not have equal sign are called the expression. The expressions are used to simplify the equations. The expressions are written to form the conditions of the problems.

According to the direction of the question the expressions are formed to solve a problem.

Set Operations

Set Operations: - The collection of the distinct objects is called the sets. For example the number 5, 7 and 11 are the distinct object when they are considered separately. Collectively they can form a set which can be written as [5, 7, 11]. The most fundamental concept of the mathematics is the sets.  The objects which are used to make the sets are called the elements. The elements of the sets may be anything, the letters of the alphabet, people or number. The set are denoted by the capital letters.  The two sets are equal when they have equal elements. The set can be represented by two methods, roster or tabular form and the sets builder form. In the roster form the elements are separated by commas. For example a sets of positive and odd integers less than or equal to seven is represented by {1, 3, 5, 7}.

The set of all the vowels of the alphabet is {a, e, I, o, and u}. The sets of the even numbers are {2, 4, 6,}. The dots indicate that the positive even integers are going to infinite. While writing the set in the roster form the element must be distinct. For example the set of all the letters in the word “SCHOOL” is {s, c, h, o, and l}. In the set builder form all the elements of the sets have the common property which is not possessed by any element which is outside of the set. For example vowels have the same property and can be written as {a, e, I, o, and u}. The set in the set builder form can be written as V = {x; x is a vowel in the English alphabet}.  A = {x; x is the natural number and 5 < x < 11}. C = {z; z is an odd natural number}.

Operations on Sets: - Like the mathematical operations, there are number of operations which are carried out by the sets. We will study them one by one. Union of sets is the union of the all the elements of a set and all the elements of the other  taking one element only once. Let A = {2, 3, 4, 5} and B = (4, 5, 6}. Therefore the union  of the two sets is written as A ∪B  = {2, 3, 4, 5, and 6}. Intersection of two sets A and B is the set of all elements which are common in both the sets. If the intersections of the two sets are equal to null then the set is called the disjoint set.

The difference of sets A and B can be defined as the A – B. It is equal to the elements which belong to set a and the element which does not belong to the set B. The operation on the sets such as union and intersection satisfy the various laws of the algebra such as the associative laws, the commutative laws, idempotent laws, identity laws, distributive laws, De Morgan’s laws, complement laws and involution law. These all the laws can be verified by the Venn diagram. 

Dispersion Statistics

Measures of Dispersion : - The dispersion is also known as the variability is the set of constant which would in a concise way explain variability or spread in a data. The four measures of dispersion or variability are the range, quartile deviations, average deviation and the standard deviation. The difference between two extreme observations in the given data is known as the range. It is denoted by R. In frequency distribution, R = (largest value –smallest value). It is used in statistical quality control studies rather widely. Median bisects the distribution. If we divide the distribution into four parts, we get what are called quartiles, Q(1 ),Q2 (median) and Q(3.)

The first quartile Q(1,) would have 25 % of the value below it and the rest above it; the third quartile would have 75% of values below it. Quartile deviation is defined as, Q. D.  = 1/2  ( Q3-Q1). If the average is chosen a, then the average deviation about A is defined as A.D. A.D. (A) = 1/n ∑|(xi- A)|  for discrete data. The Standard deviation is also called as the Root mean square deviation. The formula for the standard deviation is given as Standard deviation,σ=√(1/n ∑(xi- x ̅)^2 ) for discrete data.

The Square of the standard deviation is known as the variance. It is denoted by the square of sigma. Out of these measures, the last σ is widely used as a companion to x ̅ on who is based, when dealing with dispersion or scatter. Measure of dispersion is calculated for the data scattering. Deviation means how a value is deviated from it mean or average value. The mean of the two groups of the data may be same but their deviation may be high.

Central Tendency Measures : - The central tendency measures are also called the statistics central tendency. The clustering of data about some central value is known as the frequency distribution. The measure of central tendency is the averages or mean. The commonly used measures of central values are mean, Mode and median. The mean is the most important for it can be computed easily. The median, though more easily calculated, cannot be applied with case to theoretical analysis. Median is of advantage when there are exceptionally large and small values at the end of the distribution. The mode though easily calculated, has the least significance. It is particularly misleading in distributions which are small in numbers or highly unsymmetrical. In symmetrical distribution, the mean, median and mode coincide.

For other distributions, they are different and are known to be connected by empirical relationship. Mean – Mode = 3 (mean – median). The sum of the values of all the observations divided by the total number of observations is called the mean or average of a number of observations.  The value of the middle most observations is called the median. Therefore to calculate the median of the data, it is arranged in ascending (or descending) order. The observation which is found most frequently is known as mode.

The central tendency measures and the variability or dispersion are used in the statistical analysis of the data.

Interpolation

Interpolation is a concept that is used in numerical analysis. It means finding an intermediate value from the given data. That is, for a set of function values, sometimes a situation arises to know what the value of the function is, for some intermediate value of the variable.

Let us explain the concept with a simple example. Let the ordered pairs of the data of a function be (0, -1), (1, 7), (2, 22), (3, 40), (4, 69), (5, 98) and we need to find the function value of 3.8 of the variable. The attempt to find that value is called interpolation and there are several methods. Let us describe those one by one.

The first method is to round the given value of the variable to the nearest value in the data and take the corresponding value of the function. So for the given data, the value 3.8 of the variable can be rounded to 4 and assume the value of the function approximately as 40. But as one can easily see it is a crude method and far from accurate. This method can only be used just for guidance.

The next method is called as linear interpolation. That is, the function is assumed to be linear in the interval that contains the required value of the variable and accordingly the value of the function is determined. In the given example 3.8 falls in the interval [3, 4]. Considering the function to be linear in this interval, the slope of the function in this interval is (69 – 40)/(1) = 29. So the value of the function at x = 3.8 is, 29(0.8) + 40 = 63.2. Though this method also is not very accurate still the accuracy is much better than that in method 1. This method, even at the cost of sacrificing some accuracy, is preferred because it is easy to work with. In general the linear interpolation y at a point x  is given by the formula y = [(y_b – y_a)/(x_b – x_a)](x – x_a) + y_a, where [a, b] is the interval in which the required point occurs. This is the algorithm used in any linear interpolation calculator.

It is always possible a curve rather than a straight line could better cover the points plotted from a data. In other words a polynomial function can give a better interpolation. Finding a suitable polynomial function for a given data is called as polynomial regression.  Suppose we consider the same data, the three degree polynomial function f(x) = 3x² + 5x – 1 will be very close to the desired results. In such a case evaluating the function for x = 3.8, the value of the function is will be a very accurate interpolation. The accuracy can further be improved when a polynomial function of degree same as the number of data points is determined.

A high level interpolation polynomial can be derived by extensive methods and such an interpolation is called as Lagrange interpolation introduces by the famous Italian mathematician.