Speed, Time and Distance
- A railway engine is proceeding towards A at uniform speed of 30 km/hr. While the engine is 20 kms away from A an insect starting from A flies again and again between A and the engine relentlessly. The speed of insect is 42 km per hr. Find the distance covered by the insect till the engine reaches A.
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Relative speed of insect
= 30 + 42 = 72 km per hr.
Distance between railway engine and insect = 20 km.
Engine and insect will meet for the first time after= 20 hr. 72
Distance covered in this period= 20 × 42 = 35 km 72 3 The insect will cover 35 km in returning to A. 3
The distance covered by engine in this period= 20 × 30 = 25 km 72 3
Since, the insect when reaches A,the engine will cover 25 to A. 3
∴ Remaining distance between A and engine= 20 − 25 + 25 3 3 Again, engine and insect will meet after 10 = 5 hr. 3 × 72 108
The distance covered by the insect in this period= 5 × 42 = 35 km 108 18 tand again the insect will cover 35 km in returning. 18 ∴ Total distance covered by the insect = 70 + 70 +..... 3 18 = 70 1 + 1 +.......∞ 3 6
It is a Geometric Progression to infinity with common ratio 1/6.= 70 1 3 1 − (1/6) ∴ S∞ = a 1 − r = 70 × 1 3 (5/6) = 70 × 6 = 28 km 3 5 Correct Option: D
Relative speed of insect
= 30 + 42 = 72 km per hr.
Distance between railway engine and insect = 20 km.
Engine and insect will meet for the first time after= 20 hr. 72
Distance covered in this period= 20 × 42 = 35 km 72 3 The insect will cover 35 km in returning to A. 3
The distance covered by engine in this period= 20 × 30 = 25 km 72 3
Since, the insect when reaches A,the engine will cover 25 to A. 3
∴ Remaining distance between A and engine= 20 − 25 + 25 3 3 Again, engine and insect will meet after 10 = 5 hr. 3 × 72 108
The distance covered by the insect in this period= 5 × 42 = 35 km 108 18 tand again the insect will cover 35 km in returning. 18 ∴ Total distance covered by the insect = 70 + 70 +..... 3 18 = 70 1 + 1 +.......∞ 3 6
It is a Geometric Progression to infinity with common ratio 1/6.= 70 1 3 1 − (1/6) ∴ S∞ = a 1 − r = 70 × 1 3 (5/6) = 70 × 6 = 28 km 3 5
- A man covered a distance of 3990 km partly by air, partly by sea and remaining by land. The time spent in air, on sea and on land is in the ratio 1 : 16 : 2 and the ratio of average speed is 20 : 1 : 3 respectively. If total average speed is 42 km per hr, find the distance covered by sea.
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Total distance travelled = 3990 km
Distance = Time × Speed
Ratio of time spent = 1 : 16 : 2
Ratio of speed = 20 : 1 : 3
∴ Ratio of time × speed
= 20 × 1 : 16 × 1 : 2 × 3
= 20 : 16 : 6
Sum of the ratios
= 20 + 16 + 6 = 42
∴ Distance covered by sea= 3990 × 16 = 1520 kms 42 Correct Option: C
Total distance travelled = 3990 km
Distance = Time × Speed
Ratio of time spent = 1 : 16 : 2
Ratio of speed = 20 : 1 : 3
∴ Ratio of time × speed
= 20 × 1 : 16 × 1 : 2 × 3
= 20 : 16 : 6
Sum of the ratios
= 20 + 16 + 6 = 42
∴ Distance covered by sea= 3990 × 16 = 1520 kms 42
- A motorist and a cyclist start from A to B at the same time. AB is 18 km. The speed of motorist is 15 m per hr. more than the cyclist. After covering half the distance, the motorist rests for 30 minutes and thereafter his speed is reduced by 20%. If the motorist reaches the destination B, 15 minutes earlier than that of the cyclist, then find the speed of the cyclist.
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Let the speed of the cyclist be x km per hr.
Speed of the motorist= (x + 15) km per hr.
Time taken by the motorist to cover half of the distance= 18 = 9 hrs. 2 × (x + 15) x + 15
After covering 9 kms, the speed of motorist gets reduced by 20%∴ New speed = (x + 15) × 80 100 = 4(x + 15) km per hr. 5
Time taken by the motorist to cover the remaining half distance= 9 × 5 = 45 hrs. 4 (x + 15) 4 (x + 15)
Total time taken by the motorist= 9 + 1 + 45 hrs. x + 15 2 4 (x + 15) Total time taken by the cyclist = 18 hrs. x Motorist reaches 15 minutes, i.e., 1 hr. earlier. 4 ∴ 18 − 9 − 1 − 45 = 1 x x + 15 2 4 (x + 15) 4 ⇒ 18 × 4 (x + 15) − 36x − 2x (x + 15) − 45x = 1 4x (x + 15) 4
⇒ 72x + 1080 – 36x – 2x2 – 30x – 45x = x2 + 15
⇒ 3x2 + 54x – 1080 = 0
⇒ x2 + 18x – 360 = 0
⇒ x2 + 30x – 12x – 360 = 0
⇒ x (x + 30) – 12 (x + 30) = 0
⇒ (x + 30) (x – 12) = 0
⇒ x = – 30, 12
The speed cannot be negative.
∴ The speed of the cyclist = 12 km per hr.Correct Option: B
Let the speed of the cyclist be x km per hr.
Speed of the motorist= (x + 15) km per hr.
Time taken by the motorist to cover half of the distance= 18 = 9 hrs. 2 × (x + 15) x + 15
After covering 9 kms, the speed of motorist gets reduced by 20%∴ New speed = (x + 15) × 80 100 = 4(x + 15) km per hr. 5
Time taken by the motorist to cover the remaining half distance= 9 × 5 = 45 hrs. 4 (x + 15) 4 (x + 15)
Total time taken by the motorist= 9 + 1 + 45 hrs. x + 15 2 4 (x + 15) Total time taken by the cyclist = 18 hrs. x Motorist reaches 15 minutes, i.e., 1 hr. earlier. 4 ∴ 18 − 9 − 1 − 45 = 1 x x + 15 2 4 (x + 15) 4 ⇒ 18 × 4 (x + 15) − 36x − 2x (x + 15) − 45x = 1 4x (x + 15) 4
⇒ 72x + 1080 – 36x – 2x2 – 30x – 45x = x2 + 15
⇒ 3x2 + 54x – 1080 = 0
⇒ x2 + 18x – 360 = 0
⇒ x2 + 30x – 12x – 360 = 0
⇒ x (x + 30) – 12 (x + 30) = 0
⇒ (x + 30) (x – 12) = 0
⇒ x = – 30, 12
The speed cannot be negative.
∴ The speed of the cyclist = 12 km per hr.
- A boatman takes his boat in a river against the stream from a place A to a place B where AB is 21 km and again returns to A. Thus he takes 10 hours in all. The time taken by him downstream in going 7 km is equal to the time taken by him against stream in going 3 km. Find the speed of river.
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Let the speed of boat and river be x km per hr. and y km per hr. respectively.
Then, The speed of boatman downstream = (x + y) km per hr.
and the speed of boatman upstream = (x – y) km per hr.
Time taken by boatman in going 21 km downstream= 21 hours x + y
Time taken by boatman in going 21 km upstream= 21 hrs. x − y
According to the question,= 21 = 21 = 10 ...(i) x + y x − y Now, time taken for 7 kms downstream = 7 hrs. x + y and time taken for 3 kms upstream = 3 hrs. x − y
According to the question7 − 3 = 0 ...(ii) x + y x − y
By (ii) × 7 + (i)49 − 21 = 21 = 21 = 10 x + y x − y x + y x − y ⇒ 70 = 10 x + y
⇒ x + y = 7 ...(iii)
Putting x + y = 7 in equation (ii)
we have7 − 3 = 0 7 x − y ⇒ 1 − 3 = 0 x − y
⇒ x – y = 3 ...(iv)
On adding (iii) and (iv), we have
2x = 10
⇒ x = 5
∴ y = 7 – x = 7 – 5 = 2
∴ Speed of river = 2 km per hr.Correct Option: A
Let the speed of boat and river be x km per hr. and y km per hr. respectively.
Then, The speed of boatman downstream = (x + y) km per hr.
and the speed of boatman upstream = (x – y) km per hr.
Time taken by boatman in going 21 km downstream= 21 hours x + y
Time taken by boatman in going 21 km upstream= 21 hrs. x − y
According to the question,= 21 = 21 = 10 ...(i) x + y x − y Now, time taken for 7 kms downstream = 7 hrs. x + y and time taken for 3 kms upstream = 3 hrs. x − y
According to the question7 − 3 = 0 ...(ii) x + y x − y
By (ii) × 7 + (i)49 − 21 = 21 = 21 = 10 x + y x − y x + y x − y ⇒ 70 = 10 x + y
⇒ x + y = 7 ...(iii)
Putting x + y = 7 in equation (ii)
we have7 − 3 = 0 7 x − y ⇒ 1 − 3 = 0 x − y
⇒ x – y = 3 ...(iv)
On adding (iii) and (iv), we have
2x = 10
⇒ x = 5
∴ y = 7 – x = 7 – 5 = 2
∴ Speed of river = 2 km per hr.
- A person can row a boat 32 km upstream and 60 km downstream in 9 hours. Also, he can row 40 km upstream and 84 km downstream in 12 hours. Find the rate of the current.
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Let the upstream speed be x km per hr. and downstream speed be y km per hr.
Then, we can write,32 + 60 = 9 x y and, 40 + 84 = 12 x y Let 1 = m and 1 = n x y
The above two equations can now be written as
32 m + 60 n = 9 ...(i)
and, 40 m + 84 n = 12 ...(ii)
7 × (i) – 5 × (ii) gives 24 m = 3or m = 1 or x = 8 km per hr. 8
4 × (ii) – 5 × (i) gives 36 n = 3or n = 1 or y = 12 km per hr. 12
Rate of current= y − x = 12 − 8 = 2 km. per hr. 2 2 Correct Option: D
Let the upstream speed be x km per hr. and downstream speed be y km per hr.
Then, we can write,32 + 60 = 9 x y and, 40 + 84 = 12 x y Let 1 = m and 1 = n x y
The above two equations can now be written as
32 m + 60 n = 9 ...(i)
and, 40 m + 84 n = 12 ...(ii)
7 × (i) – 5 × (ii) gives 24 m = 3or m = 1 or x = 8 km per hr. 8
4 × (ii) – 5 × (i) gives 36 n = 3or n = 1 or y = 12 km per hr. 12
Rate of current= y − x = 12 − 8 = 2 km. per hr. 2 2