For natural numbers x, y, and z, if xy + yz = 19 and yz + xz = 51, then the minimum possible value of xyz is
xy + yz = 19, y(x + z) = 19 Now, we have two cases y = 1, x + z = 19 .....(1) y = 19, x + z = 1 which is not possible as x and z are natural numbers so they cannot take value less than 1 yz + xz = 51 z(y+x) = 51 z = 3, y + x = 17....(2) or z = 17 , y + x = 3...(3) Combining (1) and (2) together y = 1, x = 16, z = 3 then xyz = 16 × 1 × 3 = 48 Combining (1) and (3) together z = 17, y = 1, x = 2 then xyz = 2 × 1 × 17 = 34 Hence the minimum value of xyz = 34
Alex invested his savings in two parts. The simple interest earned on the first part at 15% per annum for 4 years is the same as the simple interest earned on the second part at 12% per annum for 3 years. Then, the percentage of his savings invested in the first part is
In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is
Let a, b, c and d be the number of students who like none of the drinks, exactly one drink, exactly 2 drinks, and all three drinks, respectively. Hence a + b + c + d = 100... (1). Also b + 2c + 3d = 73 + 80 + 52. So b + 2c + 3d = 205 ... (2) Subtracting (1) from (2) we get, c + 2d - a = 105. Maximum value d can take is 52 i.e. d = 52, c = 1, a = 0 The minimum value d can take is 5, i.e. t = 5, c = 95 and a = 0 (This also satisfies equation (1). Difference = 52 - 5 = 47 Hence, the difference between the minimum and maximum number of people who like all the 3 drinks is 47.
A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is
Let the lemon juice be 10 litre Let the sugar juice be 10 litre Total mixture = 20 liter New mixture sugar added be in the ratio 1:3 20 liter mixture and 60 liter sugar syrup New lemon juice = 10 liter New sugar syrup = 10 + 60 = 70 Therefore, required ratio = 10 : 70 = 1 : 7 : Option 4
Let a, b, c be non-zero real numbers such that b2 < 4ac, and f(x) = ax2 + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be
Amal buys 110 kg of syrup and 120 kg of juice, syrup being 20% less costly than juice, per kg. He sells 10 kg of syrup at 10% profit and 20 kg of juice at 20% profit. Mixing the remaining juice and syrup, Amal sells the mixture at Rs. 308.32 per kg and makes an overall profit of 64%. Then, Amal’s cost price for syrup, in rupees per kg, is
Let cost price of syrup = Rs 80 x/kg Cost price of juice = Rs 100x/ kg Total cost = 80x × 100 + 100x × 120 = 20800x Selling price = 80x×1.1 ×10 kg + 100x × 1.2×20 kg + 308.32 × 200 kg = 880x + 2400x + 61664 = 3280x + 61664 Selling price = cost price × 1.64 = 20800x × 1.64 = 34112x Equating selling price = 34112x = 3280x + 61664 30832x = 61664 x = 2 cost price of syrup = 80x = 160
In a village, the ratio of number of males to females is 5 : 4. The ratio of number of literate males to literate females is 2 : 3. The ratio of the number of illiterate males to illiterate females is 4 : 3. If 3600 males in the village are literate, then the total number of females in the village is
Ankita buys 4 kg cashews, 14 kg peanuts and 6 kg almonds when the cost of 7 kg cashews is the same as that of 30 kg peanuts or 9 kg almonds. She mixes all the three nuts and marks a price for the mixture in order to make a profit of Rs.1752. She sells 4 kg of the mixture at this marked price and the remaining at a 20% discount on the marked price, thus making a total profit of Rs. 744. Then the amount, in rupees, that she had spent in buying almonds is
Let the 1kg price of cashews = C, 1kg price of peanuts = P, 1 kg price of almonds = A 7C = 30P = 9A = 630x C = 90x × 4 = 360x P = 21x × 14 = 294x A = 70x × 6 = 420x Total cost price = 360x + 294x + 420x = 1074x Let the selling price be y Rs/ kg 4y + 20y = 1752....(1) 4y + 16y = 744.....(2) Subtract (2) – (1) 4y = 1008 24y = 6048 Total cost = 6048 – 1752 = 1074x 1074x = 4296 x = 4 Price of almonds = 420x = 420(4) = 1680 : Option 2
The average weight of students in a class increases by 600 gm when some new students join the class. If the average weight of the new students is 3 kg more than the average weight of the original students, then the ratio of the number of original students to the number of new students is
Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of Pinky to the number of persons standing behind her in the queue is 3 : 5. If the total number of persons in the queue is less than 300, then the maximum possible number of persons standing ahead of Pinky is
Let the total number of persons ahead = 3x Let the total number of persons behind = 5x Total number of persons = 3x + 5x + 1 < 300 8x < 300 – 1 < 299 x < 37.3 so x = 37 ( integral value = 37) No of persons ahead = 3x = 3(37) = 111
Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in 10 minutes while train B takes 9 minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train B to travel from station Y to station X is
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