Time and Distance: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

Let two people move in the same direction, one chasing the other. Each second, the faster one gains only the extra distance over the slower one. So relative speed = (faster speed) – (slower speed). This difference is the rate at which the gap between them shrinks.

Let the two starting points be D apart. The two people move towards each other. So the gap between them closes at the sum of their speeds. Relative speed = speed of the first + speed of the second. Time to meet = distance apart / relative speed = D / (sum of speeds).

Say each one-way leg covers distance d. Time for the first leg = d/x. Time for the second leg = d/y. Total distance = 2d. Total time = d/x + d/y = d(x+y)/(xy). Average speed = total distance / total time = 2d divided by d(x+y)/(xy). This simplifies to 2xy/(x+y), the harmonic mean of x and y. It is not the simple average, (x+y)/2.

Let the speeds of A and B be in the ratio a : b. For a fixed distance, time = distance / speed. So time is inversely proportional to speed. The time ratio is 1/a : 1/b. Multiplying both parts by ab clears the fractions. This gives b : a, the speed ratio reversed.

1 m/sec means 3600 metres every hour, since an hour has 3600 seconds. 3600 metres = 3.6 km. So 1 m/sec = 3.6 km/hr, which is the same as 18/5. To convert any m/sec value to km/hr, multiply by 18/5. This is simply the reverse of the km/hr-to-m/sec factor, 5/18.

1 km/hr means 1000 metres every 3600 seconds. 1000/3600 simplifies to 5/18. So 1 km/hr = 5/18 m/sec. To convert any km/hr value to m/sec, multiply by 5/18. Check: 18 km/hr x 5/18 = 5 m/sec, a normal jogging speed.

Let distance = D, speed = S, and time = T. Distance equals speed multiplied by time: D = S x T. Rearranging, speed = D / T. Rearranging again, time = D / S. Example: if S = 60 km/hr and T = 2 hours, D = 60 x 2 = 120 km.

Let the total distance be x km. 1/5 of x is ridden at 5 km/hr, and the rest at 8 km/hr. Time = distance/speed, so total time = (1x/5)/5 + (4x/5)/8. This simplifies to x times 0.14 = 2.8 hours, i.e. 2 hours 48 minutes. So x = 2.8 / 0.14 = 20 km. Check: time at 5 km/hr = 1/5 x 20 / 5 = 0.8 hours. Time at 8 km/hr = 4/5 x 20 / 8 = 2 hours. Total = 2.8 hours, matching 2.8.

Let the total distance be x km. 1/2 of x is ridden at 5 km/hr, and the rest at 3 km/hr. Time = distance/speed, so total time = (1x/2)/5 + (1x/2)/3. This simplifies to x times 0.27 = 3.2 hours, i.e. 3 hours 12 minutes. So x = 3.2 / 0.27 = 12 km. Check: time at 5 km/hr = 1/2 x 12 / 5 = 1.2 hours. Time at 3 km/hr = 1/2 x 12 / 3 = 2 hours. Total = 3.2 hours, matching 3.2.

Let the total distance be x km. 2/5 of x is ridden at 8 km/hr, and the rest at 4 km/hr. Time = distance/speed, so total time = (2x/5)/8 + (3x/5)/4. This simplifies to x times 0.2 = 1.2 hours, i.e. 1 hour 12 minutes. So x = 1.2 / 0.2 = 6 km. Check: time at 8 km/hr = 2/5 x 6 / 8 = 0.3 hours. Time at 4 km/hr = 3/5 x 6 / 4 = 0.9 hours. Total = 1.2 hours, matching 1.2.

Let the total distance be x km. 2/5 of x is ridden at 5 km/hr, and the rest at 6 km/hr. Time = distance/speed, so total time = (2x/5)/5 + (3x/5)/6. This simplifies to x times 0.18 = 1.8 hours, i.e. 1 hour 48 minutes. So x = 1.8 / 0.18 = 10 km. Check: time at 5 km/hr = 2/5 x 10 / 5 = 0.8 hours. Time at 6 km/hr = 3/5 x 10 / 6 = 1 hours. Total = 1.8 hours, matching 1.8.

Let the total distance be x km. 1/3 of x is ridden at 5 km/hr, and the rest at 8 km/hr. Time = distance/speed, so total time = (1x/3)/5 + (2x/3)/8. This simplifies to x times 0.15 = 1.2 hours, i.e. 1 hour 12 minutes. So x = 1.2 / 0.15 = 8 km. Check: time at 5 km/hr = 1/3 x 8 / 5 = 0.53 hours. Time at 8 km/hr = 2/3 x 8 / 8 = 0.67 hours. Total = 1.2 hours, matching 1.2.

Let the total distance be x km. 2/5 of x is ridden at 4 km/hr, and the rest at 6 km/hr. Time = distance/speed, so total time = (2x/5)/4 + (3x/5)/6. This simplifies to x times 0.2 = 1.2 hours, i.e. 1 hour 12 minutes. So x = 1.2 / 0.2 = 6 km. Check: time at 4 km/hr = 2/5 x 6 / 4 = 0.6 hours. Time at 6 km/hr = 3/5 x 6 / 6 = 0.6 hours. Total = 1.2 hours, matching 1.2.

Let the total distance be x km. 2/5 of x is ridden at 8 km/hr, and the rest at 4 km/hr. Time = distance/speed, so total time = (2x/5)/8 + (3x/5)/4. This simplifies to x times 0.2 = 2.4 hours, i.e. 2 hours 24 minutes. So x = 2.4 / 0.2 = 12 km. Check: time at 8 km/hr = 2/5 x 12 / 8 = 0.6 hours. Time at 4 km/hr = 3/5 x 12 / 4 = 1.8 hours. Total = 2.4 hours, matching 2.4.

Let the total distance be x km. 1/5 of x is ridden at 3 km/hr, and the rest at 6 km/hr. Time = distance/speed, so total time = (1x/5)/3 + (4x/5)/6. This simplifies to x times 0.2 = 3.4 hours, i.e. 3 hours 24 minutes. So x = 3.4 / 0.2 = 17 km. Check: time at 3 km/hr = 1/5 x 17 / 3 = 1.13 hours. Time at 6 km/hr = 4/5 x 17 / 6 = 2.27 hours. Total = 3.4 hours, matching 3.4.

Let the total distance be x km. 1/4 of x is ridden at 5 km/hr, and the rest at 3 km/hr. Time = distance/speed, so total time = (1x/4)/5 + (3x/4)/3. This simplifies to x times 0.3 = 1.8 hours, i.e. 1 hour 48 minutes. So x = 1.8 / 0.3 = 6 km. Check: time at 5 km/hr = 1/4 x 6 / 5 = 0.3 hours. Time at 3 km/hr = 3/4 x 6 / 3 = 1.5 hours. Total = 1.8 hours, matching 1.8.

Let the total distance be x km. 1/4 of x is ridden at 5 km/hr, and the rest at 3 km/hr. Time = distance/speed, so total time = (1x/4)/5 + (3x/4)/3. This simplifies to x times 0.3 = 3 hours, i.e. 3 hours. So x = 3 / 0.3 = 10 km. Check: time at 5 km/hr = 1/4 x 10 / 5 = 0.5 hours. Time at 3 km/hr = 3/4 x 10 / 3 = 2.5 hours. Total = 3 hours, matching 3.

Let the total distance be x km. 1/2 of x is ridden at 3 km/hr, and the rest at 5 km/hr. Time = distance/speed, so total time = (1x/2)/3 + (1x/2)/5. This simplifies to x times 0.27 = 3.2 hours, i.e. 3 hours 12 minutes. So x = 3.2 / 0.27 = 12 km. Check: time at 3 km/hr = 1/2 x 12 / 3 = 2 hours. Time at 5 km/hr = 1/2 x 12 / 5 = 1.2 hours. Total = 3.2 hours, matching 3.2.

Let the total distance be x km. 2/3 of x is ridden at 8 km/hr, and the rest at 5 km/hr. Time = distance/speed, so total time = (2x/3)/8 + (1x/3)/5. This simplifies to x times 0.15 = 1.2 hours, i.e. 1 hour 12 minutes. So x = 1.2 / 0.15 = 8 km. Check: time at 8 km/hr = 2/3 x 8 / 8 = 0.67 hours. Time at 5 km/hr = 1/3 x 8 / 5 = 0.53 hours. Total = 1.2 hours, matching 1.2.

Distance = speed x time, for each car. Distance covered by A = 25 x 6 = 150 km. Distance covered by B = 42 x 10 = 420 km. Ratio of distances = 150 : 420. Divide both terms by their HCF, 30: 150/30 : 420/30 = 5 : 14. Check: 5 x 30 = 150 and 14 x 30 = 420, so the simplified ratio is correct.

Distance = speed x time, for each car. Distance covered by A = 62 x 10 = 620 km. Distance covered by B = 32 x 9 = 288 km. Ratio of distances = 620 : 288. Divide both terms by their HCF, 4: 620/4 : 288/4 = 155 : 72. Check: 155 x 4 = 620 and 72 x 4 = 288, so the simplified ratio is correct.

Distance = speed x time, for each car. Distance covered by A = 56 x 8 = 448 km. Distance covered by B = 85 x 10 = 850 km. Ratio of distances = 448 : 850. Divide both terms by their HCF, 2: 448/2 : 850/2 = 224 : 425. Check: 224 x 2 = 448 and 425 x 2 = 850, so the simplified ratio is correct.

Distance = speed x time, for each car. Distance covered by A = 43 x 4 = 172 km. Distance covered by B = 51 x 2 = 102 km. Ratio of distances = 172 : 102. Divide both terms by their HCF, 2: 172/2 : 102/2 = 86 : 51. Check: 86 x 2 = 172 and 51 x 2 = 102, so the simplified ratio is correct.

Distance = speed x time, for each car. Distance covered by A = 75 x 7 = 525 km. Distance covered by B = 51 x 5 = 255 km. Ratio of distances = 525 : 255. Divide both terms by their HCF, 15: 525/15 : 255/15 = 35 : 17. Check: 35 x 15 = 525 and 17 x 15 = 255, so the simplified ratio is correct.

Distance = speed x time, for each car. Distance covered by A = 69 x 9 = 621 km. Distance covered by B = 83 x 2 = 166 km. Ratio of distances = 621 : 166. Divide both terms by their HCF, 1: 621/1 : 166/1 = 621 : 166. Check: 621 x 1 = 621 and 166 x 1 = 166, so the simplified ratio is correct.