31. In a 1 km race, the speeds of A and B are in the ratio of 5 : 3 and A wins by 10 seconds. Find the time taken by B to finish the race and distance covered by B when A finishes the race.
Sol:
Distance is constant
∴ Time ∝ \(\frac{1}{{Speed}}\)
⟹ \(\frac{3}{5} = \frac{t}{{t + 10}}\)
t = 15 sec
∴ time taken by B to finish the race = 15 + 10= 25 sec Answer
B takes 25 sec to cover 1000 m
∴ in 15 se B will cover = \(\frac{{15}}{{25}} \times 1000\) = 600 m Answer
32. A and B are running on a circular track in opposite direction from same time at same point with speed of 81 m/sec and 6 m/sec respectively. If the length of the circular track be 960 m, how many times distinct points they will meet.
Sol:
4 : 3 ⟹ 4 + 3 = 7 times Answer
33. A and B run a 12 km race on a circular track of length 1200 m. They complete one round in 300 sec and 400 sec respectively. After how much time from start will the faster person meet the slower person for the last time.
Sol:
Number of rounds = \(\frac{{12 \times 1000}}{{1200}}\) = 10 rounds
Speed of A = \(\frac{{1200}}{{300}}\) = 4 m/sec
Speed of B = \(\frac{{1200}}{{400}}\) = 3 m/sec
So in 1200 sec A will complete \(\frac{{1200}}{{300}}\) = 4 rounds
& in 1200 sec B will complete \(\frac{{1200}}{{400}}\) = 3 rounds
So after A’s 4 rounds they will meet first time after A’s next 4 round they will meet second time 7 number of rounds are 10 (maximum by A)
So they will meet last time after 8 rounds of A & time taken by A to complete one round = 300 se
So they will meet last time after = 8 × 300 sec
= 2400 sec Answer
34. A and B run around a circular track of length 1100 m in opposite direction with initial speed of 6 m/sec and 4 m/sec respectively. Starting from the same point whenever they meet, A’s speed halves and B’s speed doubles. After how much time from the beginning will they meet for the second time ?
Sol:
1ˢᵗ meeting = \(\frac{{1100}}{{4 + 6}}\) = 110 m
2ⁿᵈ meeting = \(\frac{{1100}}{{\frac{6}{2} + 4 \times 2}}\) = 100 sec
∴ From starting time for 2ⁿᵈ meeting = 110 + 100 = 210m Answer
35. Three persons A, B and C run along a circular track at speed of 3 km/h, 4 km/h and 6 km/h respectively. If the length of the track is 36 km, then after how much time will they meet again at the starting point.
Sol:
Time taken by A = \(\frac{{36}}{3}\) = 12 h
Time taken by A = \(\frac{{36}}{4}\) = 9 h
Time taken by A = \(\frac{{36}}{6}\) = 6 h
They will meet at the starting point after = LCM(12, 9, 6) = 36 h Answer