The document defines average and provides formulas for calculating average. It discusses how to calculate the average of consecutive even/odd numbers, what happens when quantities are added to or replaced in a group, and how to calculate averages of specific data sets like positive integers. It then provides 10 problems calculating averages based on the information and formulas provided in the definitions section. The problems include calculating averages of data sets, determining values based on changes to averages, and identifying values that would produce a given average.
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
This PPT tells you how to tackle with questions based on Average in CAT 2009. Ample of PPTs of this type on every topic of CAT 2009 are available on www.tcyonline.com
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
This PPT tells you how to tackle with questions based on Average in CAT 2009. Ample of PPTs of this type on every topic of CAT 2009 are available on www.tcyonline.com
This Slideshare presentation tells you how to tackle with questions based on number of theory. Ample of PPT of this type on every topic of CAT 2009 are available on www.tcyonline.com
Pedagogy of Mathematics - Part II (Numbers and Sequence - Ex 2.1), Numbers and Sequence, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, Euclid's Division Lemma, Euclid's Division algorithm,
Pedagogy of Mathematics - Part II (Numbers and Sequence - Ex 2.3), Numbers and Sequence, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, Modular arithmetic, congruence module, connecting euclid's lemma and modular arithmetic, Module operations,
More companies in the process of recruitment, play more emphasis in the topic of numbers in numerical aptitude. Especially for AMCAT aspirants this is very much useful.
quantitative aptitude, maths
applicable to
Common Aptitude Test (CAT)
Bank Competitive Exam
UPSC Competitive Exams
SSC Competitive Exams
Defence Competitive Exams
L.I.C/ G. I.C Competitive Exams
Railway Competitive Exam
University Grants Commission (UGC)
Career Aptitude Test (IT Companies) and etc.
1. Which of the following is the correct matrix representation .docxjackiewalcutt
1.
Which of the following is the correct matrix representation of the October Inventory of small and large t-shirts and pants?
Inventory for August
Inventory for September
Inventory for October
Small
Large
Small
Large
Small
Large
T-Shirts
2
3
4
6
7
8
Pants
1
4
3
5
5
6
Answer
1.
If find -2A.
Answer
1.
If and find A - B.
Answer
1.
If find 2A.
Answer
1.
Matrix A has dimensions 2 x 3. Matrix B has dimensions 3 x 6. These two matrices can be multiplied to find the product AB.
Answer True False
1.
Evaluate
Answer
Does not exist.
1.
Evaluate
Answer
Does not exist
1.
Evaluate
Answer
Does not exist.
1.
Find the determinant of
Answer
14
-14
1
-1
1.
Find the determinant of
Answer
-48
-56
56
48
1.
What is the determinant of the following matrix?
Answer
-10
-14
-20
-25
1.
What is the determinant of the following matrix? Does the matrix have an inverse?
Answer
5; no
5; yes
1; no
1; yes
1.
Is the following system consistent or inconsistent?
8x + y + 4 = 0
y = -8x - 4
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
7x + 6 = -2y
-14x -4y + 2 = 0
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
-2x - 2y = 6
10x + 10y = -30
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
2y = x - 7
-2x - 6y = -14
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
y = 2x + 5
-2x + y = -2
Answer
consistent
inconsistent
If a system has exactly one solution it is called _______________.
Answer
consistent
inconsistent
independent
dependent
1.
Does the following system of equations have a solution?
Answer
Yes
No
1.
What is the approximate solution of the following system of equations?
Answer
(2, -7)
(-7, 2)
(7, 2)
(-7, -2)
1.
Solve the following system of equations by using the substitution method.
3y – 2x = 11
y + 2x = 9
Answer
(2, 5)
(-2, -5)
(4, 5)
Inconsistent
1.
Solve the following system of equations by using the substitution method.
y = -3x + 5
5x – 4y = -3
Answer
(-1, -2)
(2, 1)
(3, 4)
(1, 2)
1.
Solve the following system of equations by using the elimination method.
x – y = 11
2x + y = 19
Answer
(1, 10)
(-1, -1)
(12, 2)
(10, -1)
1.
Solve the following system of equations by using the elimination method.
-4x – 2y = -12
4x + 8y = -24
Answer
(5, 3)
(-6, -6)
(3, 1)
(6, -6)
1.
Solve the following system of equations using matrices.
3x – 2y = 31
3x + 2y = -1
Answer
(5, -8)
(3, 5)
(-3, 9)
(2, 8)
1.
Solve the following system of equations using matrices.
4x + 5y = -7
3x – 6y = 24
Answer
(1, 4)
(2, -3)
(5, 6)
(3, 4)
1.
The sum of two numbers is 7. Four times the first number is one more than 5 times the second. Find the two numbe ...
This Slideshare presentation tells you how to tackle with questions based on number of theory. Ample of PPT of this type on every topic of CAT 2009 are available on www.tcyonline.com
Pedagogy of Mathematics - Part II (Numbers and Sequence - Ex 2.1), Numbers and Sequence, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, Euclid's Division Lemma, Euclid's Division algorithm,
Pedagogy of Mathematics - Part II (Numbers and Sequence - Ex 2.3), Numbers and Sequence, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, Modular arithmetic, congruence module, connecting euclid's lemma and modular arithmetic, Module operations,
More companies in the process of recruitment, play more emphasis in the topic of numbers in numerical aptitude. Especially for AMCAT aspirants this is very much useful.
quantitative aptitude, maths
applicable to
Common Aptitude Test (CAT)
Bank Competitive Exam
UPSC Competitive Exams
SSC Competitive Exams
Defence Competitive Exams
L.I.C/ G. I.C Competitive Exams
Railway Competitive Exam
University Grants Commission (UGC)
Career Aptitude Test (IT Companies) and etc.
1. Which of the following is the correct matrix representation .docxjackiewalcutt
1.
Which of the following is the correct matrix representation of the October Inventory of small and large t-shirts and pants?
Inventory for August
Inventory for September
Inventory for October
Small
Large
Small
Large
Small
Large
T-Shirts
2
3
4
6
7
8
Pants
1
4
3
5
5
6
Answer
1.
If find -2A.
Answer
1.
If and find A - B.
Answer
1.
If find 2A.
Answer
1.
Matrix A has dimensions 2 x 3. Matrix B has dimensions 3 x 6. These two matrices can be multiplied to find the product AB.
Answer True False
1.
Evaluate
Answer
Does not exist.
1.
Evaluate
Answer
Does not exist
1.
Evaluate
Answer
Does not exist.
1.
Find the determinant of
Answer
14
-14
1
-1
1.
Find the determinant of
Answer
-48
-56
56
48
1.
What is the determinant of the following matrix?
Answer
-10
-14
-20
-25
1.
What is the determinant of the following matrix? Does the matrix have an inverse?
Answer
5; no
5; yes
1; no
1; yes
1.
Is the following system consistent or inconsistent?
8x + y + 4 = 0
y = -8x - 4
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
7x + 6 = -2y
-14x -4y + 2 = 0
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
-2x - 2y = 6
10x + 10y = -30
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
2y = x - 7
-2x - 6y = -14
Answer
consistent
inconsistent
1.
Is the following system consistent or inconsistent?
y = 2x + 5
-2x + y = -2
Answer
consistent
inconsistent
If a system has exactly one solution it is called _______________.
Answer
consistent
inconsistent
independent
dependent
1.
Does the following system of equations have a solution?
Answer
Yes
No
1.
What is the approximate solution of the following system of equations?
Answer
(2, -7)
(-7, 2)
(7, 2)
(-7, -2)
1.
Solve the following system of equations by using the substitution method.
3y – 2x = 11
y + 2x = 9
Answer
(2, 5)
(-2, -5)
(4, 5)
Inconsistent
1.
Solve the following system of equations by using the substitution method.
y = -3x + 5
5x – 4y = -3
Answer
(-1, -2)
(2, 1)
(3, 4)
(1, 2)
1.
Solve the following system of equations by using the elimination method.
x – y = 11
2x + y = 19
Answer
(1, 10)
(-1, -1)
(12, 2)
(10, -1)
1.
Solve the following system of equations by using the elimination method.
-4x – 2y = -12
4x + 8y = -24
Answer
(5, 3)
(-6, -6)
(3, 1)
(6, -6)
1.
Solve the following system of equations using matrices.
3x – 2y = 31
3x + 2y = -1
Answer
(5, -8)
(3, 5)
(-3, 9)
(2, 8)
1.
Solve the following system of equations using matrices.
4x + 5y = -7
3x – 6y = 24
Answer
(1, 4)
(2, -3)
(5, 6)
(3, 4)
1.
The sum of two numbers is 7. Four times the first number is one more than 5 times the second. Find the two numbe ...
Arithmetic progression
For class 10.
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant
1Bivariate RegressionStraight Lines¾ Simple way to.docxaulasnilda
1
Bivariate Regression
Straight Lines
¾ Simple way to describe a relationship
¾ Remember the equation for a straight line?
z y = mx + b
¾ What is m? What is b?
¾ How do you compute the equation?
(x1,y1)
(x2,y2)
What if every point is
not on the line?
¾ Straight line may be good description even
if not all points are on the line
Computing the line
when points are scattered
¾ = a + bX
¾ Y-hat means predicted value of Y
¾ Computing the slope:
¾ b = 𝑋−𝑋 𝑌−𝑌
𝑋−𝑋
¾ I ill ri e/r n, b no e al o
consider variability in X and Y
Computing the intercept
¾ a = - bX
¾ Need o pl g in al e of (X, )
¾ Can e j an Y or X!
z Line would be very different depending on
which ones you chose
¾ Must have X and Y that we know are on
the line
z mean of X and mean of Y
2
Computing the intercept
¾ Regression line will always go through the
mean of X and mean of Y
¾ A = 𝑌 - b𝑋
¾ Le r it with our example from before
X
(# of kids)
Y
(hours of
housework) 𝑋 𝑋 𝑌 𝑌 𝑋 𝑋 𝑌 𝑌 𝑋 𝑋
1 1 -1.75 -2.5 4.375 3.063
1 2 -1.75 -1.5 2.625 3.063
1 3 -1.75 -0.5 0.875 3.063
2 6 -0.75 2.5 -1.875 0.563
2 4 -0.75 0.5 -0.375 0.563
2 1 -0.75 -2.5 1.875 0.563
3 5 0.25 1.5 0.375 0.063
3 0 0.25 -3.5 -0.875 0.063
4 6 1.25 2.5 3.125 1.563
4 3 1.25 -0.5 -0.625 1.563
5 7 2.25 3.5 7.875 5.063
5 4 2.25 0.5 1.125 5.063
MX=2.75 MY=3.5 = 0 = 0 = 18.5 = 24.25
Computing the equation
¾ b = .
.
.76
¾ a = 3.5 - .76(2.75)
¾ = 1.41
¾ = 1.41 + .76X
Interpreting the coefficients
¾ Slope
z For a one unit increase in X, we predict a b
unit increase in Y
What does that mean for this study?
¾ Intercept
z The predicted value of Y when X = 0
What does that mean for this study?
Interpreting the coefficients
¾ Slope
z For each additional child, we predict
parents will do an additional .76 hours of
housework per day
¾ Intercept
z For a family with zero kids, we predict they
will do 1.41 hours of housework per day
Drawing the regression line
¾ Need to plot two points
z 𝑋, 𝑌
z Y-intercept
1
Scatterplots and
Correlation
Correlation
¾ Useful tool to assess relationships
¾ Must have two variables measured on one set of
people
¾ Correlation only measures strength of linear
association
Linear relationships are
not perfect lines
¾ Variables have variability (duh)
¾ Relationships may be generally linear
even if all points are not on the line
Magnitude of r Not all relationships are linear
2
Properties of r
¾ X & Y must be quantitative
z Interval or ratio
¾ I doe n ma e hich a iable i edic o
and which is response
z rxy = ryx
Properties of r
¾ Correlation has no units
z So r can be compared for different variables
¾ Value of r is always between -1 and +1
Computing r
¾ Consider deviations around mean of X & Y
¾ (X 𝑋) (Y 𝑌)
Cross-Product
¾ To consider X & Y together, multiply their
deviations
¾ (X 𝑋)(Y 𝑌)
¾ Sign will be positive or negative
¾ Sum of cross-pr
MATHS - Linear equation in two variable (Class - X) Maharashtra Board
apptitude
1. www.sakshieducation.com
AVERAGE
Definition: Average (A) is the ratio between sum (S) of the quantities and the
number (N) of quantities.
S
A=
N
S = A× N
Formulae:
1. The average of n-consecutive even integers or n-consecutive odd integers is equal
to the middle number if n is odd.
2. The average of n-consecutive even integers or n-consecutive odd integers is equal
to the Average of middle two numbers if n is even.
Note: In the above case if the average is x the middle two numbers will be (x – 1)
and (x + 1) respectively.
3. The average of a group of n-quantities is A. If one more quantity whose value is x
is added to the group such that the average increases by ‘i’ then x = A + i × (n + 1)
Note: In the above case if there is a reduction in the average then take ‘i’ as
negative.
4. The average of a group of n-quantities is A. If a quantity whose value is y is
replaced by another quantity whose value is x such that the average increases by ‘i’.
Then x = y +i × n
Note 1: If there is a reduction in average take ‘i’ as negative.
Note 2: The original average ‘A’ has no effect on the equation.
5. The average weight of a group of n-quantities is A but while taking the values one
⎛q− p⎞
quantity ‘p’ is erroneously read as ‘q’. Then the actual average = A − ⎜ ⎟
⎝ n ⎠
n +1
6. The average of first n-positive integers =
2
7. The average of first n-positive even integers = n + 1
8. The average of first n-positive odd integers = n
( n + 1) ( 2n + 1)
9. The average of squares of first n-numbers =
6
PROBLEMS
1. Find the average of the following set of scores.
216, 463, 154, 605, 446, 336
1) 387 2) 370 3) 379 4) 380 5) None of
these
ANSWER: 2
S = 216 + 463 + 154 + 605 + 446 + 336 = 2220
N=6
S 2220
A= = = 370
N 6
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2. The average of four consecutive even numbers A, B, C and D respectively is 55.
What is the product of A and C?
1) 2652 2) 3248 3) 3024 4) 2808 5) None of
these
ANSWER: 5
The average of A, B, C and D = Average of B and C
But B and C are consecutive even numbers. Their average will be equal to the odd
number in between them (which is 55)
B = 55 – 1 = 54 C = 55 + 1 = 56
A = B – 2 = 54 – 2 = 52
A×C = 52×56 = 2912
3. Average of four consecutive odd numbers is 106. What is the third number in
ascending order?
1) 107 2) 111 3) 113 4) Cannot be determined 5) None of
these
ANSWER: 1
A, B, C and D be the four consecutive odd numbers in ascending order.
Their average = Average of B and C = the even number between B and C = 106
But B = 106 – 1 = 105 and C = 106 + 1 = 107
The third number in ascending order = C = 107
4. The average of odd numbers up to 100 is
1) 50.5 2) 50 3) 49.5 4) 49 5) None of
these
ANSWER: 2
From 1 to 100 the number of odd numbers = 50
But the sum of first n-odd numbers = n 2
n2
Their average = =n
n
The average of first 50 odd numbers = 50
5. The sum of five consecutive numbers is 190. What is the sum of the largest and
the smallest number?
1) 75 2) 77 3) 76 4) 73 5) None of these
ANSWER: 3
The sum of five consecutive numbers = 190
190
Their average = = 38
5
In a set of consecutive numbers the average of the largest and the smallest
numbers is equal to the average of all the numbers.
Average of the largest and the smallest = 38
Their sum = 38×2 = 76
6. The average of eight successive numbers is 6.5. The average of the smallest and
the greatest numbers among them will be
1) 4 2) 6.5 3) 7.5 4) 9 5) None of these
ANSWER: 2
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The average of the smallest and the largest of a set of n-consecutive numbers is
equal to the average of given set of numbers.
Average of the smallest and the largest = 6.5
7. The average of five numbers is 49. The average of the first and the second
numbers is 48 and the average of the fourth and fifth numbers is 28. What is the
third number?
1) 92 2) 91 3) 95 4) Cannot be determined 5) None of
these
ANSWER: 5
A, B, C, D and E be the five numbers
A + B + C + D + E = 49×5 = 245
The sum of A and B = A + B = 48×2 = 96
The sum of D and E = D + E = 28×2 = 56
C = (A + B + C + D + E) – (A + B + D + E) = 245 – (96 + 56) = 93
8. The average age of a man and his son is 48 years. The ratio of their ages is 2 : 1
respectively. What is the son’s age?
1) 28 years 2) 35 years 3) 24 years 4) 32 years 5) None of
these
ANSWER: 4
Total age of man and son = 48×2 = 96
Ratio of their ages = 2 : 1
96 96
Son’s age = ×1 = × 1 = 32
( 2 + 1) 3
9. Average score of Rahul, Manish and Suresh is 63. Rahul’s score is 15 less than
Ajay and 10 more than Manish. If Ajay scored 30 marks more than the average score
of Rahul, Manish and Suresh, what is the sum of Manish’s and Suresh’s score?
1) 120 2) 111 3) 117 4) Can’t be determined 5) None of
these
ANSWER: 2
Total score of Rahul, Manish and Suresh = 63×3 = 189
Ajay score is 30 more than the above average Ajay’s score = 63 + 30 = 93
Rahul’s score is 15 less than Ajay’s score, so Rahul’s score = 93 – 15 = 78
Sum of Manish’s and Suresh’s score = 189 – 78 = 111
10. The average marks of a student in seven subjects is 41. After reevaluation in one
subject the marks were changed to 42 from 14 and in remaining subjects the marks
remain unchanged. What are the new average marks?
1) 45 2) 44 3) 46 4) 47 5) None of
these
ANSWER: 1
After reevaluation, increase in total marks = 42 – 14 = 28
28
Increase in the average = =4
7
New average = 41 + 4 = 45
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11. The average of 10 numbers is calculated as 15. It is discovered later on that while
calculating the average one number, namely 36, was wrongly read as 26. The correct
average is
1) 20 2) 18 3) 16 4) 14 5) None of these
ANSWER: 3
A number 36 was wrongly read as 26
Reduction in total due to error = 36 – 26 = 10
10
Reduction due to error in average = = 1
10
Actual average = 15 + 1 = 16
12. The average age of 24 boys in a class is 11. When the teacher’s age is included, the
average increases by one. What is the age of the teacher?
1) 34 years 2) 42 years 3) 36 years 4) 48 years 5) None of
these
ANSWER: 3
Total age of 24 boys = 24×11 = 264
The average age of 24 boys and teacher = 11 + 1 = 12
Total age of 24 boys and teacher = 25×12 = 300
Teacher’s age = 300 – 264 = 36
SHORTCUT METHOD:
N = 24 A = 11 i = +1 x=?
x = A + i(N + 1) = 11 + 1×(24 + 1) = 11 + 25 = 36
13. Average weight of 25 persons is increased by 1 kg when one man weighing 60 kg
is replaced by a new person. Weight of new person is
1) 50 kg 2) 61 kg 3) 86 kg 4) 85 kg 5) None of
these
ANSWER: 4
Applying the formula discussed in theory
N = 25 i = +1 y = 60 x=?
x = y + i × N = 60 + 1 × 25 = 85 kg
14. The average of the first 100 positive integers is
1) 100 2) 51 3) 50.5 4) 49.5 5) None of
these
ANSWER: 3
n ( n + 1)
The sum of first n-positive integers =
2
n ( n + 1) n + 1
Their average = =
2n 2
100 + 1 101
The sum of first 100 positive integers = = = 50.5
2 2
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