IIM CAT Preparation Tips

CAT Functions - Questions

2011-12-09T21:04:00.000-08:00

Have given below a few questions on functions.

1. Give the domain and range of the following functions

i. f(x) = x² + 1
ii. g(x) = log (x + 1)
iii. h(x) = 2^xiv. f(x) = 1/ (x+1)
v. p(x) = | x + 1|
vi. q(x) = [2x], where [x] gives the greatest integer less than or equal to x

Happy cracking

CAT Data Sufficiency Solutions

2011-09-06T19:52:00.000-07:00

Have given below solutions to the questions on data sufficiency

1. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(1,6)?

Stmt 1. m = 3. Not sufficient.
Stmt 2. n = 2.5 Not sufficient.

Either statement alone is not sufficient. Both put together, we can answer the questions. Ans B

2. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(5,6)?

Stmt 1. m = 5. Not sufficient. It tells us that all points lie on the line x = 5, but this is not enough
Stmt 2. n = 3 Not sufficient.

Both put together, this is enough.

In the above question, would n = 1 have been sufficient? Think about that.

3. A survey of 100 people tried to find the number of people who can write with both their left and right hands. What is the maximum number of people who could write left-handed and right-handed?

Stmt 1. 50 people can write only with their left hand. 40 people can write only with their right hand. Sufficient: Maximum of 10 people could write left-handed and right-handed.
Stmt 2. 50 people can write with their left hand. 40 people can write with their right hand. Sufficient: Maximum of 40 people could write left-handed and right-handed.

Answer Choice C

4. What is the slope of a line?

Stmt 1: The line makes 135 degrees with the negative direction of x - axis. Sufficient: The line makes 45 degrees with positive x axis. This should be enough.
Stmt 2: The line makes an isosceles right triangle with the coodinate axes and the product of the intercepts is negative. Sufficient: Either both intercepts are positive and equal or negative and equal. Slope = -1

Answer Choice C

CAT Data Sufficiency,

2011-09-04T21:31:00.000-07:00

Following questions are courtesy Naveenan Ramachandran, head of 2IIM Mumbai.

Mark A) If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone
Mark B) If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together
Mark C) If the question can be answered with either statement alone
Mark D) If the question cannot be answered with the information provided

1. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(1,6)?

Stmt 1. m = 3
Stmt 2. n = 2.5

2. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(5,6)?

Stmt 1. m = 5
Stmt 2. n = 3

3. A survey of 100 people tried to find the number of people who can write with both their left and right hands. What is the maximum number of people who could write left-handed and right-handed?

Stmt 1. 50 people can write only with their left hand. 40 people can write only with their right hand.
Stmt 2. 50 people can write with their left hand. 40 people can write with their right hand.

4. What is the slope of a line?

Stmt 1: The line makes 135 degrees with the negative direction of x - axis.
Stmt 2: The line makes an isosceles right triangle with the coodinate axes and the product of the intercepts is negative.

Solutions to CAT Geometry Data Sufficiency Questions

2011-08-29T21:57:00.001-07:00

Have given below the solutions to the questions on geometry DS. The solutions are courtesy Vimal Gopinath (person incharge of 2IIM Bengaluru)

Qn 1: Is triangle ABC obtuse angled?
I) a^2 + b^2 > c^2 - Not enough. We don’t have info about b^2 or a^2
II) The circumcenter of the triangle does not lie inside the triangle - Not enough. The triangle could be right-angled as well.
Combination is also not enough, it is valid for both right angled and obtuse angled triangles.
D

Qn 2: Do the two circles with centers A and B and radii R and r intersect each other
I) AB > R - r – Not enough. May intersect, may be “parallel” or "disjoint"
II) AB > R + r – Sufficient. Circles cant intersect. Have to be separate.
A

Qn 3: Trapezium ABCD is such that AB is parallel to CD. Is this trapezium anisosceles trapezium?
I) Angle B and D are supplementary Sufficient. If two base angles are equal, then the trapezium has to be isosceles.
II) The quadrilateral is inscribed inside a circle. Sufficient. All trapeziums inscribed in circles have to be isosceles. (Think about the proof for this)
C

Qn 4: Circle C has center O, and a chord AB such that angle AOB = 80 degrees.Does point E lie inside the circle

I) Angle AEB > 50 degrees Insufficient.E could lie on the minor segment ADB or slightly outside or slightly inside the circle.
II) Angle AEB < 30 degrees. Sufficient. All the angles inside the circle will be in the range from 40 – 140 degrees. Anything less than 40 will have to be outside the circle.)
A

CAT Geometry Data Sufficiency questions

2011-08-28T19:07:00.000-07:00

For the following questions,

Mark A) If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone
Mark B) If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together
Mark C) If the question can be answered with either statement alone
Mark D) If the question cannot be answered with the information provided

Qn 1: Is triangle ABC obtuse angled
I) a^2 + b^2 > c^2
II) The circumcenter of the triangle does not lie inside the triangle

Qn 2: Do the two circles with centers A and B and radii R and r intersect each other
I) AB > R - r
II) AB > R + r

Qn 3: Trapezium ABCD is such that AB is parallel to CD. Is this trapezium an isosceles trapezium?
I) Angle B and D are supplementary
II) The quadrilateral is inscribed inside a circle

Qn 4: Circle C has center O, and a chord AB such that angle AOB = 80 degrees. Does point E lie inside the circle
I) Angle AEB > 50 degrees
II) Angle AEB < 30 degrees

CAT Geometry Solutions

2011-08-26T22:29:00.000-07:00

Have given below the solutions to the questions on basic geometry

1. Perimeter of a triangle with integer sides is equal to 15. How many such triangles are possible?

This is just a counting question, with the caveat that sum of two sides should be greater than the third. Let us assume a < b < c

a = 1, Possible triangle 1, 7, 7
a = 2, possible triangle 2, 6, 7
a = 3, possible triangles 3, 6, 6 and 3, 5, 7
a = 4, possible triangles 4, 4, 7 and 4, 5, 6

Again, from comments,
a = 5, possible triangle is 5,5,5,

There are totally 7 triangles possible

2. Triangle ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?

xz = 12

x,z can be 1, 12 or 2, 6 or 3, 4

Possible triangles
1-12-12
2-6-5; 2-6-6; 2-6-7
3-4-2; 3-4-3; 3-4-5; 3-4-6.

As pointed out in the comments section, I have missed the triangle 3-4-4.

There are totally 9 triangles.

3. Triangle has sides a^2, b^2 and c^2. Then the triangle with sides a, b, c has to be - a) Right angled b) Acute-angled c) Obtuse angled d) can be any of these three

Assuming a < b < c, we have a^2 + b^2 > c^2. This implies the triangle with sides a, b, c has to be acute-angled.

P.S: Big thanks to 'maniac' for pointing out the errors

CAT Geometry

2011-08-23T05:01:00.000-07:00

Have given below a few questions on basic geometry

1. Perimeter of a triangle with integer sides is equal to 15. How many such triangles are possible?

2. Triangle ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?

3. Triangle has sides a^2, b^2 and c^2. Then the triangle with sides a, b, c has to be - a) Right angled b) Acute-angled c) Obtuse angled d) can be any of these three

Solutions to questions on Coordinate Geometry

2011-08-22T18:23:00.000-07:00

Have given the solutions to questions on Cogeo

1. Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P?

Any point on the line x/3 + y/5 =1 will have the shortest overall distance. However, we need to have integral coordinates. So, we need to find points with integral coordinates as close as possible to the line 5x + 3y =15.

Substitute x =1, we get y = 2 or 3
Substitute x = 2, we get y = 1 or 2

Sum of distances for ( 1, 2) = sqrt(8) + sqrt (10)
Sum of distances for ( 1, 3) = sqrt(13) + sqrt (5)
Sum of distances for ( 2, 1) = sqrt(2) + sqrt (20)
Sum of distances for ( 2, 2) = sqrt(5) + sqrt (13)

sqrt(5) + sqrt (13) is the shortest distance. Sum of abscissa + ordinate = 4

2. Region R is defined as the region in the first quadrant satisfying the condition 3x + 4y < 12. Given that a point P with coordinates (r, s) lies within the region R, what is the probability that r > 2?

Line 3x + 4y =12 cuts the x-axis at (4, 0) and y axis at (0, 3)

The region in the first quadrant satisfying the condition 3x + 4y < 12 forms aright triangle with sides 3, 4 and 5. Area of this triangle = 6 sq units.

The lines x = 2 and 3x + 4y = 12 intersect at (2, 1.5). So, the region r > 2, 3x + 4y < 12 also forms a right triangle. This right triangle has base sides 2, 1.5. Area of this triangle = 1.5

Probability of the point lying in said region = 1.5/6 = 1/4

3. Region Q is defined by the equation 2x + y < 40. How many points (r, s) exist such that r is a natural number and s is a multiple of r?

When r = 1, s can take 37 values [37/1]
When r = 2, s can take 17 values [35/2]
When r = 3, s can take 11 values [33/3]
When r = 4, s can take 7 values [31/4]
When r = 5, s can take 5 values [29/5]
When r = 6, s can take 4 values [27/6]
When r = 7, s can take 3 values [25/7]
When r = 8, s can take 2 values [23/8]
When r = 9, s can take 2 values [21/9]
When r = 10, s can take 1 values [19/10]
When r = 11, 12, 13 s can take 1 values one value each

Totally, there are 92 values possible

CAT Coordinate geometry

2011-08-21T19:07:00.000-07:00

Few questions on coordinate geometry. These are slightly unconventional questions. CAT has stayed away from raw, brute-force cogeo questions for a while now. The questions that we see in CAT have not been algebraic. They have either been of the visual type or of the type where one has to do trial and error. So, the cogeo basics are important, but no point learning all kinds of fancy formula stuff.

1. Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P?

2. Region R is defined as the region in the first quadrant satisfying the condition 3x + 4y < 12. Given that a point P with coordinates (r, s) lies within the region R, what is the probability that r > 2?

3. Region Q is defined by the equation 2x + y < 40. How many points (r, s) exist such that r is a natural number and s is a multiple of r?

Solutions to questions on permutations and combinations

2011-08-21T05:23:00.000-07:00

Have given below solutions to the questions on permutations and combinations

1. a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)

Exactly one of ab, bc and ca is odd => Two are odd and one is even

abc is a multiple of 4 => the even number is a multiple of 4

The arithmetic mean of a and b is an integer => a and b are odd

and so is the arithmetic mean of a, b and c. => a+ b + c is a multiple of 3

c can be 4 or 8.
c = 4; a, b can be 3, 5 or 5, 9
c = 8; a, b can be 3, 7 or 7, 9

Four triplets are possible

2. A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?

Number should be a multiple of 3 and 4. So, the sum of the digits should be a multiple of 3. WE can either have all seven digits as 3, or have three 2's and four 3's, or six 2's and a 3. (The number of 2's should be a multiple of 3).

For the number to be a multiple of 4, the last 2 digits should be 32. Now, let us combine these two.

All seven 3's - No possibility
Three 2's and four 3's - The first 5 digits should have two 2's and three 3's in some order. No of possibilities = 5!/3!*2! = 10

Six 2's and one 3 - The first 5 digits should all be 2's. So, there is only one number 2222232

3. How many factors of 2^5 * 3^6 * 5^2 are perfect squares?

Any factor of this number should be of the form 2^a * 3^b * 5^c. For the factor to be a perfect square a,b,c have to be even. a can take values 0, 2, 4. b can take values 0,2, 4, 6 and c can take values 0,2. Total number of perfect squares = 3 * 4 * 2 = 24

4. How many factors of 2^4 * 5^3 * 7^4 are odd numbers?

Any factor of this number should be of the form 2^a * 3^b * 5^c. For the factor to be an odd number, a should be 0. b can take values 0,1, 2,3, and c can take values 0, 1, 2,3, 4. Total number of odd factors = 4 * 5 = 20

5. A number when divided by 18 leaves a remainder 7. The same number when divided by 12 leaves a remainder n. How many values can n take?

Number can be 7, 25, 43, 61, 79.
Remainders when divided by 12 are 7 and 1.

n can take exactly 2 values

Permutation and Combination (with a touch of Number Theory)

2011-08-16T20:46:00.000-07:00

Have given below a few questions on Counting that also include a bit of Number Theory. These types of questions are commonly tested.

1. a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)

2. A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?

3. How many factors of 2^5 * 3^6 * 5^2 are perfect squares?

4. How many factors of 2^4 * 5^3 * 7^4 are odd numbers?

5. A number when divided by 18 leaves a remainder 7. The same number when divided by 12 leaves a remainder n. How many values can n take?

Geometry Concept testers - Solutions

2011-06-10T20:53:00.000-07:00

Given below is the answer key to the set of True of False questions

1. Any two triangles that are congruent to each other will also be similar to each other - TRUE

2. If in a triangle with sides a, b and c, if a^2 + b^2 > c^2 the triangle has to be acute-angled - FALSE. This is true only if c is the largest side. Thinking about it differently, for any triangle, in sum of the squares of the two larger sides will be greater than the square of the smallest side. So, this would imply that any triangle has to be acute angled :)

3. Any parallelogram inscribed inside a circle has to be a rectangle - TRUE
Opposite angles of a parallelogram are equal
Opposite angles of a cyclic quadrilateral are supplementary

4. If in a triangle the orthocenter, incenter and circumcenter are collinear the triangle has to be isosceles - TRUE
In any triangle, orthocenter, circumcenter and centroid will be collinear. So, if orthocenter, incenter and circumcenter are collinear, then all 4 points lie on the same straight line, which implies the triangle has to be isosceles

5. There will be a unique circle passing through any three points - FALSE
There is a unique circle passing through any three non-collinear points

6. If two circles with centers A and B and radii r and R intersect, then AB > R - r - TRUE
If two circles intersect, then AB, R and r form sides of a triangle. => r + AB > R => AB > R - r

7. Circumradius of a triangle cannot be greater than the three sides of the triangle - FALSE
Think about this differently. Draw a large circle, draw a small triangle inside this circle in a small segment of this circle. The circle is the circumcircle for this triangle

8. For any obtuse-angled triangle, the orthocenter and circumcenter will lie outside the triangle - TRUE

9. In a scalene triangle, the sum of the three medians will be greater than the sum of the three altitudes - TRUE
Shortest distance from a point to a line is the perpendicular distance. => Altitude is always less than or equal to median. In a scalene triangle, altitude will be less than the median

10. In circumcircle and incircle are concentric, the triangle has to be equilateral - TRUE

Geometry - Concept testers

2011-06-06T22:03:00.000-07:00

For each of the statements given below, say true or false

1. Any two triangles that are congruent to each other will also be similar to each other
2. If in a triangle with sides a, b and c, if a^2 + b^2 > c^2 the triangle has to be acute-angled
3. Any parallelogram inscribed inside a circle has to be a rectangle
4. If in a triangle the orthocenter, incenter and circumcenter are collinear the triangle has to be isosceles.
5. There will be a unique circle passing through any three points
6. If two circles with centers A and B and radii r and R intersect, then AB > R - r
7. Circumradius of a triangle cannot be greater than the three sides of the triangle
8. For any obtuse-angled triangle, the orthocenter and circumcenter will lie outside the triangle
9. In a scalene triangle, the sum of the three medians will be greater than the sum of the three altitudes.
10. In circumcircle and incircle are concentric, the triangle has to be equilateral

Solutions to Number Theory and Counting questions

2011-06-03T21:10:00.001-07:00

Given below are the solutions to these Number theory questions.

1. From the digits 2,3,4,5,6 and 7, how many 5-digit numbers can be formed that have distinct digits and are multiples of 12?

Any multiple of 12 should be a multiple of 4 and 3. First, let us look at the constraint for a number being a multiple of 3. Sum of the digits should be a multiple of 3. Sum of all numbers from 2 to 7 is 27. So, if we have to drop a digit and still retain a multiple of 3, we should drop either 3 or 6.

So, the possible 5 digits are 2, 4, 5, 6, 7 or 2, 3, 4, 5, 7.

When the digits are 2, 4, 5, 6, 7. the last two digits possible for the number to be a multiple of 4 are 24, 64, 52, 72, 56, 76. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 36 different numbers

When the digits are 2, 3, 4, 5, 7. the last two digits possible for the number to be a multiple of 4 are 32, 52, 72, 24. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 24 different numbers

Overall, there are 60 different 5-digit numbers possible

2. All numbers from 1 to 200 (in decimal system) are written in base 6 and base 7 systems. How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?

If a number written in base 6 ends with a zero, it should be a multiple of 6. In other words, the question wants us to find all numbers from 1 to 200 that are not multiples of 6 or 7. There are 33 multiples of 6 less than 201. There are 28 multiples of 7 less than 201. There are 4 multiples of 6 & 7 (or multiple of 42) from 1 to 200.

So, total multiples of 6 or 7 less than 201 = 33 + 28 - 4 = 57. Number of numbers with non-zero units digit = 200-57 = 143.

3. All numbers from 1 to 150 (in decimal system) are written in base 6 notation. How many of these will not contain any zero?

Any multiple of 6 will end in a zero. There are 25 such numbers. Beyond this, we can have zero as the middle digit of a 3-digit number. This will be the case for numbers from 37-41, 73-77, 109-113 and 145-149. There are 20 such numbers. Overall, there are 45 numbers that have a zero in them.

4. How many factors of 1080 are perfect squares?

1080 = 2^3 * 3^3 * 5. For any perfect square, all the powers of the primes have to be even numbers. So, if the factor is of the form 2^a * 3^b * 5^c. The values 'a' can take are 0 and 2, b can take are 0 and 2, and c can take the value 0. Totally there are 4 possibilities. 1, 4, 9, and 36.

Few Questions on Number Theory and Counting (combined)

2011-06-02T19:12:00.000-07:00

CAT has been consistently asking questions combining basic number theory and counting. So, it is probably good practice to have a go at these.

1. From the digits 2,3,4,5,6 and 7, how many 5-digit numbers can be formed that have distinct digits and are multiples of 12?

2. All numbers from 1 to 200 (in decimal system) are written in base 6 and base 7 systems. How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?

3. All numbers from 1 to 150 (in decimal system) are written in base 6 notation. How many of these will not contain any zero?

4. How many factors of 1080 are perfect squares?

Solutions to these questions can be found here.

CAT Number Theory

2011-04-21T22:26:00.001-07:00

1. The six faces of a cube have 6 distinct natural numbers written on them. Each vertex of a cube is assigned a number that is the product of the three faces passing through the vertex. The sum of all the numbers assigned to the vertices is equal to 385. What is the sum of the numbers written on the 6 faces of the cube?

2. Six men engaged in a game. Whenever a player won a game he doubled the money of each of the players. That is, he gave each player as much money as he/she originally had in his/her pocket. They played six games and each player won one game. In the end all of them had Rs. 256 each. What was the difference between the money the person with the highest amount had and the person with the smallest amount had to start off with?

3. X and y are two distinct natural numbers such that x, y range from 1 to 15 (both inclusive). If the probability that x/y is a terminating decimal is n/420, what is the value of n?

4. How many positive integers exist such that they divide 10¹¹ but not 10 ¹⁰?

CAT Number Theory - Challenging question

2011-03-29T21:45:00.001-07:00

Just one question this time, but a fairly challenging one though.

There is a 4-digit number 'abcd' that satisfies the following property. 'abcd' = ab ^2 + cd ^2. Find abcd.

Answers to World Cup questions

2011-03-29T21:32:00.000-07:00

Have given below the answers to the world cup questions. The questions can be found here .

1. Australia, in one of their matches, scored a total that is the lowest multiple of 11 with exactly 10 factors. Against who was this?

2^4 * 11 = 176. Australia scored this against Pakistan

2. In one of the matches, the top-scorer (who scored a century) for the team batting first scored 3/8th the team's runs, but the team still did not cross 300. The HCF of the team's score and the top-scorers score is a prime number. Which match was this?

Top-scorer's score is a multiple of 3 greater than 100. So, it has to be a score like 34 * 3, 35 * 3, 36 * 3,..... The overall score is less than 300. So, the overall score has to be lower than 38 * 8. So, the overall score has to be of the form 37 * 8, 36 * 8, 35 * 8,...

The HCF is a prime number, this implies that the top scorer scored 111, the team scored 296. India vs. South Africa. Sachin Tendulkar century

3. A 'minnow' scored a total that can be made a perfect square if it is multiplied by 23. If they had scored 1 more, then their score multiplied by 13 would have yielded a perfect square. What was their score? Bonus point, what match was this?

The score should have been 23 * 4 or 23 * 9, 23 * 9 = 207. 207 + 1 = 208, which is 13 * 16.
India vs. Ireland. Ireland scored 207

4. In a particular match, team A batting first scored twice a perfect square. Team B batting second, also scored twice a perfect square. The total score of both the teams put together was also a perfect square. What match are we talking about here?

India 338, England 338 (169 * 2). Total 676 (26* 26)

5. Team scored a three digit score 'abc' in a match. a and b are prime numbers. a + 2b and a + 2c are also prime. b + 2a and b + 2c are also primes. a + c and b + c are also primes. Two digit numbers 'ab' and 'ba' are both primes. Which match was this?

a + c and b + c are also primes => a, b should be odd, c should be even
a,b cannot be 5 as ab is also prime. So, abh has to be 37.
c can only be zero.
abc = 370 (India vs. Bangladesh)

World Cup Special - Few maths questions

2011-03-26T07:18:00.000-07:00

My colleague mentioned that he had asked his class 1-2 questions on numbers related to the WC. Just thought it might be a good idea to follow that.

1. Australia, in one of their matches, scored a total that is the lowest multiple of 11 with exactly 10 factors. Against who was this?

2. In one of the matches, the top-scorer (who scored a century) for the team batting first scored 3/8th the team's runs, but the team still did not cross 300. The HCF of the team's score and the top-scorers score is a prime number. Which match was this?

3. A 'minnow' scored a total that can be made a perfect square if it is multiplied by 23. If they had scored 1 more, then their score multiplied by 13 would have yielded a perfect square. What was their score? Bonus point, what match was this?

4. In a particular match, team A batting first scored twice a perfect square. Team B batting second, also scored twice a perfect square. The total score of both the teams put together was also a perfect square. What match are we talking about here?

5. Team scored a three digit score 'abc' in a match. a and b are prime numbers. a + 2b and a + 2c are also prime. b + 2a and b + 2c are also primes. a + c and b + c are also primes. Two digit numbers 'ab' and 'ba' are both primes. Which match was this?

Solutions to questions from assorted topics

2011-03-15T21:03:00.000-07:00

Have given below solutions to the questions given here .

1. A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.

Base 6 representation ends with 10 zeroes, or the number is a multiple of 6^10. If n! has to be a multiple of 6^10, it has to be a multiple of 3^10. The smallest factorial that is a multiple of 3^10 is 24!. So, when n = 24, 25 or 26, n! will be a multiple of 6^10 (but not 6^11).

Similarly, for the second part, we need to find n! such that it is a multiple of 2 ^ 21, but not 2 ^ 24. When n = 24, n! is a multiple of 2^22. S0, when n = 24, 25, 26, 27, n! will be a multiple of 2 ^ 21 but not 2 ^ 24.

The smallest n that satisfies the above conditions is 24. n = 24, 25 or 26 will satisfy the above conditions.

2. [x] is the greatest integer less than or equal to x. Find the number of positive integers n such that [n/11] = [n/13].

This is a classic case of brute-force counting.
When n = 1, 2, 3....10, both values will be equal to 0. 10 possibilities
When n = 13,14, ....21, both values will be equal to 1. 9 possibilities
When n = 26,27, ....32, both values will be equal to 2. 7 possibilities
When n = 39,27, ....43, both values will be equal to 3. 5 possibilities
When n = 52, 53, 54, both values will be equal to 4. 3 possibilities
when n = 65, both will be equal to 5. 1 possibility

So, totally there are 1 + 3 + 5 + 7 + 9 +10 = 35 possibilities.

3. Positive numbers 1 to 55, inclusive are placed in 5 groups of 11 numbers each. What is the maximum possible average of the medians of the 5 groups?

We need to maximise each median in order to have the overall maximum median possible.

The highest possible median is 50 as they should be 5 numbers higher than the median in the group of 11. So, if we have a set that has a, b, c, d, e, 50, 51, 52, 53, 54, 55, the median will be 50. In this set, it is best not to waste any high values on a, b, c, d or e as these do not affect the median. So, a set that reads as 1, 2, 3, 4, 5, 50, 51, 52, 53, 54, 55 will also have median 50.

The next set can be 6, 7, 8, 9, 10, 44, 45, 46, 47, 48, 49. The median will be 44.

The medians of the 5 groups can be 50, 44, 38, 32, 26. The highest possible average of the medians will be 38.

Few more questions - assorted topics

2011-03-08T03:52:00.000-08:00

Have given below a few more questions from assorted topics

1. A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.

2. [x] is the greatest integer less than or equal to x. Find the number of positive integers n such that [n/11] = [n/13].

3. Positive numbers 1 to 55, inclusive are placed in 5 groups of 11 numbers each. What is the maximum possible average of the medians of the 5 groups?

Solutions - Mixed bag

2011-03-04T22:40:00.000-08:00

Have given solutions to the questions posted here

1. Average of 6 distinct positive integers is 33. The median of the three largest numbers is 43. What is the difference between the highest and lowest possible median of the 6 numbers?

Let a, b, c, d, e, f be the six numbers in ascending order

e = 43; a + b + c + d + e + f = 198

Median of 6 integers = (c +d)/2

(c +d)/2 is maximum when c and d are maximum since e = 43, d_max = 42 and c_max = 41

(c +d)/2 is minimum when c and d are minimum a_min = 1, b_min = 2, c_min = 3 d_min = 4

Therefore, = (c +d)/2 min = 3.5

Max – min = 41.5 – 3.5 = 38

2. In class A, the ratio of boys to girls is 2:3. In class B the ratio of boys to girls is 4 : 5. If the ratio of boys to girls in both classes put together is 3 : 4, what is the ratio of number of girls in class A to number of girls in class B?

Let us assume class A has 2x boys and 3x girls, class B has 4y boys and 5y girls.

(2x + 4y)/(3x + 5y) = ¾

8x + 16y = 9x + 15y

x = y

Reqd ratio = 3x/5y (since x = y) = 3/5

3. N is an 80-digit positive integer (in the decimal scale). All digits except the 44th digit (from the left) are 2. If N is divisible by 13, find the 26th digit.

To begin with, the question should read "find the 44th digit".

Any number of the form abcabc is a multiple of 1001. 1001 is 7 * 11 * 13. So, any number of the form abcabc is a multiple of 13.

So, a number comprising 42 2's would be a multiple of 13, so would a number comprising 36 2's. So, in effect, we are left with a two digit number 2a, where a is the 44th digit. 26 is a multiple of 13, so the 44th digit should be 6.

4. A page is torn from a novel. The sum of the remaining digits is 10000. What is the sum of the two page-numbers on the torn page of this novel?

n(n+1)/2 should be nearby 10,000, so n(n+1) is somewhere near 20,000. So n should be around sqrt(20,000) about 141. Try 142

142*143 = 10153

141 *142 = 10011

So, the missing pages are either 76 and 77 or 5 & 6

Few more questions - Mixed bag

2011-02-23T23:29:00.000-08:00

Have given below four questions across topics. Will discuss the answers shortly.

1. Average of 6 distinct positive integers is 33. The median of the three largest numbers is 43. What is the difference between the highest and lowest possible median of the 6 numbers?

3. N is an 80-digit positive integer (in the decimal scale). All digits except the 44th digit (from the left) are 2. If N is divisible by 13, find the 26th digit.

4. A page is torn from a novel. The sum of the remaining digits is 10000. What is the sum of the two page-numbers on the torn page of this novel?

Pipes Cisterns - Solutions

2011-01-04T20:30:00.000-08:00

Have given below solutions to the questions on Pipes Cisterns

1) Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. Find t?

A is kept open for all t minutes and fills one-third the tank. Or, A should be able to fill the entire tank in '3t' minutes.

A and B together can fill the tank completely in t minutes. A alone can fill it in 3t minutes.
A and B together can fill 1/t of the tank in a minute. A alone can fill 1/3t of the tank in a minute. So, in a minute, B can fill 1/t - 1/3t = 2/3t. Or, B takes 3t/2 minutes to fill an entire tank

To fill one-third the tank, B will take t/2 minutes. B is kept open for t - 10 minutes.

t/2 = t - 10, t = 20 minutes.

A takes 60 mins to fill the entire tank, B takes 30 minutes to fill the entire tank. A is kept open for all 20 minutes. B is kept open for 10 minutes.

C, which is kept open for 8 minutes also fills one-third the tank. Or, c alone can fill the tank in 24 minutes.

2) Pipe A fills a tank at the rate of 100lit/min, Pipe B fills at the rate of 25 lit/min, pipe C drains at the rate of 50 lit/min. The three pipes are kept open for one minute each, one after the other. If the capacity of the tank is 7000 liters, how long will it take to fill the tank if

i) A is kept open first, followed by B and then C
Each cycle of 3 minutes, 75 liters get filled. 100 + 25 - 50. So, after 3 minutes the tank would have 75 liters
6 mins - 150 liters
9 mins - 225 liters
30 mins - 750 liters
270 mins - 6750 liters
273 mins - 6825 liters
279 mins - 6975 liters

In the 280th minute, pipe A would be open and it would fill the remaining 25 liters in 15 seconds. So, it would take 279 mins and 15 seconds to fill the tank.

The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.

ii) B first, followed by A, and then C

Each cycle of 3 minutes, 75 liters get filled. 25 + 100 - 50. So, after 3 minutes the tank would have 75 liters
6 mins - 150 liters
9 mins - 225 liters
30 mins - 750 liters
270 mins - 6750 liters
273 mins - 6825 liters
279 mins - 6975 liters

In the 280th minute, pipe B would be open and it would fill the remaining 25 liters in one minutes. So, it would take 280 mins.

iii) B first, followed by C, and then A
In this case also, Each cycle of 3 minutes, 75 liters get filled. 25 -50 + 100. But there is a small catch here. In the first set of 3 minutes, we would fill up to about 100 liters. After 1 minute, we would be at 25 liters, after 2 minutes, we would be at 0 liters and in the third minute, the tank would be 100 liters full.

6 mins - 170 liters
9 mins - 250 liters
30 mins - 775 liters
270 mins - 6775 liters
273 mins - 6850 liters
276 mins - 6925 liters
279 minutes - 7000 liters

So, it would take 279 mins and 15 seconds to fill the tank.

The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.

3) 4 men and 6 women complete a task in 24 days. If the women are at least half as efficient as the men, but not more efficient than the men, what is the range of the number of days for 6 women and 2 men to complete a task twice as difficult as this one?

4m and 6w finish in 24 days.
In one day, 4m + 6w = 1/24 of task

In these questions, just substitute extreme values to get the whole range
If a woman is half as efficient as man
4m + 3m = 1/24, 7m = 1/24, m = 1/168
6w + 2m = 3m + 2m = 5m, 5m will take 168/5 days = 33.6 days

If a woman is as efficient as a man
4m + 6w finish in 24 days
10m finish 1/24 of task in a day
6w + 2m = 8m, 8m will take 240/8 = 30 days to finish the task.

So, the range = 30 to 33.6 days. The new team will take 30 to 33.6 days to finish the task.

4) A fill pipe can fill a tank in 20 hours, a drain pipe can drain a tank in 30 hours. If a system of n pipes (fill pipes and drain pipes put together) can fill the tank in exactly 5 hours, which of the following are possible values of n (More than one option could be correct)?
1) 32 2) 54 3) 29 4) 40

3 fill pipes cancel out 2 drain pipes. Plus, you need an additional 4 fill pipes fill the tank in 5 hours. so the answer has to be 5k + 4. Both 54 and 29 are possible.

Pipes Cisterns - Questions

2010-12-27T21:24:00.000-08:00

Have given below a few questions on pipes and cisterns.

1) Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. Find t?

2) Pipe A fills a tank at the rate of 100lit/min, Pipe B fills at the rate of 25 lit/min, pipe C drains at the rate of 50 lit/min. The three pipes are kept open for one minute each, one after the other. If the capacity of the tank is 7000 liters, how long will it take to fill the tank if

i) A is kept open first, followed by B and then C
ii) B first, followed by A, and then C
iii) B first, followed by C, and then A

3) 4 men and 6 women complete a task in 24 days. If the women are at least half as efficient as the men, but not more efficient than the men, what is the range of the number of days for 6 women and 2 men to complete a task twice as difficult as this one?

4) A fill pipe can fill a tank in 20 hours, a drain pipe can drain a tank in 30 hours. If a system of n pipes (fill pipes and drain pipes put together) can fill the tank in exactly 5 hours, which of the following are possible values of n (More than one option could be correct)?
1) 32 2) 54 3) 29 4) 40

CAT Pipes and Cisterns - Basic Questions

2010-12-24T01:14:00.001-08:00

Have given below a few basic questions on Pipes and Cisterns.

1. Pipe A can fill a tank in 6 hours. On account of a crack at the bottom of the tank through which water leaks, it takes an hour more for the tank to overflow. How long will the crack take to empty a full tank when pipe A is kept shut?

2. Pipe A that can fill a tank in 40 minutes and pipe B that can fill a tank in 20 minutes are opened when the tank is empty. Pipe B is shut 16 minutes before the tank overflows. How long did it take for the tank to overflow?

3. Pipe A can fill a tank in 36 minutes. Pipe B can fill the same tank in 72 minutes. There is a drain pipe, pipe C, at the bottom of the tank. A man opens pipe A and pipe B when the tank is empty and returns back to shut them when the tank should overflow. However, he finds that he had kept pipe C also open and shuts it immediately. The tank overflows in the next 6 minutes. How long will pipe C take to drain the tank, if pipes A and B are kept shut?

4. A tank is fitted with 22 pipes. Some of these are fill pipes that fill the tank and others waste pipes that drain the tank. If each fill pipe can fill the tank in 24 hours and each drain pipe can drain the tank in 16 hours, and if all the pipes are kept open simultaneously when the tank is empty, the tank overflows in 12 hours. How many of these are waste pipes?

5. Adrain completes 60% of a task in 72 days. He realizes that to meet his deadline, he needs additional help and gets Allen who is thrice as efficient as Adrain to help him. They complete the task with four days to spare. What was the deadline for the project from the day Adrain started work on it?

6. Working together A and B can complete a task in 18 days. If A worked alone, completed the first half of the task and then B took over from A, the task will be completed in 48 days. If A is more efficient than B, how long will A alone take to complete the task?

7. Working alone, Shiv will take 27 more hours to complete a task than if he worked with George to complete the task. Working alone, George will take 3 more hours to complete the task than if he worked with Shiv. How long will George take to complete the task, if he worked alone?

8. 5 men and 7 women can complete a task in 12 days. However, if 11 men and 9 women worked on the same task, the work will be completed only in 8 days. How long will a man and woman take to complete the task?

Percentages Solutions

2010-12-16T22:21:00.000-08:00

Have given below solution to the final question on percentages .

6. A, B, C and D share a loot. A gets a% of the total. B gets b% of the remaining (after A has taken his share). C gets c% of the remaining and D gets the rest. D gets a% less than what A gets. B and C get equal amounts. b = 2a.

i) What percentage of what A got did C get?

ii) If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?

Let total amount = T.

A would have got T * a. After A takes his share, there would be T (1-a) left
B would have got T (1-a) * b
C would have got T (1-a)(1-b) * c
D would have got T (1-a)(1-b)(1-c)

D gets a % less than what A gets, or, T *a (1-a) = T (1-a)(1-b)(1-c), or

a = (1-b)(1-c) -------- Eqn 1

B and C get the same amounts, or (1-a) * b = (1-a)*(1-b)*c, or b = (1-b) c, or, b = c - cb, or

b = c/c+1 ------------ Eqn 2
b = 2a -----------------Eqn 3

Substituting, eqn 2 in and 3 in eqn 1, we get

c/2(c+1) = (1-c)/(c+1)
Or, c = 2 -2c, or c = 2/3

b = c/(c+1) = 2/5, a = 1/5
a gets 20% of the loot, b gets 40% of the remaining 80%, or 32% of the loot. C gets 66.6% of 48%, or 32% of the overall loot. And d gets the final 16% of the overall amount (20% lesser than A's share)

Now, to the questions

i) What percentage of what A got did C get?

A got 20% of overall, C got 32% of overall. Or, C got 160% of A

ii) If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?
A would have got Rs. 200, D Rs. 160. A would have got Rs. 40 more than D.

Solutions to questions on percentages

2010-12-14T01:39:00.000-08:00

Have given below solutions to the questions 4 and 5 on percentages .

4. In a class, if 50% of the boys were girls, then there would be 50% more girls than boys. What percentage of the overall class is girls?

For questions of this type, it is best to go from the final step. In the final state, the number of girls should be 1.5 * the number of boys.

When 50% of the boys are taken as girls, let the number of boys = x
Number of girls = 1.5x
total number of students = 2.5x

Original number of boys = 2x (50% of boys = x)
Original number of girls = 0.5x

Girls form 20% of the overall class.

5. Class B has 50% more students than class A. Number of girls in class A is equal to number of boys in class B. The percentage of girls is the same in both classes. What percentage of the student group are boys?

Let number of girls in class A = x
Let number of boys in class A = y
Total number of students = x + y
Proportion of girls = x/(x+y)

Number of boys in class B = x.
Total number of students in class B = 1.5( x + y)
Proportion of girls = 1 - x/1.5(x+y)

Percentage of boys in the overall student community = (x+y)/2.5* (x + y) * 100 = 40%

Percentages - Solutions

2010-12-09T19:31:00.001-08:00

Have given below solutions to the first three questions on Percentages

1. a is x % of b, b is x% more than a. Find x?

a = bx,
b = a (1+x), substituting this in the previous equation, we get
x(x+1) = 1
x^2 + x -1 = 0

or x is approximately 0.62 or 62%

2. A two digit number ab is 60% of x. The two-digit number formed by reversing the digits of ab is 60% more than x. Find x?

10a + b = 0.6x
10b + a = 1.6x

Subtracting one from the other, we have

9b - 9a = x or x = 9( b-a)
x should be a multiple of 9.

10a + b = 3x/5, this implies that x should also be a multiple of 5.Or, x should be a multiple of 45

x should be equal to 45, ab = 27, ba = 72

3. A is x % more than B. B is y% less than C. If A,B and C are positive and A is greater than C, find the relation between x and y

a = (1 +x)b
b = c(1-y)
c = b /(1-y)

a is greater than c

Or, b (1+x) > b/(1-y)
1 + x > 1/(1-y)
(1+x)(1-y) > 1
1 + x -y - xy > 1
x > y (1+x)
y < x/(1+x)

Percentages - Solutions to basic questions

2010-12-08T22:50:00.000-08:00

Have given below solutions to the basic questions on Percentages.

Percentages online revised_final

View more presentations from rajeshb1980.

Percentages - Questions

2010-12-07T20:44:00.000-08:00

1. a is x % of b, b is x% more than a. Find x?

2. A two digit number ab is 60% of x. The two-digit number formed by reversing the digits of ab is 60% more than x. Find x?

3. A is x % more than B. B is y% less than C. If A,B and C are positive and A > C, find the relation between x and y

4. In a class, if 50% of the boys were girls, then there would be 50% more girls than boys. What percentage of the overall class is girls?

i) What percentage of what A got did C get?

ii) If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?

Percentages - Basic Questions

2010-12-05T20:08:00.000-08:00

1. The price of a mobile phone decreased by 25% to Rs. 100. What was the price before the decrease and by what amount did it decrease?

2. The price of a mobile phone decreased by 25% from Rs. 100. What was the price after the decrease and by what amount did it decrease?

3. The price of a mobile phone decreased by 25% or Rs. 100. What was the price before the decrease and what is after the decrease?

4. In an election contested by two candidates, the winning candidate got 24% of the total votes more than the losing candidate. If the losing candidate got 76,000 votes, how many votes did the winning candidate get? Assume that there were no invalid votes

5. The price of Ford Fusion is 25% less than that of Honda City. By what percentage is a Honda City more expensive than a Ford Fusion?

6. 40% of the people in a gathering speak English. Of the remaining people, 60% speak Hindi. If 96 people speak neither Hindi nor English, how many people were there in the gathering?

7. Ann and Ron are a married couple and Ann earned twice as much as Ron in the month of January. In Feb Ann got a rise of 20% while Ron got a rise of only 10%. If both of them saved half their wages and the combined savings of the couple in the month of February is Rs.52,500, what is the salary of Ron in February?

8. 50% of the students in a class that has 40% women students have got jobs during the recently concluded placement week. If 60% of the men in the class have got placements, what percentage of the women in class are yet to be placed?

9. A sample of 40 liters of diesel is found to be adulterated to the extent of 20%. How many liters of pure diesel be added to the above mixture so that level of impurity stands reduced at 5%?

10. What is the percentage change in the perimeter of a rectangle whose length increases by 5% and width decreases by 15%?

Solutions to questions on mean median

2010-12-02T21:34:00.000-08:00

Have given below solutions to the questions on mean and median .

1. The median of n distinct numbers is greater than the average, does this mean that there are more terms above the average than below it?

No.

If there are an odd number of terms, say, 2n+1, then the median is the middle term. And if average is lesser than the middle term, there will at least be n +1 terms greater than the average. So, there will be more terms above the average than below it.

However, this need not be the case when there are an even number of terms. When there are 2n distinct terms, n of them will be greater than the median and n will be lesser than the median. The average of these two terms can be such that there are n terms above the average and n below it.

For instance, if the numbers are 0, 1, 7, 7.5. The median is 4, average is 3.875. Average is less than the median. And there are more 2 numbers above the average and 2 below the average.

2. In a sequence of 25 terms, can 20 terms be below the average? Can 20 terms be between median and average?

Yes. We can have 24 zeroes and 1500 as the 25 numbers. In this case, there are 24 numbers below the average.

No. In a sequence of 25 numbers, 12 will be greater than or equal to the median and 12 will be lesser than or equal to the median. We cannot have 20 terms in between the average and median

3. From 10 numbers, a,b,c,...j, all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the average of the 10 numbers?

Yes. All the terms appear the same number of times when we select them 4 at a time. So, the average of averages will be equal to the overall average.

Mean Median - Final few questions

2010-12-01T19:57:00.001-08:00

Few more questions on Mean, median. Have just tried to construct a few Yes/No questions to try to build some intuition. These questions are better-answered by constructing examples/counter-examples.

1. The median of n distinct numbers is greater than the average, does this mean that there are more terms above the average than below it?

2. In a sequence of 25 terms, can 20 terms be below the average? Can 20 terms be between median and average?

3. From 10 numbers, a,b,c,...j, all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the average of the 10 numbers?

There is one further question that does not come under this classification, but an interesting one nevertheless

Q. The 64 squares of a chessboard are filled with natural numbers from 1 to 100. Multiples cells can have the same number (Any number of cells can have the same number). If the value in any square is equal to the average of the numbers in all the squares around it (either 3, 5 or 8 squares, depending on the position of the square), in how many ways can the chessboard be filled?

This is a fairly difficult question, and not exactly a question for CAT prep, but a fun question nevertheless. (This question is an adaptation from one of the Olympiad questions. Have not been able to locate the exact one yet.Will send in a post once I locate the correct Olympiad)

Happy cracking.

Mean median - Solutions

2010-12-01T19:29:00.001-08:00

Have given below solutions to the two questions on mean and median.

1. Consider three classes A, B and C with 20, 30 and 50 students respectively. The averages score in maths of students in class B is 16 more than that of those in class C and the average score of those in class A is 2 more than the overall average of scores in A, B and C. What is the difference between average of class C and class A

Let average in class C = x. Average of class B = x +16. Average of class A = y
y = 2 + (20y + 30(x+16) + 50x) /100

(y - 2) * 10 = 2y + 3x + 48 + 5x
10y -20 = 2y + 8x + 48
8y - 8x = 68
y - x = 8.5. This is the difference between average of class A and class C

2. Natural numbers 1 to 25 (both inclusive) are split into 5 groups of 5 numbers each. The medians of these 5 groups are A, B, C, D and E. If the average of these medians is m, what are the smallest and the largest values m can take?

Let us try to construct a scenario for getting the smallest value of the average. We need to have the minimum value for each of the 5 medians.

Now, what is the lowest value the median of any of the 5 groups can take?

The median is the middle term among the 5 terms. So, it cannot be 1 or 2, but it can be 3. If the set were 1, 2, 3, 4, 5, the median would be 3. If we choose a sub-group such as this, the smallest median fro the next group will be 8. The next group could be 6, 7, 8, 9, 10. The idea is to create sub-groups in such a way so that the second group can have a smaller median than 8.

Now, even a sub-group that has the numbers 1, 2, 3, 24, 25 would have the median as 3. So, if we want to keep the medians as small as possible, we should choose sub-groups that have small numbers till the median, and then very large numbers. Because, these large numbers do not affect the median, we will still have small numbers to choose from for the next group. So, choose sub-groups such that the first 3 numbers are small, the final two are as large as possible.

1, 2, 3, 24, 25
4, 5, 6, 22, 23
7, 8, 9, 20, 21
10, 11, 12, 18, 19
13, 14, 15, 16, 17

would be an ideal set of groups. The medians would be 3, 6, 9, 12, 15, and the average of the medians would be 9.

Similar set of groups can be found to find the highest value of the average. The medians would be 23, 20, 17, 14, 11 and the average would be 17.

Mean median - Two more questions

2010-11-29T23:25:00.000-08:00

Have given below two more questions on Averages.

1. Consider three classes A, B and C with 20, 30 and 50 students respectively. The averages score in maths of students in class B is 16 more than that of those in class C and the average score of those in class A is 2 more than the overall average of scores in A, B and C. What is the difference between average of class C and class A

2. Natural numbers 1 to 25 (both inclusive) are split into 5 groups of 5 numbers each. The medians of these 5 groups are A, B, C, D and E. If the average of these medians is m, what are the smallest and the largest values m can take?

Mean Median (Averages) - Solutions

2010-11-29T04:06:00.001-08:00

Have given below the solutions to the questions on mean and median .

Q. Consider 4 numbers a, b, c and d. Ram figures that the smallest average of some three of these four numbers is 30 and the largest average of some three of these 4 is 40. What is the range of values the average of all 4 numbers can take?

We can assume a, b,c d are in ascending order (with the caveat that numbers can be equal to each other)

a + b + c = 90
b + c + d = 120

We need to find the maximum and minimum value of a + b + c + d.

a + b + c + d = 120 + a. So, this will be minimum when a is minimum. Given a + b + c = 90. a is minimum when b + c is maximum. If b + c is maximum, d should be minimum. Given that b + c + d = 120, the minimum value d can take is 40 as d cannot be less than b or c. The highest value b +c can take is 80, when b = c= d = 40. When b = c = d = 40, a = 10. a + b + c + d = 130. Average = 32.5

Similarly, a + b + c + d = 90 + d. So, this will be maximum when d is maximum. Given b + c + d = 120. d is maximum when b + c is minimum. If b + c is minimum, a should be maximum. Given that a + b + c = 90, the maximum value a can take is 30 as a cannot be greater than b or c. The lowest value b +c can take is 60, when a = b= c = 30. When a = b = c = 30, d = 60. a + b + c + d = 150. Average = 37.5

So, the average has to range from 32.5 to 37.5

Q. The average of 5 distinct positive integers if 33. What are the maximum and minimum possible values of the median of the 5 numbers if the average of the three largest numbers within this set is 39?

Let the numbers be a, b, c, d, e in ascending order. a + b + c + d + e = 165. Average of the three largest numbers is 39, so c + d + e = 117, or a + b = 48.

We need to find the maximum and minimum possible values of c.

For minimum value, a and b have to be minimum. a +b = 48. Let us assume a =23, b =25, ca can be as low as 26.
23, 25, 26, 45, 46 is a possible sequence that satisfies the conditions.

For maximum value, we need d + e to be minimum as c + d + e = 117. we can have c = 38, d =39 and e =40. Or, the maximum value c can take = 38

23, 25, 38, 39, 40 is a possible sequence that satisfies the conditions specified

Q. Consider 5 distinct positive numbers a, b, c, d, and e. The average of these numbers is k. If we remove b from this set, the average drops to m (m is less than k). Average of c, b, d and e is K. We also know that c is less than d and e is less than k. The difference between c and b is equal to the difference between e and d. Average of a, b, c and e is greater than m. Write down a, d, c, d and e in ascending order.

Average of a, b, c, d and e is k, Average of b, c, d and e is also k, this implies that a = k
If we remove b, the average drops, this implies that b is higher than the average
e is less than k, or e is less than a. c is less than d.

e < a < b, c < d

Average of a, b, c and e is greater than average a, c, d and e. This tells us that d < b

From this, we get that b is the largest number

b - c = d - e
b + e = c + d
a + b + c +d + e = 5a
Or, b + c + d + e = 4a

b + e = 2a, c + d = 2a. Or, e, a, b is an AP, c, a, d is an AP.

e is the smallest number (as b is the largest number)

Or, the ascending order should be e c a d b

Mean Median - Solutions to Basic Questionsn

2010-11-29T03:16:00.000-08:00

Have given below the solutions to the basic questions in Mean Median .

Averages online

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Averages - Questions

2010-11-27T17:33:00.000-08:00

Have given below few questions from averages. These are slightly more challenging than the basic questions

Consider 4 numbers a, b, c and d. Ram figures that the smallest average of some three of these four numbers is 30 and the largest average of some three of these 4 is 40. What is the range of values the average of all 4 numbers can take?
The average of 5 distinct positive integers if 33. What are the maximum and minimum possible values of the median of the 5 numbers if the average of the three largest numbers within this set is 39?
Consider 5 distinct positive numbers a, b, c, d, and e. The average of these numbers is k. If we remove b from this set, the average drops to m (m is less than k). Average of c, b, d and e is K. We also know that c is less than d and e is less than k. The difference between c and b is equal to the difference between e and d. Average of a, b, c and e is greater than m. Write down a, d, c, d and e in ascending order.

Mean Median Basic questions

2010-11-26T22:25:00.000-08:00

Have given a few very simple questions in mean, median. Will post more challenging questions through the week.

1. The average revenue of company A in a year was Rs.30,000 a month. During the first 5 months, the company saw an average revenue of Rs. 25,000 per month and in the last 6 months, the company saw an average of Rs.33,000 per month. What was the revenue level in the 6th month?

2. The captain of a cricket squad of 14 players is 28 years old. The wicket-keeper is 32 years old. If these two players are replaced by two younger players, the average age of the team drops by 1 year. What is the average age of the two new players.?

3. The average age of the students of class A comprising 25 students is 19 years and that of the students of class B comprising 15 students is 11 years. What is the average age of the students of the two classes taken together?

4. The average weight of a class comprising 29 students is 40 kgs. If the weight of the teacher is also included, the average weight increases by 0.5 kgs. What is the weight of the teacher?

5. 30 friends go out for dinner. 12 of them spend Rs.32.5 each for the dinner while the rest of them spend Rs.3 more than the average expense of all 30. What was the total money spent?

6. The range of the height of male members in a team is 9 cms, and that of the female members in the team is 8 cms. What is the range of the height of the team if the shortest man in the team is 3 cms taller than the shortest woman in the team?

7. The median of 68, 93, 109, x, y, and 97 is 97. What is the least possible average of x and y?

Happy cracking

Races - Solutions

2010-11-25T20:32:00.000-08:00

Have given below solutions to the races questions that can be found here.

1. Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

Let track length be equal to T. Time taken to meet for the first time = T / relative speed
= T/(6-b) or T/(b-6)

Time taken for a lap for A = T/6
Time taken for a lap for A = T/b

So, time taken to meet for the first time at the starting point = LCM (T/6, T/b) = T / HCF (6,b)

Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b). For a more detailed discussion on this have a look at the last few slides in this presentation.

So, in essence we have to find values for b such that 6-b/ HCF(6,b) = 2 or b-6/ HCF(6,b) = 2

b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.

2. Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A,B and C?

Let track length be equal to T. When a completes a lap, let us assume B has run a distance of (t-d). At this time, C should have run a distance of (t-2d)

After three laps C would have traveled a distance of 3 * (t-2d) = 3t - 6d.

After 3 laps C is in the same position as B was at the end one lap. So, the position after 3t-6d should be the same as t-d. Or, C should be at a distance of d from the end of the lap. C will have completed less than 3 laps (as he is slower than A), so he could have traveled a distance of either t-d or 2t-d.

=> 3t-6d = t-d => 2t = 5d => d = 0.4t => The distances covered by A,B and C when A completes a lap will be t, 0.6t and 0.2t respectively. Or, the ratio of their speeds is 5:3:1

In the second scenario, 3t-6d = 2t-d => t = 5d=> d = 0.2t => The distances covered by A,B and C when A completes a lap will be t, 0.8t and 0.6t respectively. Or, the ratio of their speeds is 5:4:3

The ratio of the speeds of A, B and C is either 5:3:1 or 5:4:3

Races - Two Questions

2010-11-24T22:51:00.000-08:00

Have given below two questions on races. For a recap of the basics on Races, see questions here and the solutions here

1. Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

2. Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A,B and C?

Races Basic Questions Solutions

2010-11-24T20:52:00.001-08:00

Have given below the solutions to the basic questions on Races.

Races

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Races - Basic Questions

2010-11-23T22:50:00.000-08:00

Have given below a few basic questions in Races

1. In a 100m race, A can give B a start of 12 meters or 3 seconds. What is the speed of B in m/sec and how long will A take to run 100m?

2. A can give B a start of 30 in a km race and A can give C a start of 50m in a km race. How much start will B be able to give C in a km race?

3. A runs a 400m race at the speed of 8 m/sec. He gives B a start of 40m and still beats him by 10 seconds. What is B's speed?

4. If A gives B a start of 20m in a race, A beats him by 10 seconds. On the contrary, if A gives B a start of 40m, they finish the race in a dead heat. If the length of the race is 400m, how long will A take to run the race?

5. A and B start around a circular track of length 600m at speeds of 6m/s and 9 m/sec from the same point, simultaneously and in the same direction. When will they meet for the first time? How far from the starting point will they meet, when they meet for the first time?

6. A and B run around a circular track at speeds of 8m/sec and 5m/sec starting from the same point, simultaneously and in the same direction. How many points with respect to the starting point on the circular track will they meet, if the run at the above mentioned constant speeds endlessly?

Solution to Speed Time Distance LR question

2010-11-23T20:03:00.000-08:00

Have given below the s0lution to the question on Speed Time Distance LR .

The question is as follows

Station X of length 900 meters has two station masters A and B. But as the station is not a busy one, they are mostly jobless and decide to conduct an experiment. They stand at either end of the station and decide to note the exact time when trains cross the stationmasters. They synchronize their watches and proceed to either end of the station. Two trains P and Q go past the station (neither train stops here), and after having taken down their readings, the station masters sit down to have a chat

1. A: Train P entered the station at exactly 8:00:00
2. B: Train Q entered the station at exactly 8:00:10 (10 seconds past 8)
3. A: The last carriage of train P crossed me by at 8:00:20, and precisely two seconds after this, the engines of the two trains went past each other. (Engines are at the front of the train)
4. B: The last carriage of train Q crossed me 22 seconds after the engine of P went past me.
5. A: After the last carriage of train P crossed by me, it took 35 seconds for the engine of train Q to cross me.
6. B: I got bored and I came here.

Now, let us try to jot down the points in a slightly different format and see if we can make any inferences. Let us assume speed of train P = p, speed of train Q = q, length of train P = L, length of train Q = M
Take statements 1 and 3
Train P crosses the stationmaster A entirely in 20 seconds (enters station at 8:00:00 and last carriage passes at 8:00:20) => Length of train = 20p => L = 20p

Statement 5 tells us that engine of train Q crosses station master A at 8:00:55. Train Q enters the station at 8:00:10, so the train takes 45 seconds to cross the entire station. Or, it takes 45 seconds to travel 900 meters => Speed to train Q, q = 20m/s.

Statement 3 tells us that the two trains cross each other at 8:00:22. This implies train P has traveled for 22 seconds since entering the station and train Q has traveled for 12 seconds since entering the station before they cross each other. The cumulative distance traveled by the two trains should be equal to the length of the station = 900m.

=> 900 = 22p + 12q
q = 20m/s => p = 30m/s.

So, train P_ takes 900/p = 30 seconds to cross the station. So, engine of train P will cross stationmaster B at 8:00:30.

Statement 4 states that the last carriage of Q went past 22 seconds after the engine of P went by. Or, the last carriage of Q went by at 8:00:52.
Engine of train Q went by at 8:00:10, last carriage went by at 8:00:52, or train Q took 42 seconds to cross station master B. Train Q travels at 20m/s. => Length of train Q = 840m

Now, to the questions

1. What is the length of train Q?
Train Q is 840m long

2. At what time do the rear ends of the two trains cross each other?
The engines cross each other at 8:00:22. The relative speed of the two trains = 20+30 = 50m/s. The relative distance traveled by two trains from the time the engines cross each other to the times the rear ends cross each other = Sum of the two lengths = 600 + 840 =1440. Time taken = 1440/50 = 28.8 seconds past 8:00:22, or at 8:00:50.8 seconds.

3. How far from station master A do the rear ends of the two trains cross each other?
At 8:00:50.8, P would have traveled 50.8 * 30m/s post entering the station. Or, train P would have traveled 1524m. The rear end of train P would be at a point 1524-600m = 924m from stationmaster A (or 24 meters from stationmaster B and outside the station)

4. You are told that the two trains enter the station at the same times mentioned and the length of the two trains are unchanged. Furthermore, train P continues to travel at the same speed (as computed above). At what minimum speed should train Q travel such that the rear ends of the two trains cross each other at a point within the length of the platform?

Rear end of train P crosses the station completely at 8:00:50. (Train P takes 30 seconds to travel the station and 20 seconds to travel a distance equal to its length). Train Q should have traveled 840m by this time. => Train Q should travel 840 within 40 seconds.

Or, minimum speed of Q = 840/40 = 21m/s.

Speed Time Distance - Logical Reasoning Question

2010-11-22T03:43:00.000-08:00

Station X of length 900 meters has two station masters A and B. But as the station is not a busy one, they are mostly jobless and decide to conduct an experiment. They stand at either end of the station and decide to note the exact time when trains cross the stationmasters. They synchronize their watches and proceed to either end of the station. Two trains P and Q go past the station (neither train stops here), and after having taken down their readings, the station masters sit down to have a chat

A: Train P entered the station at exactly 8:00:00
B: Train Q entered the station at exactly 8:00:10 (10 seconds past 8)
A: The last carriage of train P crossed me by at 8:00:20, and precisely two seconds after this, the engines of the two trains went past each other. (Engines are at the front of the train)
B: The last carriage of train Q crossed me 22 seconds after the engine of P went past me.
A: After the last carriage of train P crossed by me, it took 35 seconds for the engine of train Q to cross me.
B: I got bored and I came here.

1. What is the length of train Q?

2. At what time do the rear ends of the two trains cross each other?

3. How far from station master A do the rear ends of the two trains cross each other?

4. You are told that the two trains enter the station at the same times mentioned and the length of the two trains are unchanged. Furthermore, train P continues to travel at the same speed (as computed above). At what minimum speed should train Q travel such that the rear ends of the two trains cross each other at a point within the length of the platform?

Happy cracking

Speed Time Distance - Solutions

2010-11-21T19:06:00.000-08:00

Have given below the solutions to the two questions on Speed Time Distance .

1. Two friends A and B leave City P and City Q simultaneously and travel towards Q and P at constant speeds. They meet at a point in between the two cities and then proceed to their respective destinations in 54 minutes and 24 minutes respectively. How long did B take to cover the entire journey between City Q and City P?

Let us assume Car A travels at a speed of a and Car B travels at a speed of b. Further, let us assume that they meet after t minutes.

Distance traveled by car A before meeting car B = a*t. Likewise distance traveled by car B before meeting car A = b * t.

Distance traveled by car A after meeting car B = a *54. Distance traveled by car B after meeting car A = 24* b

Distance traveled by car A after crossing car B = distance traveled by car B before crossing car A (and vice versa)

=> at = 54b -------- 1
and bt = 24a -------- 2

Multiplying equations 1 and 2
we have ab * t² = 54 * 24 * ab

=> t² = 54 * 24 => t = 36.

So, both cars would have traveled 36 minutes prior to crossing each other. Or, B would have taken 36 + 24 = 60 minutes to travel the whole distance.

2. Car A trails car B by 50 meters. Car B travels at 45km/hr. Car C travels from the opposite direction at 54km/hr. Car C is at a distance of 220 meters from Car B. If car A decides to overtake Car B before cars B and C cross each other, what is the minimum speed at which car A must travel?

To begin with, let us ignore car A. Car B and car C travel in opposite directions. Their relative speed = Sum of the two speeds = 45 + 54 kmph. = 99kmph. = 99 * 5/18 m/s = 55/2 m/s = 27.5m/s.

The relative distance = 220m. So, time they will take to cross each other = 220/27.5 = 8 seconds.

Now, car A has to overtake car B within 8 seconds. The relative distance = 50m
=> Relative speed should be at least 50/8m/s. = 6.25m/s
= 6.25 * 18/5 kmph = 22.5kmph.

Car B travels at 45kmph, so car A should travel at at least 45 + 22.5 = 67.5kmph.

Speed Time Distance - Two more questions.

2010-11-19T22:01:00.000-08:00

Two more questions from Speed Time Distance. Reasonably straightforward ones.

1. Two friends A and B leave City P and City Q simultaneously and travel towards Q and P at constant speeds. They meet at a point in between the two cities and then proceed to their respective destinations in 54 minutes and 24 minutes respectively. How long did B take to cover the entire journey between City Q and City P?

2. Car A trails car B by 50 meters. Car B travels at 45km/hr. Car C travels from the opposite direction at 54km/hr. Car C is at a distance of 220 meters from Car B. If car A decides to overtake Car B before cars B and C cross each other, what is the minimum speed at which car A must travel?

Speed Time Distance - Solutions

2010-11-19T03:40:00.000-08:00

Have given below solutions to the two questions on Speed Time Distance.

1. City A to City B is a downstream journey on a stream which flows at a speed of 5km/hr. Boats P and Q run a shuttle service between the two cities that are 300 kms apart. Boat P, which starts from City A has a still-water speed of 25km/hr, while boat Q, which starts from city B at the same time has a still-water speed of 15km/hr. When will the two boats meet for the first time? (this part is easy) When and where will they meet for the second time?

When boat P travels downstream, it will effectively have a speed to 30kmph. Likewise, Q will have an effective speed of 10kmph. The relative speed = 40kmph. So, the two boats will meet for the first time after 300/40 hours (Distance/relative speed) = 7.5 hours (Actually, for this part we do not need the speed of the stream)

The second part is more interesting, because the speed of the boats change when they change direction. Boat P is quicker, so it will reach the destination sooner. Boat P will reach City B in 10 hours (300/30). When boat P reaches city B, boat Q will be at a point 100kms from city B.

After 10 hours, both P and Q will be traveling upstream

P's speed = 20km/hr
Q's speed = 10 km/hr => Relative speed = 10kmph

Q is ahead of P by 100 kms

P will catch up with Q after 10 more hours (Relative Distance/relative speed - 100/10).

So, P and Q will meet after 20 hours at a point 200 kms from city B

2. Cities M and N are 600km apart. Bus A starts from city M towards N at 9AM and bus B starts from city N towards M at the same time. Bus A travels the first one-third of the distance at a speed of 40kmph, the second one-third at 50kmph and the third one-third at 60km/hr. Bus B travels the first one-third of the total time taken at a speed of 40kmph, the second one-third at 50kmph and the third one-third at 60km/hr. When and where will the two buses cross each other?

Bus A
Travels 200km at 40kmph
the next 200km @ 50kmph and
the final 200km @ 60kmph

So, Bus A will be at a distance of 200km from city M after 5 hours, and at a distance of 400km after 9 hours, and reach N after 12 hours and 20 mins

Bus B
Travels at an overall average speed of 50kmph, so will take 12 hours for the entire trip
So, Bus B will travel
160kms in the first 4 hours
200 kms in the next 4
and 240 in the final 4

So, both buses cross each other when they are in their middle legs.

After 5 hours, bus A will be at a position 200kms from city M. At the same time, bus B will be at a distance 210kms from city N (4*40+50).

The distance between them will be 190kms (600-200-210). Relative speed = Sum of the the two speeds = 50+50 = 100 kmph.

Time taken = 190/100 = 1.9 hours. = 1 hour and 54 minutes. So, the two buses will meet after 6 hours and 54 minutes. Bus B will have travelled 210 + 95 = 305 kms. So, the two buses will meet at a point that is 305 kms from City N and 295 kms from city A.

Speed Time Distance - Two Questions

2010-11-18T08:32:00.001-08:00

Have given below two questions on Speed Time Distance. These are slightly difficult questions. For a recap of the basics, have a look at questions here and solutions here .

Solutions to Speed Time Distance

2010-11-18T04:58:00.000-08:00

Have given below the solutions to the basic questions on Speed Time Distance. Have also added some basic theory on Speed Time distances in the PPT. Will start posting tougher questions on Speed Time Distance beginning next week.

The answer key for the questions is as follows

1. 80 minutes (There was an error in this question. Have now fixed the error)
2. 6 km
3. 50 km/hr
4. 45km/hr
5. 200km from Bangalore
6. 520m
7. 350m
8. 18kmph

The solutions are given in the PPT below

Speed time distance

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