maths for practice

In how many different ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only the odd positions ? ​

(1)(2)(3)(4)(5)(6)(7) Now, 3 vowels can be placed at any of the three places, out of the four marked 1, 3, 5,7. Number of ways of arranging the vowels = 4P3 = (4 *3 * 2) = 24. Also, the 4 consonants at the remaining 4 positions may be arranged in = 4P4 = 4 ! = 24 ways. Required number of ways = (24 * 24) = 576. ​

Water flows through a cylindrical pipe of internal diameter 7cm at 5m/s. The time, in minutes, the pipe would take to fill an empty rectangular tank 4m×3m×2.31m. ​

(400*300*231)/[π*(7/2)²*500] =>1440 secs ​

A train 400m long overtook a man walking along the line in the same direction as the train, at the rate of 5 kmph and passed him in 40 seconds. The train reached the station in 20 minutes after passing the man. In what time did the man reach the station ? ​

(x-5)5/18*40=400 =>x=41 in 60 min.....41 in 20min.......41/3 41/3+4/10=211/15 km time=((211/15)/5)*60 =>2 hrs 48 min 48 sec. ​

A team of workers was employed by a contractor who undertook to finish 360 pieces of an article in a certain number of days. If the team were to make four more pieces per day than was planned, they would complete the job a day ahead of the schedule. How many days did the team take to complete the job at the original rate? ​

10 days, let the number of days taken be x. so, number of articles made far day = 360/x. ​

A trader professes to sell his goods at a nominal gain percentage but actually earns 37½% profit by using false weight. If for a kg he uses a weight of 800 gm, what is the nominal gain percentage at which he claims to be selling his goods ? ​

10%

The work done by a women in 8 hours is equal to the work done by a man in 6 hours and by a boy in 12 hours. If working 6 hours per day 9 men can complete a work in 6 days, then in how many days can 12 men, 12 women and 12 boys together finish the same work, working 8 hours per day ? ​

3/2 days. 6x=8y=12z=k

In covering a distance of 30 km, Abhay takes 2 hours more than Sameer. If Abhay doubles his speed, then he would take 1 hour less than Sameer. Abhay's speed is ​

30/x -30/2x = 3 x = 5

Two pipes A and B can fill a tank in 6 hours and 4 hours respectively. If they are opened on alternate hours and if pipe A is opened first, then in how many hours, the tank shall be full? ​

5 hours

Two guns are fired from the same place at an interval of 6 minutes. A person approaching the place observes that 5 minutes 52 seconds have elapsed between the hearings of the sound of the two guns. If the velocity of the sound is 330 m/sec, then at what speed the person was approaching that place?

Difference of time = 6 min − 5 min 52 sec = 8 sec Distance covered by person in 5 min 52 sec = Distance covered by sound in 8 sec = 330 × 8 = 2640 m

A pipe can empty a tank in 40 minutes. A second pipe with diameter twice as much as that of the first is also attached with the tank to empty it. The two together can empty the tank in... ​

Here, the diameter of the second pipe is twice that of first pipe. ∴ Volume of water emptied by the second pipe will be 4 times to that of first pipe. Hence, time taken will be 1/4 of the first pipe. Second pipe will empty the tank in ¼×40 = 10 minutes Hence, the tank will be emptied in 8 minutes. ​

The sum and difference of the LCM and HCF of two numbers are 592 and 518 respectively, and the sum is 296. What are the numbers? ​

L+H=592 L-H=518 L=555 H=37 x(296-x)=555×37 x=185/111 ​

A booster pump can be used for filling as well as emptying a tank. The capacity of the tank is 2400m³. The emptying capacity of the tank is 10m³/min higher than its filling capacity and the pump needs 8 min lesser to empty the tank than it need to fill it. What is the filling capacity of the pump? ​

Let the filled capacity of the tank x m³/min Then emptied capacity of the tank=(x+10)m³/min 2400/x-2400/(x+10)=8 =>x=50 ​

Two pipes A and B can fill a cistern in 12 min and 16 min respectively. Both the pipes are opened together for a certain time but due to some obstruction the flow of water was restricted to 7/8 of full flow in pipe A and 5/6 of full in pipe B. This obstruction is removed after some time and tank is now filled in 3 min from that moment. How long was it before the full flow. ​

Let the obstruction remain for x min. Hence, Part of cistern filled in X min + part of cistern filled in 3 min = full cistern => x(7/8×1/12+5/6×1/16)+3(1/12+1/16)=1 => x=4.5 ​

The sum of two numbers is 462 and their highest common factor is 22. What is the minimum number of pair that satisfy these conditions? ​

Let the required number be 22a and 22b. Then, 22a+22b=462 =a+b=21. Now, co-primes with sum 21 are (1,20),(2,19),(4,17),(5,16),(8,13)and (10,11) ∴ Required numbers are (22×1,22×20),(22×2,22×19),(22×4,22×17),(22×5,22×16),(22×8,22×13), and (22×10,22×11). Clearly, the number of such pairs is 6 ​

Two pipes A and B can fill a tank in 20 and 30 hours respectively. Both the pipes are opened to fill the tank but when the tank is one - third full, a leak develops in the tank through which one - third water supplied by both the pipes gose out. The total time taken to fill the tank is? ​

Part filled by (A + B) in 1 hour = (1/20+1/30)=1/12 So, A and B together can fill the tank in 12 hrs, 1/3 part is filled by (A + B) in (1/3×12) = 4 hrs Since the leak empties one - third water, so time taken to fill the tank = Time taken by (A + B) to fill the whole tank + Time taken by (A + B) to fill one - third tank = (12 + 4) = 16 hours ​

Working together, Asha and Sudha can complete an assigned task in 20 days. However, if Asha worked alone and completed half the work and then Sudha takes over the task and completes the second half of the task, the task will be completed in 45 days. How long will Asha take to complete the if she worked alone ? Assume that Sudha is more efficient than Asha.

Suppose, Asha takes x days to complete the task alone while Sudha takes y days to complete it alone. Since Sudha is more efficient than Asha, we have x > y. Asha's 1 day's work=1/x. Sudha's 1 day's work=1/y. (Asha+Sudha)'s 1 day's work=1/x+1/y=x+y/xy. If Asha and Sudha each does half of the work alone, time taken =(x/2+y/2) days =(x+y/2) days. ∴x+y/2=45⇒x+y=90

A does half as much work as B in one -sixth of the time.If together they take 10 days to complete a work, how much time shall B take to do it alone? ​

To do half of the work in one sixth of the time means A is 3 times as efficient as B. Now they together complete the work in 10 days. B is working with someone who is 3 times as efficient as himself. That means 4 people of B's efficiency finished the work in 10 days. So B alone would have done it in 40 days. A alone would have taken one third of this time. ​

A tank is 7 metre long and 4 metre wide. At what speed should water run through a pipe 5 cm broad and 4 cm deep so that in 6 hours and 18 minutes water level in the tank rises by 4.5 metre ? ​

Volume of water that flowed in the tank in 6 hours 18 minutes. = 60 * 60 + 18 minutes = 378 minutes = 2x * 378 cu.cm According to question, 20x 378 = 700 * 400 * 450 x = 700 * 400 * 450/ 20 * 378 cm/minutes x = 700 * 400 * 450 * 60/10000 * 20 * 378 km/hr x = 10 km/hr ​

A leak in the bottom of a tank can empty the full tank in 8 hour. An inlet pipe fills water at the rate of 6 liters a minutes. When the tank is full the inlet is opened and due to leak the tank is empty in 12 hours. How many liters does the cistern hold ? ​

Work done by the inlet in 1 hour = 1/8-1/12=1/24 Then, work done by the inlet in 1 minute = 1/24*1/60=1/1440 1/1440 parts = 6 litres 1 part =6×1440=8640 liters. OR, capacity of tank be x litres. in 1 min.....6 litres in 60 " ....360 " x/8-x/12=360 x=8640

A bath can be filled by the cold water pipe in 10 min and by hot water pipe in 15 min (independently each). A person leaves the bathroom after turning on both pipes simultaneously and returns at the moments when the bath should be full. Finding however, that the waste pipe has been open he now closes it. In 4 min more, bath is full. In what time would be the waste pipe empty it? ​

Work done together in 1 minute = 1/6 Time taken to fill when both are open = 6 minutes. He returns at the time when the bath should have been full, means that he returns in 6 minutes but finds that the waste pipe also has been opened for 6 minutes. Let the waste pipe empty the bath in x min. Work done by cold pipe in 10 minutes (6 + 4) = 1 Work done by Hot pipe in 10 minutes = 2/3 Work done by waste pipe in 6 minutes = 6/x Now, 1+2/3+6/x=1 x = 9 minutes ​

In what time would a cistern be filled by three pipes whose diameters are 1cm , 4/3 cm , 2 cm,running together, when the largest alone will fill it in 61 minutes, the amount of water flowing in by each pipe being proportional to square of its diameter ?? ​

from given info dia === flow rate 1====1 4/3====16/9 2====4 so largest dia 2 cm ; flow rate=4 cm2/min vol of tank=4×61=244 cm^3 since all three pipes are open together so total combined flow rate=1+16/9+4 = 61/9 time taken=61/9*x = 244 x= 36 mins ​

A cistern has 3 pipes A, B and C. A and B can fill it in 3 hours and 4 hours respectively While C can empty the completely filled cistern in 1 hour. If the Pipe are opened in order at 3, 4 and 5 p.m respectively, at what time will the cistern be empty ? ​

let the cistern be emptied t hours after 3 p.m. t/3+(t-1)/4-(t-2)/1=0 t=4⅕ so the cistern will be emptied 4 hours 12 minutes after 3 p.m. i. e. 7 : 12 p.m. ​

Two trains start simultaneously (with uniform speeds) from two stations 270 km apart, each to the opposite station; they reach their destinations in 6¼ hours and 4 hours after they meet. The rate at which the slower train travels is.... ​

speed ratio=√4:√25/4 =>4:5 Let the speeds of the two trains be 4x and 5x km/hr respectively Then time taken by trains to meet each other=270/(4x+5x)hr =(30x)hr Time taken by slower train to travel 270 km =(270/4x)hr ∴270/4x=30/x+25/4 =>x=6 Hence speed of slower train=4x=24km/hr ​

At his usual rowing rate, Rahul can travel 12 miles downstream in a certain river in 6 hours less than it takes him to travel the same distance upstream. But if he could double his usual rowing rate for his 24 -mile round trip, the downstream 12 miles would then take only one hour less than the upstream 12 miles. What is the speed of the current in miles per hour?

Let the speed in still water be x mph and the speed of the current be y mph. Speed of upstream =(x−y) Speed of downstream =(x+y) 12/(x-y)-12/(x+y)=6. x²=4y-y²......(i) 12/(2x-y)-12/(2x-y)=1 x²=(24y+y²)/4

What is the least number that when divided by 3,5,6,8,10 and 12 leaves a remainder of 2 but when divided by 13 leaves no remainder? ​

the required number is of the form 120x+2 (since it leaves a reminder of 2) Now the number is a multiple of 13. so (120x+2) or (117x+3x+2) is a multiple of 13. The least value of x for which this holds true is 8 (3*8+2 = 26) Therefore the least such number is 120*8+2 = 962 ​

When 15% is lost in grinding wheat, a country can export 30 lakh tons of wheat. On the other hand, if 10% is lost in grinding, it can export 40 lakh tons of wheat. The production of wheat in the country is? ​

5%=10 lakh 100%=200 lakh

The cost of manufacturing an article rose by 18% a result of the increase in the cost of raw material. A manufacturer revised the selling price of the article so as to maintain the same profit percent as before. However, he found that he now got Rs.9 more than the earlier profit by selling each article. What was the earlier profit per article? ​

50

A garrison had provision for a certain number of days. After 10 days, 1/5 of the men desert and it is found that the provisions will now last just as long as before. How long was that ? ​

50 days. 5m(d-10) = 4md

A tank has two outlets A and B , which together take 6 hours to empty a full tank when they are opened simuntaneously. The tank was initially half-full and both the outlets were opened. After an hour, an inlet pipe X was also opened. If the inlet alone can fill an empty tank in 4 hours , how much time will it now take to fill the tank completely ? ​

Part of the tank filled by inlet in 1 h = 1/4 part of the tank emptied by outlets A and B together in 1h = 1/6 Let the time taken to fill the tank completely = ah (a - 1)/4 - a/6 = 1/2 6a - 6 - 4a/24 = 1/2 2a - 6 = 12 2a = 18 a = 9 hour. ​

Two pipes can fill cistern in 14 hours and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom it took 32 minutes more to fill the cistern. When the cistern is full, in what time will leak empty it? ​

Work done by the two pipes in 1 hour =(1/14)+(1/16)=(15/112). Time taken by these pipes to fill the tank = (112/15) hrs = 7 hrs 28 min. Due to leakage, time taken = 7 hrs 28 min + 32 min = 8 hrs Work done by (two pipes + leak) in 1 hour = (1/8). Work done by the leak in 1 hour =(15/112)-(1/8)=(1/112). Leak will empty the full cistern in 112 hours. ​

In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. The duration of the flight is: ​

let, the duration of the fight be the x hours, x = 1