Time, Speed and Distance: SSC CGL Quantitative Aptitude

Formulas (3)

• Distance equals speed times time · Speed, distance and time
  $$d = s \times t \qquad s = \frac{d}{t} \qquad t = \frac{d}{s}$$
• Changing km/h to m/s and back · Changing units of speed
  $$1 \text{ km/h} = \frac{5}{18} \text{ m/s} \qquad 1 \text{ m/s} = \frac{18}{5} \text{ km/h}$$
• Speed ratios when the time or the distance is fixed · Ratio of two speeds
  $$\frac{s_1}{s_2} = \frac{d_1}{d_2} \times \frac{t_2}{t_1}$$

Watch out for (6)

• Reading 2 hours 30 minutes as 2.3 hours
  → Distance equals speed times time
• Counting clock time across noon
  → Distance equals speed times time
• Turning the fraction upside down
  → Changing km/h to m/s and back
• Mixing seconds and minutes with m/s
  → Changing km/h to m/s and back
• Using the speed ratio as the time ratio
  → Speed ratios when the time or the distance is fixed
• Multiplying the two ratios instead of dividing
  → Speed ratios when the time or the distance is fixed

Formulas (2)

• Average speed of a journey in parts · Average speed
  $$\bar{s} = \frac{d_1 + d_2 + d_3}{t_1 + t_2 + t_3} \qquad \bar{s} = \frac{s_1t_1 + s_2t_2}{t_1 + t_2}$$
• Average speed over equal distances · Average speed over equal distances
  $$\bar{s} = \frac{2ab}{a + b} \qquad \bar{s} = \frac{3abc}{ab + bc + ca}$$

Watch out for (4)

• Averaging the speeds
  → Average speed of a journey in parts
• Leaving a halt out of the time
  → Average speed of a journey in parts
• Using 2ab/(a + b) when the times are equal
  → Average speed over equal distances
• Taking the average speed as the one-way distance
  → Average speed over equal distances

Formulas (3)

• Speed and time on the same route · Fixed distance
  $$s_1 t_1 = s_2 t_2 \qquad \frac{t_2}{t_1} = \frac{s_1}{s_2}$$
• Two speeds over the same distance and the time between them · Distance from a time gap
  $$d = \frac{s_1 s_2 \, \Delta t}{s_2 - s_1}$$
• Halts, rests and the speed needed to arrive on time · Rests after every k km
  $$\text{number of rests} = \frac{d}{k} - 1 \quad (k \text{ divides } d)$$

Watch out for (6)

• Taking a 25% rise in speed as a 25% cut in time
  → Speed and time on the same route
• Giving the new speed when the increase is asked
  → Speed and time on the same route
• Leaving the time gap in minutes
  → Two speeds over the same distance and the time between them
• Subtracting a late time from an early time
  → Two speeds over the same distance and the time between them
• Counting a rest at the finish
  → Halts, rests and the speed needed to arrive on time
• Using the whole trip to find the catching up speed
  → Halts, rests and the speed needed to arrive on time

Formulas (4)

• Closing a gap at the relative speed · Catching up
  $$t = \frac{\text{gap}}{s_1 - s_2}$$
• How far each one runs before the catch · Distance run before the catch
  $$d_{\text{ahead}} = \frac{s_2 \times \text{gap}}{s_1 - s_2} \qquad d_{\text{chaser}} = \frac{s_1 \times \text{gap}}{s_1 - s_2}$$
• A chase where the chaser starts later · Chase after a late start
  $$t = \frac{s_2 \times \text{delay}}{s_1 - s_2}$$
• Moving towards each other · Meeting head on
  $$t = \frac{d}{s_1 + s_2} \qquad s_1 + s_2 = \frac{d}{t_1} \qquad s_1 - s_2 = \frac{d}{t_2}$$

Watch out for (9)

• Adding the speeds in a chase
  → Closing a gap at the relative speed
• Dividing metres by km/h
  → Closing a gap at the relative speed
• Giving the chaser's distance when the other's is asked
  → How far each one runs before the catch
• Measuring the catch point from the wrong start
  → How far each one runs before the catch
• Using the delay as the chase time
  → A chase where the chaser starts later
• Adding the chase time to the wrong start
  → A chase where the chaser starts later
• Subtracting the speeds for movers coming towards each other
  → Moving towards each other
• Forgetting the early starter's lead
  → Moving towards each other
• Out and back: closing only the one-way distance
  → Moving towards each other

Formulas (2)

• A train passing a pole, a platform or a standing train · Train passing a fixed object
  $$t = \frac{L_{\text{train}} + L_{\text{object}}}{s} \qquad s = \frac{p}{t_2 - t_1}$$
• Trains passing moving trains or people · Two trains crossing
  $$t_{\text{same}} = \frac{L_1 + L_2}{s_1 - s_2} \qquad t_{\text{opp}} = \frac{L_1 + L_2}{s_1 + s_2}$$

Watch out for (4)

• Leaving out the train's own length
  → A train passing a pole, a platform or a standing train
• Adding a length for a pole or a standing man
  → A train passing a pole, a platform or a standing train
• Using one train's length for two moving trains
  → Trains passing moving trains or people
• Subtracting the speed of a walker coming towards the train
  → Trains passing moving trains or people

Formulas (2)

• Downstream and upstream speeds · Boats and streams
  $$D = b + s \qquad U = b - s \qquad b = \frac{D + U}{2} \qquad s = \frac{D - U}{2}$$
• Boat and stream speeds from a ratio · Boat to stream from the two times
  $$\frac{b}{s} = \frac{t_U + t_D}{t_U - t_D}$$

Watch out for (5)

• Forgetting to halve
  → Downstream and upstream speeds
• Giving the downstream speed as the boat's speed
  → Downstream and upstream speeds
• Putting the time ratio straight in as the speed ratio
  → Boat and stream speeds from a ratio
• Taking the speeds themselves as the unknowns
  → Two journeys with two unknown speeds
• Stopping at the downstream and upstream speeds
  → Two journeys with two unknown speeds

Formulas (2)

• Beating by a distance or by a time · Beating margins
  $$\frac{s_A}{s_B} = \frac{d}{d - x} \qquad s_B = \frac{x}{t}$$
• Head starts and dead heats · Dead heat with a head start
  $$\frac{d}{d - x} = \frac{s_A}{s_B} \qquad d = \frac{kx}{k - 1}$$

Watch out for (4)

• Taking the margin as the loser's distance
  → Beating by a distance or by a time
• Subtracting the two margins
  → Beating by a distance or by a time
• Adding the head start to the leader's distance
  → Head starts and dead heats
• Treating a dead heat as a win
  → Head starts and dead heats

Formulas (3)

• The first meeting on a circular track · First meeting on a circular track
  $$t_{\text{same}} = \frac{L}{s_1 - s_2} \qquad t_{\text{opp}} = \frac{L}{s_1 + s_2}$$
• Back at the starting point together · Together at the starting point
  $$t = \text{LCM}\left(\frac{L}{s_1}, \frac{L}{s_2}\right)$$
• Distinct meeting points on a circular track · Number of distinct meeting points
  $$\text{same direction: } a - b \qquad \text{opposite directions: } a + b$$

Watch out for (7)

• Using the difference of speeds for opposite directions
  → The first meeting on a circular track
• Waiting for a full lap when one starts ahead
  → The first meeting on a circular track
• Counting a meeting after the race is over
  → The first meeting on a circular track
• Taking the LCM of the speeds
  → Back at the starting point together
• Giving the first meeting when the start is asked
  → Back at the starting point together
• Using the speeds before reducing the ratio
  → Distinct meeting points on a circular track
• Rounding up a part of a meeting
  → Distinct meeting points on a circular track