Problems related to consecutive integers

Many a times in arithmetic we face problems related to finding consecutive integers the sum of which is given to us. The said integers may be odd or even or neither. In this article we’ll try to understand how to solve such type of problems.

Formula for sum of consecutive integers:
As such there is no specific formula for finding sum of consecutive integers. Such problems are solved using algebra and arithmetic approach. The methods may however, vary from problem to problem. Let us look at the following example to understand this better.

Example 1: Three consecutive integers are there whose sum is 66. Find consecutive integers.
Solution: Suppose we assume that the middle or the second integer is n, then the first integer would be n-1 and the third integer would be n+1, because the integers are said to be consecutive, and we know that consecutive integers differ by 1. The three integers are therefore now: n-1, n and n+1. The sum of these integers is given to be 66. Therefore,
n-1 + n + n+1 = 66
3n = 66
n = 66/3
n = 22.
Since n was our middle integer, the other two would be n-1 = 21 and n+1 = 23. So the three consecutive integers are 21, 22 and 23.
Odd consecutive integers:
Odd consecutive integers are such that each of the integer in the list is an odd number. It may be negative or positive, but has to be an odd number. For example, …. -11, -9, -7, -5, -3, -1,  1, 3,  5,  7,  9…These are odd consecutive integers. They go up to infinity in both directions.

Example 2: The sum of three consecutive odd integers is -39. Find the integers.
Solution: Here again, assume that the middle integer is n. Then the odd integer before n would be n-2 and the odd integer after n would be n+2. So the sum
= n-2 + n + n+2
= 3n = -39
n = -39/3
n = -13.
Therefore the other two odd integers would be -15 and -11. So solution: -15, -13, -11 are the required three odd integers.
Even consecutive integers:
The method of solving problems of even consecutive integers is exactly same as that for odd integers.

Example 3: Find the three even integers whose sum is 0.
Solution: If the middle integer is n, then the other two would be n-2 and n+2. Therefore the sum
= n-2 + n + n+2
= 3n = 0
n = 0/3
n = 0.
So the integers would be -2, 0, and 2

Integers: Positive and Negative

Integers are the numbers consisting of positive numbers, zero and also the negative numbers.  The positive numbers and negative numbers are called as positive negative integers. Positive integers are the integers with positive sign for example, 2, 3, 1, 5 etc. and negative integers are the integers with negative sign. Examples of Negative Integers are -21, -3, -1, -5 etc. On a number line the integers are shown as given below

The following are the Positive and Negative Integers Rules:
Addition rules of Integers: 
When two positive integers are added the sum would be a positive integer; (3+5=8)
When two negative integers are added the sum would be a negative integer; [(-3) + (-5)= -(3+5)= -8]
When a positive integer and a negative integer are added, the numbers are subtracted and given the sign of the larger number;[ (3)+(-5)= -2]

Subtracting Positive and Negative Integers:
A positive integer subtracted from a negative integer would give a negative integer; (-7)-(4)=-7+4= -3
A negative integer subtracted from a positive integer would give a positive integer, here the numbers are added and given the positive sign; (5)- (-7)=(5+7) = 12
When two negative numbers are subtracted it gives a negative and a positive integer. The numbers are subtracted and given the sign of the larger number; (-5) – (-7)=-5+7= 2; -7 – (-5)= -7+5= -2

Multiplication Rules of Integers:
A  Positive Integer multiplied with a positive integer, the product would be a positive integer
3x5=15
A positive integer multiplied with a negative integer, the product would be a negative integer
3x (-5)=-15
A negative integer multiplied with a positive integer, the product would be a negative integer
(-7)x3=-21
A negative integer multiplied with a negative integer, the product would be a positive integer
(-7)x(-5)= 35

Dividing Positive and Negative Integers:
A  Positive Integer divided by a positive integer gives a positive integer, 15/5=3
A positive integer divided by a negative integer gives a negative integer, 15/(-3) = -5
A negative integer divided by a positive integer gives a negative integer, -21/3 = -7
A negative integer divided by a negative integer gives a positive integer, (-21)/(-7) = 3

Positive and Negative Integers Word Problems
The temperatures recorded in the Sahara Desert and Thar Desert are 138 degrees Fahrenheit and -55 degrees Fahrenheit respectively. Calculate the difference in the temperatures.
Sahara Desert Temperature= 138 degrees Fahrenheit
Thar Desert Temperature= (-55) degrees Fahrenheit
Difference in temperatures = Temperature in Sahara Desert – Temperature in Thar Desert
      = 138 – (-55) = 138 +55 = 193 degrees Fahrenheit (Answer)

Standard Deviation

Standard deviation is denoted by a symbol of Greek letter (s) and is shows how much variation exists from the mean or average or expected value. If the standard deviation is low, that will indicate that the point tents to very close to the mean, whereas standard deviation is high, the data points are spread out over a large range of values. Standard deviation has very useful important properties such as unlike variance and is expressed in the same units as the data. Mean standard deviation is mainly used in the statistic conclusion to measure confidence. Also used to find how set of data spread out.
Standard deviation (s) is the square root of its variance (s2), which is the average of the squared difference from the average of the mean. Mean (µ) is simple average value of given set of data.
For example just take different height of dogs, find out mean, variance, and standard deviation.
Height of dogs = 700mm, 570mm, 180mm, 530mm, 400mm
To find the mean
Mean = 700+570+180+530+400/5
= 2380/5 = 476
The mean or average height of the dog is 476mm.
Then to calculate the variance first subtracts the value of mean from the every height of the dog and then square the resulting values. Add the sum of squared values and divide with total number of dogs, resulting value gives the variance.
Variance = (700-476)2 + (570-476)2+ (180-476)2 + (530-476)2 + (400-476)2 / 5
  = 50176+8836+87616+2916+5776 / 5
= 155320/5 = 31064
The square root of the variance gives the standard deviation for the height of the dogs.
Standard deviation (s) = vvariance
= v31064
= 176.24 mm


Finding the standard deviation for population and sample, following formulas are used,
Standard deviation for population (s) = v(1/n ?_(i=1)^n¦(xi- µ)2)
Standard deviation for sample (s) = v(1/(n-1) ?_(i=1)^n¦(xi- " " )2)
Where,
µ,   - mean
When we have n value of data, if we are calculating variance, we should divide by n for the population and divided by n-1 for a sample. Standard deviation is used to measure the investment volatility, in finance. It is also called as historical volatility.

Mean and Standard Deviation
Mean and standard deviation is mainly used to find the center of the data set. Mean is defined as; it is the simple average value of the given data set and is represented by a symbol of Greek letter (µ).
Mean for population (µ) = 1/n ?_(i=0)^(n-1)¦xi
Mean for sample ( ) = 1/n ?_(i=1)^n¦x

Where,
n - Size of the sample or number of item in the sample
x, xi - Observed value or set of value

Finding Standard Deviation
Finding standard deviation the following steps should be followed.
First calculate the mean of given set of data by sum of given data divided by total number of data.
Then subtract the mean from each observed value.
Square the each difference and then add all the squared values to get their total sum. The resulting value divided by one less then the number of data in the data set.
The resulting value gives the variance.
Finally standard deviation can get from square root of the variance.

Find Standard Deviation
Find standard deviation for the list of numbers, 1, 3, 4, 6, 9, 8
Mean (µ) = 1+3+4+6+9+8/6
= 5.16
Variance (s2) = (1-5.16)2 + (3-5.16)2 + (4-5.16)2 + (6-5.16)2 + (9-5.16)2 + (8-5.16)2  / ( 6 – 1)
= 17.31+4.66+1.35+0.71+8.07+14.75 / 5
= 46.85 / 5 = 9.37
Variance (s2) is 9.37
We know that standard deviation is the square root of the variance
So the standard deviation (s) = vs2
= v9.37
s = 7.81

Introduction To Parabola

Graphing of a Parabola  : Let us understand  how to Graph Parabola . For graphing parabola we will first draw rough sketches of parabola and various terms associated to them are given below:
1. Equation: y^2 = 4ax, vertex = (0, 0), focus = (a, 0), Latustrectum = 4a, Directrix: x = -a.
2. Equation: y^2 = -4ax, vertex = (0, 0), focus = (-a, 0), Latustrectum = 4a, Directrix: x = a.
3. Equation: x^2 = 4ay, vertex = (0, 0), focus = (0, a), Latustrectum = 4a, Directrix: y = -a.
4. Equation: x^2 = -4ay, vertex = (0, 0), focus = (0, -a), Latustrectum = 4a, Directrix: y = a.

Sketching of curves represented by y = ax^2 + bx + c. For graphing a parabola equation y = ax^2 + bx + c always represents Vertex of Parabola (-b/2a, -D/4a) and axis x = -b/2a.

The parabola graph opens upwards or downward according as a > 0 or < 0. It meets x-axis at (alpha, 0) and (beta, 0), where alpha and beta are the roots of the equation ax^2 + bx + c = 0.

If the roots of this equation are not real, then the parabola does not cross x-axis. In order to draw rough sketch of the parabolas given by the equations of the form y = ax^2 + bx + c, we may follow the following algorithm.

Algorithm: 
Step 1: Obtain the equation and observe the sign of the coefficient of x^2 in it.
Step 2: Put y = 0 in the given equation and get the values of x. Let the values be alpha and beta.
Step 3: Mark the points A(alpha, 0) and B(beta, 0) on x-axis.
Step 4: Draw a parabola passing through points A and B having its vertex on x = -b/2a = (alpha + beta)/2 and opening upward and downward according as the coefficient of x^2 in the given equation is positive or negative.

In the above algorithm, if the values of alpha and beta are imaginary, then the equation y = ax^2 + bx + c represents a parabola having vertex at (-b/2a, -D/4a) and opens upward or downward according as a > 0 or a < 0.

Sketching of curves represented by x = ay^2 + by + c. For a graphing a parabola the equation x = ay^2 + by + c also represents a Parabola Vertex at (-D/4a, -b/2a) axis y = -b/2a and so the parabola graph opens leftward or rightward according as a < 0 or > 0.

It crosses y –axis at (0, alpha) and (0, beta), where alpha and beta are the roots of the equation ay^2 + by + c = 0.
If alpha and beta are not real, then the parabola does not cross y-axis and it opens rightward if a > 0 and leftward if a < 0.
In order to draw a rough sketch of the parabolas given by the equations of the form x = ay^2 + by + c, we may follow the following algorithm.

Algorithm:
Step 1: Obtain the equation and observe the sign of the coefficient of y^2 in it.
Step 2: Put x = 0 in the given equation and get the values of y. Let the values be alpha and beta.
Step 3: Mark the points A(0, alpha) and B(0, beta) on y-axis.
Step 4: Draw a parabola passing through points A and B having its vertex on y = -b/2a = (alpha + beta)/2 and opening upward and downward according as a > 0 or a < 0.

What are lines?

Lines are more related to geometry than to math and could be considered as under math lines. Therefore if the above title was ‘geometry lines’, also it would have been apt.
What are lines?
A line is a set of points that has only one dimension, length.

A line AB is shown in the given diagram. The arrow heads on the line AB show that it is extending endlessly in both the directions and has no end points. Hence a line has no fixed length. Or we can say that the length of a line is infinite. We used the two points A and B to define the line. Therefore in general we can say that two distinct points in a plane determine a line. In other words, if there is only one point under consideration, then there can be infinitely many lines passing through that point as shown in the figure below:

However if we have two points to describe a line, then there can be only one line that passes through both these points.

In co-ordinate geometry, a line can be defined as a set of ordered pairs with a definite property. For example, if we have an ordered pair (x,y) such that y = 2x + 1, that means that all those values of  x and y that satisfy the above equation, would be a part of the line defined by that equation.

Some special subsets of a line:
1. Line segment: A line segment is simply a part of a line that has a specific length and specific end points as shown in the diagram below:

Here CD and PN are both line segments. Note that the end points are dots and not arrows as was the case with line.

2. Ray: A ray is a part of a line that has only one end point. It extends endlessly in one direction. See figure below:

In the above figure AF is a ray. It has one end point at A and extends endlessly in the other direction. Therefore it has a dot at one end and an arrow at the other end.

Two lines can be related to each other in four different ways:

1. Lines that have just one point in common are called intersecting lines.

2. Lines that lie the same plane but never intersect even if produced endlessly in both directions are called parallel lines.

3. Two intersecting lines that form a right angle are called perpendicular lines.

4. Lines that are not in the same plane and do not intersect are called skewed lines.

What is a line plot?

Define line plot:
A line plot is a method of data representation. Of the many methods to represent statistical data, one is graphical representation. A line plot is a type of graphical representation of data. When frequency of data is plotted along a number line it is called a line plot. A line plot makes sense only when the number of observations is lesser than 25. For small number of observations a line plot gives a very good visual representation of data. Following are some examples of line plots:

Line plot graph:

The first step for any data representation is collection of data. Data can be collected from various sources such as, the internet, person to person observations, questionnaire, survey records etc. After collecting the data we can proceed to make the line plot. For making a line plot, first we need to come to a suitable scale for plotting the data. As we can see in the two examples above, in one the scale 1 unit = 5 marks and each unit is subdivided to 5 marks. That is because, the marks are all lying between 30 to 50. For the second example we see that the scale is 1 unit = one day, since we are interested in number of cars sold each day.

Once we decide on the scale, we can actually make the line and mark the numbers as per our scale. Ensure that the distance between consecutive points on the line is proportional to the number the point represents.

Next take the data and list it. Now for each observation, mark a X over the number on the number line that is equal to the observation value. For example, if we have 4 cars sold on Monday, then we mark four Xs over the Monday mark of the number line. Similarly we go on for each of the observation from the data available with us.
See the example chart below;

Understanding the line plot:

We have seen above how to make a line plot. But now the question is that if such a plot is given to us, how do we read that plot (or understand the line plot)? A line plot easily brings out the outlier. An outlier is an observation that is very large or very small as compared to all of the other observations. A line plot also helps us identify gaps and clusters in observations.

Guided reading answers

The guided reading answers are nothing but getting help from others to reading the subjects.In online only we can get the help of tutor for reading any subjects.Initially the math problems can be solved by using some arithmetic operations  like addition,subtraction,division and multiplications and these can be denoted by (+ ,`xx` ,`-` ,÷ ).The following article shows some guided reading answers.

Solved Math Problems with some Guided Reading Answers

Problem 1:

Solve the 56x + 47y = 2632 given equation on the x and y intercepts.

Given:

56x + 47y = 2632

Solution:

56x + 47y = 2632

To find the x intercept of y = 0 and solve for x.

56x + 47(0) = 2632

Solve the value of x.

  x = `2632/56`

  x = 47

  To find the y intercept of x=0 and solve for y.

56(0) + 47y = 2632

Solve the value of y

47y = 2632

  y = `2632/47`

  y = 56.

The equations of x intercept on (47,0) and y intercept on (0,56).

Problem 2:

Solve the problem 185 + 35 ( 40 + 27 ) ÷ 67 – 60 in method of order of operation
Solution:

Given:

 `=>`  185 + 35 ( 40 + 27 ) ÷ 67 – 60

Step 1: we need to simplify the parentheses

 `=>` 185 + 35 `xx` 67 ÷ 67 – 60

Step 2: We need to simplify the multiplication

 `=>` 185 + 2345 ÷ 67 – 60

Step 3: We need to simplify the division

 `=>` 185 + 35 – 60

Step 4: We need to simplify the addition

 `=>`220 –  60

Step 5: We need to simplify the subtraction

 `=>` 160

Answer: 185 + 35 ( 40 + 27 ) ÷ 67 – 60 = 160

Solved more Math Problems with some Guided Reading Answers

Problem 3:

Solve the given problem 11(s – 9) – 6s ` - ` 26 = 13(s + 33)
The Solutions follows below:

Step 1: Given expression is,

11(s – 9) – 6s ` - ` 26 = 13(s + 33)

Step 2: Multiplying the integer terms

11s – 99 – 6s – 26 = 13s + 429.

Step 3: Grouping the above terms

5s –125 = 13s + 429

Step 4:  Add 125 on both sides

5s –125 + 125 = 13s + 429 + 125

Step 5: Grouping the above terms

5s = 13s + 554

Step 6: Subtract 13s by on both sides

5s `-` 13s = 13s `-` 13s + 554

Step 7: Grouping the above terms

–8s = 554

S = `- 554/8`

The required answers is

S = `- 554/8`

Problem 4:

Solve given problem (156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20)
 Solution:

The problem can be solved in simplifying method .

Step 1: (156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20)

Step 2: 156x2 – 141x – 91 + 213x2 – 181x – 144 + 916x2 `-`    41x `-` 20

Step 3:  1285x2 – 363x – 255      

The required answer is

(156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20) = 1285x2 – 363x – 255