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Introduction

The quantitative section of the GRE contains 28 multiple choice-questions in a 45-minute period. The questions include two formats:

The math topics include arithmetic, basic algebra and geometry (no proofs). Trigonometry and calculus are NOT included. The test writers carefully choose questions to eliminate biases toward candidates with specific majors: all test takers will be on a level playing field. The section is designed to test your ability to solve problems, rather than your mathematical knowledge. Questions lean heavily toward word problems and applying mathematical formulas in typical real-world applications, such as:

• Calculating interest on a loan
• Calculating the percentage drop of a stock price
• Determining a salary increase
• Determining travel times and speeds
• Determining work schedules

While test writers vary their question types from year to year, topics tend to appear with similar frequency. Recent exam questions fell into the following categories:

Nearly every test question has a simple solution and can be solved with a minimum of calculations. In fact, quantitative comparison questions often require NO calculating, asking you to simply determine whether the quantity in Column A or Column B is greater. The trick is to correctly assess each question and apply the correct formulas to get the right answer. For standard multiple-choice questions, you have the advantage of the correct answer being right in front of you. You KNOW it is one of the five listed choices.

Factors, Multiples, HCF and LCM

• If the HCF of two numbers is H, the numbers can be assumed as H × a and H × b such that a and b are coprime.

• While finding the largest/smallest number that leaves particular remainders when divided by certain numbers, do not forget to first eliminate options based on the remainder and divisor combinations, e.g. a number leaving a remainder of 5 on division by 8 has to be odd; a number leaving a remainder of 3 on division by 15 has to have the unit digit as 3 or 8; etc.

• LCM of multiples of 5 would end with just 0 or 5.

• HCF of a set of numbers will be odd if even one of the number in the given set is odd.

• Keep in mind- HCF will be lesser than or equal to the least of the numbers and LCM will be greater than or equal to the greatest of the numbers.

• LCM of fraction= LCM of numerators / HCF of denominators

• HCF of fraction= HCF of numerator / LCM of denominators

• ERROR PRONE AREA: While calculating HCF or LCM of fractions, the fractions should be in the most reducible form.

• The HCF of a set of numbers has to be a factor of the LCM of the set of numbers OR in other words, the LCM of a set of numbers must be a multiple of the HCF of the set of numbers. Thus if HCF of a set of numbers is a multiple of 5, the LCM of the set of numbers must also be a multiple of 5.

• Do not forget that for two numbers, the product of numbers = HCF × LCM

• If a series of numbers is of the type a × N + b, then consecutive numbers of this series differ by N. Conversely any series of numbers differing by N (basically an AP with common difference N) can be represented as a multiple of N ± x.In any block of N consecutive numbers, there will be one and ONLY one number belonging to the above series. Thus from 1 to N, there exists one and ONLY one number belonging to the series. And so also from 101 to 100+N, or any such block of N numbers.The above property can be used to identify the range of the smallest or largest n-digit number of the series. E.g. the smallest 4 digit number of the series 52N + 15 will be within 1000 and 1051 and the largest 3 digit number such number will lie within 948 and 999.

• Whenever the number of factors is the focus of any questions, train your mind to think in the following direction…to find the number of factors of any given number, factorize the number i.e. write it as powers of prime numbers 2a × 3b × 5c × 7d × 11e × …Now the number of factors is (a + 1)(b + 1)(c + 1)(d + 1)(e + 1)…

• The number of ways a given number be written as a product of two numbers = ½ × the number of factor (if number of factors is even) OR ½ × (number of factors + 1) (if number of factors is odd). This will contain one way of writing the number as a product of similar numbers i.e. square of the square root of the given number.

Highest power dividing a product / number

• To find the highest power of x that divides N!, keep dividing N successively by x and the addition of all the quotients is your answer. (Successive division means dividing the quotient of the earlier division). While this is the process of getting the answer, do understand the concept behind find the highest power as it may be used in other application. The interpretation of the highest power of x that divides a factorial is that if from 1 × 2 × 3 × 4 × 5 × 6 ×…× N, if all the powers of x is segregated, how many will they amount to. E.g. From 19!, if we segregate all powers of 3 we will have something as follows:
  1×2×3×4×5×(3×2)×7×8×(3×3)×10×11×(3×4)×13×
  14× (3×5)×16×17×(3×3×2)×19i.e. 38 × N, where N will not be a multiple of 3.

The same logic as above can be used for any product and not necessarily a factorial. Thus if the questions is what is the highest power of 3 that can divide the product of squares of all odd numbers from 1 to 20…one should identify that the powers of 3 would appear only in the squares of 3, 9 and 15. Thus the largest power of 3 dividing the given product is 2 + 4 + 2 = 8.

Consider another method
Suppose, we need to find the highest power of 5 that divides 100!
Solution: 100 / 5 = 20
20 / 5 = 4
4 / 5 = 0
Hence the total contribution of the powers of 5 is 24. Or the number 100! is divisible by 5^24.

• When the number of zeroes at the end of a product of series of numbers is asked, think of the highest power of 2 and 5 in the product.

• If any number is expressed as 10n × m, where m is not a multiple of 10, then n is the number of zeroes at the end of the given number. n is also the highest power of 10 that divides the given number.

• The above property is not just limited to 10. If a number N can be expressed as 7n × m, where m is not a multiple of 7, then in this case also n is the largest power of 7 that can divide the given number.

• The above rules can be used effectively to factorize a factorial. Thus if one needs to find the number of ways in which 15! can be written as a product of 2 numbers…Recollect from the earlier matter that to find the number of ways a number can be written as a product of two natural numbers, one has to find the number of factors of the given number. Also recollect that to find the number of factors of any given number one has to factorize the given number. Factorizing the given number means writing it in the form of 2a × 3b × 5c × 7d × 11e ×…i.e. writing it in powers of prime numbers. Thus we need to find the highest power of prime numbers that can divide the given number. Thus in our case 15! can be written as 211 × 36 × 53 × 71 × 111 × 131. Now one can easily find the number of factors and then half them to get to the answer.

Converting recurring number to p/q form

Non-terminating but recurring numbers are rational and hence can be expressed in the form p/q.

Now, let us see how can we find thee recurring form of .4444444……
Let x=0.44444…..
Therefore, 10x=4.4444…

Subtracting the two of them now we get

Try the same thing with 0.6666… you’ll get the p/q form as 2 / 3.

If x=.43434343…….
Then 100x= 43.43434343…….
Subtracting the two again we get
99x=43
=> x=43 / 99

Thus for a purely recurring number we can identify the procedure as
The p/q form = The recurring part written once / As many 9s as the number of digits in the recurring part

What if the number is like 0.12333333…
Let x=.1233333…….
& 100x=12.33333….
& 1000x=123.333….

subtracting, 900x=123-12
=> x=111 / 900

Cosider following examples for your practice now
1) 4.33333….
2) 0.126666…..

CYCLICITY

Take any two numbers say 39 & 47.
If they are multiplied, the last digit of the product is same as the last digit of 9 x 7.
Hence, it is 3.

This concept could be extended to a host of situations. An interesting pattern emerges when we look at the exponents of the numbers. We would find conclusions as given below.

The last digits of the exponents of all numbers have cyclicity i.e. every Nth power of the base shall have the same last digit, if N is the cyclicity of the number. All numbers ending with 2, 3, 7, 8 have a cyclicity of 4.

For instance,
2^1 ends with 2
2^2 ends with 4
2^3 ends with 8
2^4 ends with 6
2^5 end with 2 again.

The same set of the last digits shall be repeated for the subsequent powers. So, if we want to find the last digit of (say) 2^45, divide 45 by 4.
The remainder is 1
So the last digit would be the same as last digit of 2^1, which is 2

Working out similarly for all other digits we get

DIGIT CYCLICITY
0, 1, 5 & 6 1
2, 3, 7 & 8 4
4 & 9 2

Walking on the given foot steps try out the following examples:

Find out the last digit of
1) 3^57
2) 7^23 x 8^13
3) 235^1000

1. If (32) ^ (x-2) = 64 / (8^x). Find the value of x.

(a) - 2
(b) 3
(c) 2
(d) - 3

2. The largest number that always divides the product of 3 consecutive multiples of 2 is.

(a) 8
(b) 16
(c) 24
(d) 48

3. By what smallest number, 21600 must be divided or multiplies in order to make it a perfect square.

(a) 6
(b) 5
(c) 8
(d) 10

4. How many numbers are there between 500 & 600 in which 9 occurs only once?

(a) 19
(b) 20
(c) 21
(d) 18

5. How many even integers n, where 100<=n<=200 are divisible neither by 7 nor by 9.

(a) 40
(b) 37
(c) 39
(d) 38

6. What is the total number of positive integer solutions that satisfy the equation 4x + 3y = 120?

(a) 9
(b) 10
(c) 8
(d) None of these

7. How many 2-digit natural numbers are there so that ten’s digit is never less than the unit’s digit?

(a) 44
(b) 55
(c) 54
(d) None of these

8. If P is a prime number, then the LCM of P & (P+1) is.

(a) P (P+1)
(b) (P+2) P
(c) (P+1)(P-1)
(d) None of these

9. What is the remainder when 17^23 is divided by 16?

(a) 0
(b) 1
(c) 2
(d) 3

10. The product of two numbers is 16200. If their LCM is 216, find their HCF.

(a) 75
(b) 70
(c) 80
(d) Data Inconsistent

11. The LCM of two numbers is 72 & their HCF is 12. If one of the numbers is 24, what is the other number?

(a) 38
(b) 26
(c) 36
(d) 42

12. Dividing by 3/8 and then multiplying by 5/6 is equivalent to dividing by what number?

(a) 5 / 16
(b) 16 / 40
(c) 9 / 20
(d) 40 / 18

13. How many zeros are there at the end of the product 33 x 175 x 180 x 12 x 44 x 80 x 66?

(a) 2
(b) 4
(c) 5
(d) 6

14. Convert 1153 from base 10 to base 15.

(a) 51D
(b) 51E
(c) 51C
(d) None of these

15. What is the highest power of 31 in 1000!

(a) 31
(b) 32
(c) 33
(d) None of these

16. Convert 413 from base 7 to base 8

(a) 613
(b) 361
(c) 316
(d) None of these

17. (9^6) + 1 when divided by 8, would have a remainder

(a) 0
(b) 1
(c) 2
(d) 3

18. What is the highest power of 54 that divides 31! completely?

(a) 0
(b) 1
(c) 4
(d) None of these

19. The positive integer nearest to 1000 & divisible by 2, 3, 4, 5 & 6 is

(a) 1020
(b) 1040
(c) 960
(d) 1030

20. If the operation,^ is defined by the equation x ^ y = 2x + y,what is the value of a in 2 ^ a = a ^ 3

(a) 0
(b) 1
(c) -1
(d) 4

1. If the operation, ^ is defined by the equation x ^ y = 2x + y, what is the value of a in 2 ^ a = a ^ 3.

(a) 0
(b) 1
(c) 1
(d)4

2. If 13 = 13w/(1-w) , then (2w)2 =

(a) 1/4
(b) 1/2
(c) 1
(d) 2

3. If 9x-3y=12 and 3x-5y=7 then 6x-2y =?

(a) -5
(b) 4
(c) 2
(d) 8

4. If the value of x lies between 0 & 1 which of the following is the largest?

(a) x
(b) x^2
(c) -x
(d) 1/x

5. Which of the following fractions is less than 1/3

(a) 22/62
(b) 15/46
(c) 2/3
(d) 1

6. Find (7x + 4y) / (x-2y) if x/2y = 3/2 ?

(a) 6
(b) 8
(c) 7
(d) data insufficient

7. x% of y is y% of ?

(a) x/y
(b) 2y
(c) x
(d) can’t be determined

8. A man spends half of his salary on household expenses; 1/4th for rent, 1/5th for travel expenses, and the man deposits the rest in a bank. If his monthly deposits in the bank amount 50, what is his monthly salary?

(a) Rs.500
(b) Rs.1500
(c) Rs.1000
(d) Rs. 900

9. What is the smallest number by which 2880 must be divided in order to make it into a perfect square?

(a) 3
(b) 4
(c) 5
(d) 6

10. In objective test a correct answer score 4 marks and on a wrong answer 2 marks are deducted. A student scores 480 marks from 150 questions. How many answer were correct?

(a) 120
(b) 130
(c) 110
(d) 150

11. If a=2/3b, b=2/3c, and c=2/3d what part of d is b?

(a) 8/27
(b) 4/9
(c) 2/3
(d) 75%
(e) 4/3

12. The petrol tank of an automobile can hold g liters. If a liters was removed when the tank was full, what part of the full tank was removed?

(a) g-a
(b) g/a
(c) a/g
(d) (g-a)/a
(e) (g-a)/g

13. If a and b are positive integers and (a-b)/3.5 = 4/7, then

(a) b < a
(b) b > a
(c) b = a
(d) b >= a

14. A can have a piece of work done in 8 days, B can work three times faster than the A, C can work five times faster than A. How many days will they take to do the work together?

(a) 3 days
(b) 8/9 days
(c) 4 days
(d) Can’t say

15. A coffee shop blends 2 kinds of coffee, putting in 2 parts of a 33p. a gm. grade to 1 part of a 24p. a gm. If the mixture is changed to 1 part of the 33p. a gm. to 2 parts of the less expensive grade, how much will the shop save in blending 100 gms.

(a) Rs.90
(b) Rs.1.00
(c) Rs.3.00
(d) Rs.8.00

Answers

1. b
2. c
3. d
4. d
5. b
6. c
7. c
8. c
9. c
10. b
11. b
12. c
13. a
14. b
15. c

Q1. One tank is filled in 6 min at the rate of 3cu ft/min, length of tank is 4 ft and the width is 1/2 of length, what is the depth of the tank?

Q2. Pipe A will fill the tank in 6hr and B in 4hr,if the pipes are opened alternatively, in which A is opened first, then how many Hours it will take to fill the tank?

Q3. A pipe A takes 4 hrs and B 6 hrs to fill a tank and a
leakage empties the tank at the rate of 6 litres per hour. It takes 40 hrs to fill up the tank with all the pipes and the leakage working together. Find the volume of the tank?

Q4. If a square is formed by the diagonal of the square as an edge, what is the ratio between the area of new and Old Square?

Q5. 1/3 of girls, 1/2 of boys go to canteen. What factor and total number of classmates go to canteen?

Q6. An equilateral triangle of sides 3 inch each is given. How many equilateral triangles of side 1 inch can be formed from it?

Q7. Perimeter of front wheel =30, back wheel = 20. If front wheel revolves 240 times how many revolutions will the back wheel take?

Q8. City A’s population is 68000, decreasing at a rate of 80 people per year. City B having population 42000 is increasing at a rate of 120 people per year. In how many years both the cities will have same population?

Q9. There are two candles of equal lengths and of different thickness. The thicker one lasts for six hours, the thinner 2 hours less than the thicker one. Ramesh lights the two candles at the same time. When he went to bed he saw the thicker one is twice the length of the thinner one. How long ago did Ramesh light the two candles?

Q10. A worker is paid Rs.20/- for a full days work. He works 1,1/3,2/3,1/8.3/4 days in a week. What is the total amount paid for that worker?

Answers

1: 3 ft 7.5 inches
2: 5hr
3: 19.4594
4: 2
5: Cannot be determined.
6: 9
7: 360 times
8: 130 years
9: 3 hours.
10: 57.50

1. A, B, and C went to have a meal in the hotel, A has paid 50% more than the amount that B has paid and C has paid 5/6 of that of A has paid. Also B has paid $2 more than that of C has paid. Then what is the total amount that all the three paid for their meal?

2. If the cost of an article is x, first discount given is y% of cost, second discount given is z% of cost. The selling price of x is…?

3. In June a baseball team that played 60 games had won 30% of its game played. After a phenomenal winning streak this team raised its average to 50% .How many games must the team have won in a row to attain this average?

4. The price of a product is reduced by 30%. By what percentage should it be increased to make it 100%?

5. If a man buys 1 litre of milk for Rs.12 and mixes it with 20% water and sells it for Rs.15, then what is the percentage of gain?

6. If on an item a company gives 25% discount, they earn 25% profit. If they now give 10% discount then what is the profit percentage?

7. If an item costs Rs.3 in ‘99 and Rs.203 in ‘00.What is the % increase in price?

8. A student gets 70% in one subject, 80% in the other. To get an overall of 75% how much should get in third subject?

9. The cost of an item is Rs 12.60. If the profit is 10% over selling price what is the selling price?

10. A man bought a horse and a cart. If he sold the horse at 10 % loss and the cart at 20 % gain, he would not lose anything; but if he sold the horse at 5% loss and the cart at 5% gain, he would lose Rs. 10 in the bargain. The amount paid by him was Rs._______ for the horse and Rs.________ for the cart.

Answers

1. $30
2. x (1-y/100)(1-z/100)
3. 24
4. 42.857%
5. 44%
6. 30%
7. 200/3 %
8. 75%
9. Rs 13.86/-
10. Cost price of horse = Rs. 400 & the cost price of cart = 200.

Q1. Two trains start at the same time from Mumbai and Delhi and proceed towards each other at 80 km/hr and 95 km/hr respectively. When they meet it is found that one of the trains has traveled 180 kms more than the other. Find the distance between Mumbai and Delhi?

a) 210 km
b) 2000 km
c) 2100 km
d) none of these

Q2. If a train 110m long passes a telegraph pole in 3 sec, then the time taken by it to cross a railway platform of 165 m long is?

a) 110/3 sec
b) 55 sec
c) 7.5 sec
d) 55/3 sec

Q3. Rashi and Richa start simultaneously from P and Q towards Q and P respectively. They meet on the way at t, which is at a distance of 120 m from P. If Rashi and Richa take 16 sec and 25 sec respectively to reach to their respective destinations from T, then what is the distance of P and Q?

a) 214 m
b) 200 m
c) 240 m
d) 216 m

Q4. Two cars driven by A and B. They were 580 miles apart and they drove towards each other. A’s car has travelled 20mph, 4 hr/day, for 5 days, when it had met B’s car. If B had driven 3 hr/day for 5 days, what was B’s speed?

a) 8mph
b) 9mph
c) 10mph
d) 12mph

Q5. Shankar traveled first 2 hrs at a speed of 40 km/hr. Next 2hrs he traveled at a speed of 50 km/hr and the next 2 hrs at 90 km/hr. Find his average speed.

a) 45 km/hr
b) 60 km/hr
c) 54 km/hr
d) none of these

Q6. A thief escaped from police custody. Since he was a sprinter, he could run at a speed of 40 km/hr. The police realized it after 3 hr and started chasing him in the same direction at 50km/hr. The police had a dog, which could run at 60 km/hr. The dog would run to the thief and then return back to the police and then would turn back towards the thief. It kept on doing so till the police caught the thief. Find the total distance traveled by the dog in the direction of the thief?

a) 720 km
b) 600 km
c) 660 km
d) 360 km

Q7. If a boy walks from his house to his school at the rate of 4 km/hr, he reaches his school 10 mins earlier than the usual time. However, if he walks at the rate of 3km/hr, he reaches the school 10 mins later than the scheduled time. Find the distance of the school from his house?

a) 6km
b) 4.5km
c) 4km
d) 3km

Q8. A train crosses a platform which is 250m long .The speed of train is 36 km/hr. The total time taken to cross the platform is 35 sec. Find the length of the train?

a) 120 m
b) 100m
c) 120 km
d) 100 km

Q9. Ram and Shyam start simultaneously from the opposite ends of a pool, which is 50m long. They pass each other reach the respective ends and immediately turn back. Now they meet at a distance of 15 m from where Ram started, 10 sec after the start. Find the speed of Shyam?

a) 6.5 m/s
b) 7.5 m/s
c) 8.5 m/s
d) 5 m/s

Q10. Two trains are running on parallel rails and they travel at the rates of 25mph and 30mph. If the first train leaves an hour earlier than the second, how long will it take for the second train to catch up with the first train?

a) 3 hr
b) 11 hr
c) 6hr
d) 5hr