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multiple choice gmat maths question and answers
GMAT PAPER- III 1. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. What will be the cost of the second necklace? 1 (1) The cost of the first necklace is 5 more than the second and the cost of the third necklace is 2 5 more than the second. The total cost of all the three necklaces is $120000. 2 5 more than the second. The cost of the third necklace is the least and (2) The cost of the first necklace is (a) (b) (c) (d) (e) Solution. total cost of all the three necklaces is $1,20,000. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. EACH statement ALONE is sufficient Statements (1) and (2) TOGETHER are NOT sufficient. (1) Þ Ratio of the costs of first, second and third necklace is 6 : 5 : 7 and total cost is given. Hence the price of second necklace can be calculated. (2) Þ N1 = N2 + , N3 is least. and N1 + N2 + N3 = 120000 Since we don't know N3. Therefore statement (2) is not sufficient to answer the question. (a) 2. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. A, B, C and D made their project presentation, one on each day, on four consecutive days but not necessarily in that order. On which day did 'C' make his presentation? (1) The first presentation was made on 23rd, Tuesday and was followed by 'D's presentation. (2) 'A' did not make his presentation on 25th and one of them made his presentation, between A's and B's. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. (c) From (1) : We get First presentation : 23rd (Tuesday) Second presentation : 24th (Wednesday) : D Third presentation : 25th (Thursday) Fourth presentation : 26th (Friday) 'A' did not make his presentation on 26th also, because 'D' made presentation on 24th. Hence, 'A' made presentation on 23rd, 'B' on 25th and 'C' on 26th. Solution. (1) and (2) Þ 3. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. If the perimeter of the triangle ABC is 3 + 3 3 then what is its area ? B A (1) (2) (a) (b) (c) (d) (e) Solution. C Side AC ¹ Side AB Angle ABC = 30º Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. EACH statement ALONE is sufficient Statements (1) and (2) TOGETHER are NOT sufficient. (b) (1) Þ It is not sufficient (2) Þ Angles of DABC are 30º, 60º and 90º. Þ The lengths of the sides are in the ratio 1 : 2 : 3 ( The ratio between the lengths of the sides in a 30º, 60º, 90º triangle is always constant and it is 1: 2 : 3 ) Since perimeter = 3 + 3 3 ⇒ x + 2 x + 3x = 3 + 3 3 ⇒ x= 3 \ Sides of D are and 3 Also since ÐB = 30º and height = 3 Hence, area of triangle can be determined using statement (2) alone. 4. In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together? (a) 120 (b) 720 (c) 4320 (d) 2160 (e) None of these (b) The word 'OPTICAL' contains 7 different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, 5 letters can be arranged in 5 ! = 120 ways. The vowels (OIA) can be arranged among themselves in 3 ! = 6 ways Required number of ways = (120 × 6) = 720. ⇒ base = 3 Solution. 5. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. How many marks did Prakash obtain in Mathematics? (1) Prakash secured on an average 55 per cent marks in Mathematics, Physics and Chemistry together. (2) Prakash secured 10 per cent more than the average in Mathematics. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient. (e) Statements (1) and (2) TOGETHER are NOT sufficient. Solution. (1) M + P + C = 165% (2) Pr M + 10% (average) Both statements together are not sufficient to answer the question. 6. The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is: (a) 15 (b) 16 (c) 18 (d) 25 Solution. (b) Let Cost price of each article be $1 Þ Cost price. of x articles = $x and Selling price. of x articles = $ 20. Þ Profit = $ (20 – x). Now, 7. 20 − x x × 100 = 25 2000 – 100x = 25x 125x = 2000 x = 16. The average of six numbers is 3.95. The average of two of them is 3.4, while the average of the other two is 3.85. What is the average of the remaining two numbers? (a) 4.5 (b) 4.6 (c) 4.7 (d) 4.8 (e) 4.2 Solution. (b) Sum of the remaining two numbers = (3.95 × 6) – [(3.4 × 2) + (3.85 × 2)] = 23.70 – (6.8 + 7.7) = 23.70 – 14.5 = 9.20. ⎛ 9.2 ⎞ ⎜ ⎟ Required average = ⎝ 2 ⎠ = 4.6. 8. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. What is the value of m – n ÷ 37? (1) m is the largest possible six-digit number and n is the smallest possible six-digit number. (2) The difference between m and n is known. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. (1) Þ m = 999999, n = 100000 We can find the value of m – n ÷ 37 (2) Þ m – n = known, but neither the value of 'm' is known nor the value of 'n' is known. So, we cannot find the values of m – n ÷ 37 . (a) Solution. 9. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. What is the sum of the perimeter of the circle denoted by centre O and the triangle AFG shown in the given figure? E B F D O I H G C A (1) (2) (a) (b) (c) (d) (e) Solution. Lengths of AD and DE are known. Lengths of AC and ED are known. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. EACH statement ALONE is sufficient Statements (1) and (2) TOGETHER are NOT sufficient. (d) Here AB and AC are tangent segments and ADE is a secant to the circle intersecting it at D and E. FB, FH, HG and GC are also tangent segments. Since, the lengths of two tangents drawn from an external point to a circle are equal. AB = AC, BF = FH and HG = CG ....(i) Also, we know that Þ AD × AE = AC2 ....(ii) Þ (AE – ED) × AE = AC2 Þ AE2 – (AE × ED) = AC2 Þ AE2 – AC2 = AE × ED [Q ∠ECA = 90°] Þ EC2 = AE × ED [ EC = diameter] Þ Radius of the circle = Þ Perimeter of the circle can be determined if AE and ED are known. Perimeter of the triangle AFG = AF + FH + HG + GA = AF + BF + CG + GA [from (i)] = AB + AC = 2AC [from (ii)] = 2 AD × AE Þ Perimeter of DAFG can be determined if either AC or AD and DE is known. Hence, to answer the question, any two of AD, DE and AC are required. Each statement alone is sufficient to answer the question. AE × ED 2 [from (i)] 10. A tin funnel consists of two parts; one part is conical with the slant side is 12 cm, the circumference of the one end is 40 cm and of the other end is 2.5 cm; the other part is cylindrical, the circumference being 2.5 cm and length 16 cm. The area of tin used in making 20 such funnels (in sq. cm) is (a) 6,000 (b) 10,200 (c) 5,100 (d) 5,900 (e) 6,200 2πl(R + r) 2 2πlR + 2πrl 2 40 × 12 + 2.5 × 12 2 Solution. (d) Area of the frustum of cone = = 480 + 30 2 510 = = = 2 = 255 sq. cm. R r 16 Area of cylinder = 2prh = 2.5 × 16 = 40 sq. cm Area of tin = 255 + 40 = 295 sq. cm Hence, area of 20 such funnels = 295 × 20 = 5900 sq. cm 11. Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. At this rate, how many bottles could 10 such machines produce in 4 minutes? (a) 648 (b) 1800 (c) 2700 (d) 10800 (e) 3600 (b) Let the required number of bottles be x. More machines, More bottles (Direct Proportion) More minutes, More bottles (Direct Proportion) Machines Time (in Minutes) 6 : 10 :: 270 : x 1: 4 Solution. } 10 × 4 × 270 6 Þ x = 1800. 6 × 1 × x = 10 × 4 × 270 Þ x = 12. A positive integer, which when added to 1000, gives a sum which is greater than when it is multiplied by 1000. This positive integer is: (a) 1 (b) 3 (c) 5 (d) 7 (e) 1 2 Solution. 13. (a) We have, (1000 + N) > (1000N). Only, N = 1 satisfies this inequality. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. How many candidates were interviewed everyday by the panel 'A' out of the three panels A, B and C? (1) The three panels on an average can interview 15 candidates every day. (2) Out of a total of 45 candidates interviewed everyday by the three panels, the no. of candidates interviewed by panel 'A' is more by 2 than the candidates interviewed by panel 'C' and is less by 1 than the candidates interviewed by panel 'B'. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. (b) (1) is not sufficient. (2) Þ A = C + 2 = B – 1 and A + B + C = 45. Solving these, we can get the value of A. Solution. 14. A trader mixes 26 kg of rice of $20 per kg with 30 kg of rice of other variety at $36 per kg and sells the mixture at $30 per kg. His profit percent is: (a) No profit, no loss (b) 5% (c) 8% (d) 10% (e) None of these 12 c m Solution. (b) Cost price. of 56 kg rice = $ (26 × 20 + 30 × 36) = $ (520 + 1080) = $1600. Selling price. of 56 kg rice = $(56 × 30) = $1680. ⎛ 80 ⎞ ⎜ 1600 × 100 ⎟ % ⎝ ⎠ Þ Profit % = = 5%. 15. Sameer spends 24% of his monthly income on food and 15% on the education of his children. Of the remaining salary, he spends 25% on entertainment and 20% on conveyance. He is now left with $ 10,736. What is the monthly salary of Sameer ? (a) $ 27,600 (b) $28,000 (c) $31,200 (d) $32,000 (e) $30,500 (d) Let the monthly salary of Sameer be $ x. Then, [100 – (25 + 20)]% of [100 – (24 + 15)]% of x = 10736, or 55% of 61% of x = 10736 or ⎛ 10736 × 100 × 100 ⎞ 55 61 ⎜ ⎟ 55 × 61 ⎠ = $32000. 100 × 100 × x = 10736 or x = ⎝ Solution. 16. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. Is x an integer? (1) (2) (a) (b) (c) (d) (e) 19 x + 1 is an integer. x +1 5 is an integer. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. EACH statement ALONE is sufficient Statements (1) and (2) TOGETHER are NOT sufficient. 19 x +1 Þ Solution. (b) (1) = an integer, then 19 = (x + 1) × an integer and x + 1 = Þ x may or may not be an integer. Statement (1) alone is not sufficient to answer the question. x +1 (2) Þ 5 = an integer 19 an integer x + 1 = a integer × 5 = an integer x = an integer – 1 = an integer Hence, statement (2) alone is sufficient to answer the question. 17. What percentage of numbers from 1 to 70 have squares that end in the digit 1? (a) 1 (b) 14 (c) 20 (d) 21 (e) 18 (c) Clearly, the numbers which have 1 or 9 in the units digit, have squares that end in the digit 1. Such numbers from 1 to 70 are 1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69. Number of such numbers = 14. ⎛ 14 ⎞ ⎜ 70 × 100 ⎟ % ⎝ ⎠ = 20% Required percentage = Solution. 18. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. What was the total compound interest on a sum after three years? (1) The interest after one year was $. 100 and the sum was $ 1000. (2) The difference between simple and compound interest on a sum of $1000 at the end of two years was $10. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. R ⎞ ⎛ Q Amount = P⎜1 + ⎟ ⎝ 100 ⎠ n Solution. (d) (1) Þ For P = 1000 and n = 1, A = $1100 R = 10% Hence, C.I. after 3 years can be calculated. Statement (1) alone is sufficient to answer the question. ⎛ R ⎞ = P×⎜ ⎟ ⎝ 100 ⎠ Now, since difference between S.I. and C.I. for 2 years 2 From (2), R can be calculated. Hence, C.I. after 3 years can be determined. Therefore, each statement alone is sufficient to answer the question. 19. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. M M 3 an odd integer? (You may assume that 3 is an integer) Is the number (1) (2) (a) (b) (c) (d) (e) M = 3K, where K is an integer. M = 6J + 3, where J is an interger. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. EACH statement ALONE is sufficient Statements (1) and (2) TOGETHER are NOT sufficient. (1) ⇒ M =K 3 Solution. (b) ∴ M 3 may be odd or even. Therefore it is not sufficient to answer the question. M (2) 3 = 2J + 1, which is always odd because J is an integer. Statement (2) alone is sufficient to answer the question. 20. The total wage for a piece of work if done by A and B together (in equal proportions) is $ 17,300. A can do 80% of this work in 64 days and B can do 70% of the same work in 59.5 days. The wage rate of A is 110% of that of B. A can be employed for a maximum of 70% of the work and B for 60% of the work. The minimum cost (in $) for getting the work done would be (a) 17,130 (b) 17,200 (c) 17,240 (d) 17,050 (e) 17,350 Solution. (c) 11x Rs 10 Let B charges $ x for one day of work. Þ A changes for the one day of work. Now, wages [50% work by A + 50% work by B] = 17,300 5 11x 5 ⇒ × 64 × + × 59.5 × x = 17,300 8 10 7 Þ x [44 + 42.5] = 17,300 ⇒ x= 17,300 = Rs 200 86.5 = 6 4 11× 200 × 59.5 × 200 + × 64 × 7 8 10 Now, minimum cost of the work is when B completes 60% of the work and A completes the remaining. i.e. Minimum cost = 10,200 + 7,040 = $ 17,240 21. If (a + b + 2c + 3d) (a – b – 2c + 3d) = (a – b + 2c – 3d) (a + b – 2c – 3d), then 2bc is equal to: (a) (e) 3 2 (b) 2 3a 2d (c) 3ad (d) a2d2 2a 3d 2 Solution. (c) (a + b + 2c + 3d) (a – b – 2c + 3d) = (a – b + 2c – 3d) (a + b – 2c – 3d) [(a + b) + (2c + 3d)] [(a – b) – (2c – 3d)] = [(a – b) + (2c – 3d] [(a + b) – (2c + 3d)] (a + b) (a – b) – (a + b) (2c – 3d) + (a – b) (2c + 3d) – (2c + 3d) (2c – 3d) = (a – b) (a + b) – (a – b) (2c + 3d) + (a + b) (2c – 3d) – (2c + 3d) (2c – 3d) (a + b) (2c – 3d) = (a – b) (2c + 3d) 2ac – 3ad + 2bc – 3bd = 2ac + 3ad – 2bc – 3bd 4bc = 6ad 2bc = 3ad. 1 22. At an International Dinner, 5 th of the people attending were French men. If the number of French women at 2 the dinner was 3 times greater than the number of French men, and there were no other French people at the dinner, then what fraction of the people at the dinner was not French? 1 5 2 5 2 3 7 15 (a) (e) None of these (b) (c) (d) Solution. ⎛1 2 1⎞ 1 1 5 ⎜ + × ⎟ 5 ; French women = ⎝ 5 3 5 ⎠ = 15 = 3 . (d) French men = ⎛1 1⎞ 8 ⎜ + ⎟ ⎝ 5 3 ⎠ = 15 . French people = 8⎞ ⎛ 7 ⎜1 − ⎟ ⎝ 15 ⎠ = 15 . Þ Non-French = 2 1 5 1 23. A ship 77 km from the shore springs a leak which admits 4 tonnes of water in 2 minutes. 92 tonnes of water would sink her. But the pumps can throw out 12 tonnes per hour. Find the average rate of sailing so that she may just reach the shore as she starts to sink. (a) 10.5 kmph (b) 11 kmph (c) 10 kmph (d) 12.5 kmph (e) 15 kmph 9× 2 Solution. (a) In one hour water admitted through leak = 4 × 11 270 − 12 = 138 tonnes 11 × 60 = 270 tonnes 11 Water deposited in ship in one hour = 11 92× Time required to deposit 92 tonnes water = 77 ×138 11 138 hr Average speed = 92 ×11 24. = 10626 = 10.5 km / hr 1012 In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 runs? (a) 6.25 (b) 6.5 (c) 6.75 (d) 7 (e) 8 282 − (3.2 ×10) 250 40 = 40 = 6.25. Required run rate = Solution. 25. (a) DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. Mahesh's flat is on which floor of the five-floor apartment? (1) His flat is exactly above Ganesh's flat whose flat is exactly above Nitin's first -floor flat. (2) Jeevan's flat, which is adjacent to Mahesh's flat, is exactly below Ahmed's flat, who is on fourth floor. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. (d) (1) Þ 3rd floor - Mahesh, 2nd floor - Ganesh, 1st floor - Nitin i.e. Mahesh's flat is on the 3rd floor. (2) Þ 4th floor – Ahmed ¯ 3rd floor – Jeevan – Mahesh i.e. Mahesh's flat is on the 3rd floor. Hence, each statement alone is sufficient to answer the question. Solution. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. What number is opposite the face bearing number 5 of the same dice when it is thrown by different persons? Note that the dice bears the numbers 1 to 6 on different faces of the dice. (1) When Renu throws the dice on a table, she observes that the visible surfaces bear the numbers 3, 6 and 5, while, when her sister throws the dice, she observes that the visible surfaces are 1, 4 and 5. (2) When Tulika throws the same dice, she finds herself unable to see the three faces bearing the numbers 1, 3 and 5 while when her brother Shivendra throws the dice he finds himself able to see the three faces bearing the numbers 6, 4 and 5. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. Solution. (d) (1) Þ The faces bearing the numbers 1, 3, 4 and 6 are adjacent surfaces of the surface bearing the number 5. Hence 2 is opposite to 5. 26. (2) Þ Tulika finds herself able to see the surface bearing the number 2, 4 and 6. While Shivendra can see 6, 4 and 5. From this we can conclude 2 is opposite to 5. 27. If the digit in the units place of a two-digit number is halved and the digit in the tens place is doubled, the number thus obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true? (a) Sum of the digits is a two-digit number. (b) Digit in the units place is twice the digit in the tens place. (c) Digit in the units place and the tens place are equal. (d) Digit in the units place is half of the digit in the tens place. (e) None of these (b) Then, 10 × 2x + 19 2 1 y 2 Solution. Let the tens digit be x and units digit be y. = 10y + x Þ 20x – x = 10y – y 2 Þ 19x = y Þ y = 2x. Thus, the units digit is twice the tens digit. 28. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. Towards which direction is C from Q? (1) C and Q are opposite to each other. L is equidistant from C and Q. (2) L is neither towards north-east nor towards south-west of C and Q respectively. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. (e) Information given in both the statements does not lead to specific direction. Hence, even both (1) and (2) together are not sufficient. Solution. 29. There are five boxes in a cargo hold. The weight of the first box is 200 kg and the weight of the second box is 20% higher than the weight of the third box, whose weight is 25% higher than the first box's weight. The fourth box at 350 kg is 30% lighter than the fifth box. The difference in the average weight of the four heavily boxes and the four lightest boxes is (a) 51.5 kg (b) 75 kg (c) 37.5 kg (d) 112.5 kg (e) None of these (b) Weight of first box = 200 kg. 200 × 125 = 250 kg 100 120 = 300 kg 100 Solution. Weight of third box = 250 × Weight of second box = Weight of fourth box = 350 kg 350 × Weight of fifth box = Required difference = 30. 100 = 500 kg 70 500 + 350 + 300 + 250 200 + 250 + 300 + 350 300 − = = 75 kg 4 4 4 DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. The area of a square is equal to that of a circle. What is the circumference of the circle? (1) The diagonal of the square is x inches. (2) The side of the square is y inches. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. Solution. (d) ⇒ r= 2 2 Given : πr = a (where r = radius of circle; a = side of square) a π Hence, to find the circumference of circle, one should know 'a' the side of square. diagonal 2 (1) Þ Side of square = \ Statement (1) alone is sufficient. Also, as stated above, statement (2) alone is sufficient. 31. A can do a job alone in 18 days and B can do it in 30 days. A and B working together will finish twice the amount of work in 2 days (a) (e) None of these 12 1 15 (b) 1 2 22 days (c) 17 days (d) 1 2 days = Solution. (d) Units of work done by A and B together in one day = 1 1 5+3 8 + = = 18 30 90 90 Þ days required to complete the work together 45 90 45 = 8 4 Þ Twice the work is completed in 4 32. ×2 = 1 45 = 22 days 2 2 DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. The area of a playground is 1600 square metres. What is its perimeter? (1) It is a perfect square playground (2) It costs $3200 to put a fence around the playground at the rate of $20 per metre. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. (d) From (1) : Area Side Perimeter From (2) : Since, Perimeter × rate of fencing per metre = Total cost (in rupees) Hence, each, statement alone is sufficient. Solution. 33. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. What is the value of a three-digit number n? (1) When digits at units place and hundreds place are interchanged, number increases by 693 and in both cases sum of digits of n remains 9. (2) When digits at unit's place and hundreds place are interchanged, number decreases by 693 and in both cases sum of digits of n remains 10. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. Solution. (d) Suppose number is 100x + 10y + z. After interchanging digits at unit's place and at hundred's place, number = 100z + 10y + x Now (1) ⇒z–x= 693 =7 ⇒ z = 7 + x where x ≠ 0 99 If x = 1 then z = 8 and it is given that x + y + z = 9. \y=0 Þ x, y, z have determined. Therefore statement (1) alone is sufficient to answer the question. 99 (2) Þ x = z + 7 and it is given that x + y + z = 10 \ z + 7 + y + z = 10 Þ 2z + y = 3 \ When z = 1; y = 1 and when z = 0, y = 3. But z = 0 cannot be possible because after interchanging the above-said digits, it becomes a two-digit number. Þ x, y, z have determined. Therefore statement (2) alone is sufficient to answer the question. x–z= 693 =7 34. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. What was the population of State 'A' in 1999? (1) Population of the State increases every year by 20% and its population in 1997 was 1,20,000. (2) Population of State A in 1997 was twice that of State B in the same year. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. (1) ÞPopulation in 1999 = 120% of population in 1998 & Population in 1998 = 120% of 1,20,000. (2) Þ We have no data of population of state B \ Statement (2) is not sufficient. Hence, only statement (1) is sufficient to answer the question. (a) Solution. 35. DIRECTIONS: Each of the questions below consists of a question and two statements numbered (1) and (2) given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer. Who scored highest among A, B, C, D and E? (1) B scored more than D, but not as much as C. (2) E scored more than C, but not more than A. (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (b) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient (c) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. (d) EACH statement ALONE is sufficient (e) Statements (1) and (2) TOGETHER are NOT sufficient. Solution. (c) (1) ⇒ C > B > D (2) ⇒ A > E > C Combining both, we get Hence both statements together are necessary. 36. Which of the following has most number of divisors ? (a) 99 (b) 101 (c) 176 (e) both 182 and 99 have equal no. of divisors which is the most (c) 99 = 1 × 3 × 3 × 11; 101 = 1 × 101; 176 = 1 × 2 × 2 × 2 × 2 × 11; 182 = 1 × 2 × 7 × 13. Thus, the divisors of 99 are 1, 3, 9, 11, 33 and 99; the divisors of 101 are 1 and 101; the divisors of 176 are 1, 2, 4, 8, 16, 22, 44, 88 and 176; and the divisors of 182 are 1, 2, 7, 13, 14, 26, 91 and 182. Hence, 176 has the most number of divisors. (d) 182 Solution. 37. A motor car starts with the speed of 70 km/hr with its speed increasing every two hours by 10 kmph. In how many hours will it cover 345 km? 4 hrs (a) (b) 4 hrs 5 min (d) Cannot be determined 2 1 2 hrs (c) (e) None of these 4 1 Solution. (c) Distance covered in first 2 hours = (70 × 2) = 140 km. Distance covered in next 2 hours = (80 × 2) = 160 km. Remaining distance = 345 – (140 + 160) = 45 km. Speed in the fifth hour = 90 km/hr. ⎛ 45 ⎞ 1 hr ⎜ 90 ⎟ hr ⎝ ⎠ Time taken to cover 45 km = = 2 1⎞ ⎛ 1 4 ⎜2+ 2+ 2 ⎟ ⎝ ⎠ 2 Total time taken = = hrs.