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41. Two horses started simultaneously towards each other and meet each other 3 hr 20 mins later. How much time will it take the slower horse to cover the whole distance if the first arrived at the place of departure of the second 5 hours later than the second arrived at the point of departure of the first?

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Solution:
Since, the two horses meet after 200 minutes; they cover 0.5% of the distance per minute (combined) or 30% distance per hour.
This condition is satisfied only if the slower rider takes 10 hours (thereby covering 10% per hour) and the faster rider takes 5 hours.
42. Two ports A and B are 300 km apart. Two ships leave A for B such that the second leaves 8 hours after the first. The ships arrive at B simultaneously. Find the time the slower ship spent on the trip if the speed of one of them is 10 km/h higher than that of the other.

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Solution:
A__________300 km_______B
Let speed of the 1st ship is s kmph and speed of 2nd ship is (s + 10) kmph.
Let 2nd ship takes time t to cover the distance 300 km then 1st ship takes (t + 8) hours.

As distance is constant, we have,
ss+10=tt+8
Or, st + 8s = st + 10t
Or, 8s = 10t

If the slower ship took 20 hours (option D) the faster ship would take 12 hours and their respective speeds would be 15 and 25 kmph.
43. Two boats, travelling at 5 km/h and 10 km/h, head directly towards each other. They begin at a distance of 20 km from each other. How far apart are they (in km) one minute before they collide?

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Solution:
In the final minute of collision, the two boats travel 560 km and 1060 km.
As they moves towards each other relative distance covered in 1 minute,
=112+16=14km
44. Karan and Arjun run a 100 metre race, where Karan beats Arjun by 10 metre. To do a favour to Arjun, Karan starts 10 metre behind the starting line in a second 100 race. They both run at their earlier speeds. Which of the following is true in connection with the race?

Discuss
Solution:
In the 1st race when Karan runs 100 metre, Arjun runs only 90 metre.
Hence, the ratio of speeds of Arjun and Karan is 90 : 100 = 9 : 10
In 2nd race, Karan has run 110 metre.
When he finishes the race, Arjun would have run 910×110=99metre
∴ Karan beats Arjun by 1 metre
45. A ship leaves on a long voyage. When it is 18 miles from shore, a sea plane, whose speed is ten times that of the ship, is sent to deliver mail. How far from the shore does the sea plane catch up with the ship?

Discuss
Solution:
A _______18miles_____B ___x(let)___C
Let the speed of the ship be 1 mile/h and then speed of the plane becomes 10 miles/h.
Let time t is taken to cover the distance x from point B to point C by ship.
Hence, in time t plane will cover (18 + x) miles.

Now,
x1=18+x10
Or, 10x = 18 + x
Or, x = 2 miles.
Hence, ship will cover 2 miles in 2 hours and plane will cover 20 miles in same time.

Short method:
Relative speed = (10 - 1) = 9 miles/h.
hence, time taken to catch up the ship is 2 hours so distance covered by the ship = 20 miles.
46. Two planes move along a circle of circumference 1.2 km with constant speeds. When they move in different directions, they meet every 15 seconds and when they move in the same direction, one plane overtakes the other every 60 seconds. Find the speed of the slower plane.

Discuss
Solution:
The sum of speeds would be 0.8 m/s (relative speed in opposite direction).
Also if we go by option (B) the speeds will be 0.03 and 0.05 m/s respectively.
At this speed the overlapping would occur in every 60 second.

Alternate :
Let their speeds be x m/sec and y m/sec respectively.
Then,
1200x+y=15⇒x+y=80.....(i)
And,
1200x−y=60⇒x−y=20.....(ii)
Adding (i) and (ii), we get :
2x = 100 or x = 50
Putting x = 50 in (i), we get : y = 30
Hence, speed of slower plane :
= 30 m/sec
= 0.03 km/sec
47. Two joggers left Delhi for Noida simultaneously. The first jogger stopped 42 min later when he was 1 km short of Noida and the other one stopped 52 min later when he was 2 km short of Noida. If the first jogger jogged as many kilometers as the second, and the second as kilometers as first, the first one would need 17 min less than the second. Find the distance between Delhi and Noida?

Discuss
Solution:
Speed of first Jogger=x−142×60kmphSpeed of 2ndjogger=x−252×60kmphThen,x−2sb−x−1sb
Now, we use option checking method which gives us that option (B) is correct.
48. An ant moved for several seconds and covered 3 mm in the first second and 4 mm more in each successive second than its predecessor. If the ant had covered 1 mm in the first second and 8 mm more in each successive second, then the difference between the path it would cover during the same time and actual path would be more than 6 mm but less than 30 mm. find the time for which the ant moved (in seconds).

Discuss
Solution:
3 + 7 + 11 + 15 + . . . . . (1)
1 + 9 + 17 + 25 + . . . . . (2)
The condition is satisfied for the 4 seconds in 2nd journey, ant covered 17 m more than 1st one.
49. Two trains start from the same point simultaneously and in the same direction. The first train travels at 40 km /h, and the speed of the second train is 25% more than the speed of first train. Thirty minutes later, a third train starts from same point and in the same direction. It over takes the second train 90 minutes later than it overtook the first train. What is the speed of the third train?

Discuss
Solution:
A _______ B ________ C
Let both the trains start from point A.
At point B third train overtook the first train.
To overtake the first train by third train, Third train needs to cover,
Distance covered by first train in 112 h = distance covered by third train in 1 h.
[as third train has started 30 minutes later]
In this situation distance is constant,
then from,
s × t = d; we get, S α 1t
Now,
t+12t=s402t+1t=s40........(1)
From equation (1),
It is clear that time is in the ratio 3 : 2 then speed will be in 2 : 3 ratios.
Hence, Speed of the Third train will be 60 km/h.
50. A racetrack is in the form of a right triangle. The longer of the legs of track is 2 km more than the shorter of the legs (both these legs being on a highway). The start and end points are also connected to each other through a side road. The escort vehicle for the race took the side road and rode with a speed of 30 km/h and then covered the two intervals along the highway during the same time with a speed of 42 km/h. find the length of the race track.

Discuss
Solution:
The given conditions are met on a Pythagoras triplet 6, 8, 10.
Since, the racetrack only consists of the legs of the right angle triangle the length must be,
= 6 + 8 = 14 km.

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