Speed, Time and Distance: AFCAT Numerical Ability

Formulas (3)

• Speed, distance, time and their units · Speed and the unit change
  $$\text{speed} = \dfrac{\text{distance}}{\text{time}}, \qquad 1 \text{ km/h} = \tfrac{5}{18} \text{ m/s}$$
• Going and coming back at two speeds · Distance from a round trip
  $$\dfrac{d}{a} + \dfrac{d}{b} = T \;\Rightarrow\; d = \dfrac{abT}{a + b}$$
• One journey at two speeds, and rests on the way · A journey in two parts
  $$\dfrac{x}{a} + \dfrac{D - x}{b} = T$$

Watch out for (6)

• Change the units before you divide
  → Speed, distance, time and their units
• Minutes are not decimal places
  → Speed, distance, time and their units
• The total time covers both legs
  → Going and coming back at two speeds
• Turn the minutes into hours
  → Going and coming back at two speeds
• The two parts are x and what is left
  → One journey at two speeds, and rests on the way
• No rest after the last stretch
  → One journey at two speeds, and rests on the way

Formulas (5)

• Total distance over total time · Average speed
  $$\bar v = \dfrac{d_1 + d_2 + \dots}{t_1 + t_2 + \dots}$$
• Speeds held for given times · Average over given times
  $$\bar v = \dfrac{v_1 t_1 + v_2 t_2}{t_1 + t_2}$$
• Equal distances: there and back, or equal stretches · Average over two equal distances
  $$\bar v = \dfrac{2ab}{a + b}$$
• Fractions of the journey, and working back from an average · Average over fractions of the way
  $$\bar v = \dfrac{1}{\tfrac pa + \tfrac qb} \quad (p + q = 1)$$
• Stoppages and breaks · Minutes stopped per hour
  $$\text{stop per hour} = \dfrac{v_1 - v_2}{v_1} \times 60 \text{ min}$$

Watch out for (7)

• Do not average the speeds
  → Total distance over total time
• Weight by time
  → Speeds held for given times
• The mean is too high
  → Equal distances: there and back, or equal stretches
• Work out a changed speed first
  → Equal distances: there and back, or equal stretches
• Fractions of distance weight the times, not the speeds
  → Fractions of the journey, and working back from an average
• Divide by the running speed
  → Stoppages and breaks
• Include the break, or leave it out, as asked
  → Stoppages and breaks

Formulas (3)

• Speed ratios turn into time ratios · Same distance
  $$\dfrac{t_1}{t_2} = \dfrac{v_2}{v_1}$$
• Through the distance: a new speed or a new time · Keeping to a schedule
  $$\text{required speed} = \dfrac{\text{distance left}}{\text{time left}}$$
• A percent change in speed and the change in time · Speed and time move inversely
  $$\text{speed} \times \tfrac{a}{b} \;\Rightarrow\; \text{time} \times \tfrac{b}{a}$$

Watch out for (3)

• Turn the ratio upside down
  → Speed ratios turn into time ratios
• Use the distance left, not the whole route
  → Through the distance: a new speed or a new time
• The two percentages differ
  → A percent change in speed and the change in time

Formulas (3)

• Slower than usual: the delay gives the usual time · The delay at a slower speed
  $$\text{delay} = \left(\dfrac qp - 1\right) t$$
• Faster than usual: the time saved, and a late start · The saving at a faster speed
  $$\text{saving} = \left(1 - \dfrac ba\right) t$$
• Two speeds and the gap between their times · The gap between two travel times
  $$\dfrac da - \dfrac db = \text{gap in hours}$$

Watch out for (6)

• The fraction is of the usual time
  → Slower than usual: the delay gives the usual time
• Answer what is asked
  → Slower than usual: the delay gives the usual time
• The saving is a fraction of the usual time
  → Faster than usual: the time saved, and a late start
• A late start is a saving
  → Faster than usual: the time saved, and a late start
• Late and early add
  → Two speeds and the gap between their times
• Minutes into hours before solving
  → Two speeds and the gap between their times

Formulas (6)

• Adding or subtracting the speeds · Relative speed
  $$v_{\text{rel}} = a + b \;(\text{opposite}), \qquad v_{\text{rel}} = a - b \;(\text{same way})$$
• Meeting head on: when and where · Time and place of meeting
  $$t = \dfrac{D}{a + b}, \qquad d_A = \dfrac{a}{a + b} D$$
• Meeting when one starts later · A late second start
  $$t' = \dfrac{D - a\,t_0}{a + b}$$
• Chasing: catching up from behind · Catch-up time
  $$t = \dfrac{\text{head start}}{v_{\text{chaser}} - v_{\text{leader}}}$$
• A bird flying to and fro · The bird's distance
  $$\text{bird's distance} = v_{\text{bird}} \times \dfrac{D}{a + b}$$
• Speed ratios from meeting and passing times · Times after meeting
  $$\dfrac{v_A}{v_B} = \sqrt{\dfrac{t_2}{t_1}}, \qquad t_{\text{meet}} = \sqrt{t_1 t_2}$$

Watch out for (7)

• Moving apart still adds
  → Adding or subtracting the speeds
• The faster one covers more of the gap
  → Meeting head on: when and where
• Count from the right start
  → Meeting when one starts later
• The two runners cover different distances
  → Chasing: catching up from behind
• A head start in time is not a distance
  → Chasing: catching up from behind
• Do not follow the zigzags
  → A bird flying to and fro
• The quicker finish belongs to the faster train
  → Speed ratios from meeting and passing times

Formulas (5)

• Passing a pole or a standing man · Passing a point
  $$t = \dfrac{L}{v}$$
• Crossing a platform or a bridge · Crossing a platform
  $$t = \dfrac{L + P}{v}$$
• Length and speed from two crossings · Speed from two crossings
  $$v = \dfrac{P_2 - P_1}{t_2 - t_1}$$
• Passing someone who is moving · Passing a moving person
  $$t = \dfrac{L}{v \mp u}$$
• Two trains crossing or overtaking · Two trains
  $$t = \dfrac{L_1 + L_2}{v_1 \pm v_2}$$

Watch out for (5)

• Metres with metres per second
  → Passing a pole or a standing man
• Add the train's own length
  → Crossing a platform or a bridge
• The man on the platform is a pole
  → Length and speed from two crossings
• The man in the train
  → Passing someone who is moving
• Both lengths, and subtract for the same direction
  → Two trains crossing or overtaking

Formulas (6)

• Downstream and upstream speeds · With and against the stream
  $$v_{\text{down}} = b + s, \qquad v_{\text{up}} = b - s$$
• Still water and stream from the two speeds · Half the sum, half the difference
  $$b = \dfrac{d + u}{2}, \qquad s = \dfrac{d - u}{2}$$
• Equal times and ratios of time · Equal times
  $$\dfrac{d_1}{b + s} = \dfrac{d_2}{b - s}$$
• Out and back in a total time · A round trip on a river
  $$\dfrac{d}{b + s} + \dfrac{d}{b - s} = T$$
• Two trips, two equations · Two trips
  $$p_1 x + q_1 y = T_1, \quad p_2 x + q_2 y = T_2, \quad x = \tfrac1u,\; y = \tfrac1d$$
• Boats meeting, chasing and crossing the river · Two boats closing
  $$v_{\text{close}} = (b_1 + s) + (b_2 - s) = b_1 + b_2$$

Watch out for (6)

• Still water is not downstream
  → Downstream and upstream speeds
• Half the difference
  → Still water and stream from the two speeds
• Twice the time is half the speed
  → Equal times and ratios of time
• The round trip average is below b
  → Out and back in a total time
• Solve for the reciprocals
  → Two trips, two equations
• The stream cancels for boats that meet
  → Boats meeting, chasing and crossing the river

Formulas (3)

• Beating by a distance or by a time · Speed ratio from a margin
  $$\dfrac{v_A}{v_B} = \dfrac{D}{D - x}$$
• Head starts and dead heats · Start for a dead heat
  $$\text{start} = D\left(1 - \dfrac{v_B}{v_A}\right)$$
• Three runners: chaining the results · C's distance when A finishes
  $$\dfrac{(D - x)(D - y)}{D}$$

Watch out for (3)

• Beaten by a distance, or by a time
  → Beating by a distance or by a time
• Count the start once
  → Head starts and dead heats
• Distance margins do not add
  → Three runners: chaining the results

Formulas (3)

• The first meeting: opposite ways or the same way · First meeting on a track
  $$t_{\text{opp}} = \dfrac{L}{a + b}, \qquad t_{\text{same}} = \dfrac{L}{a - b}$$
• Counting meetings and laps · Meetings in a time T
  $$n = \dfrac{T\,v_{\text{rel}}}{L} \;\;(\text{whole meetings only})$$
• Back at the start together · Together at the start
  $$T_{\text{start}} = \text{LCM}(t_1, t_2, t_3)$$

Watch out for (3)

• The same way needs a whole lap
  → The first meeting: opposite ways or the same way
• Whole meetings only
  → Counting meetings and laps
• Meeting is not meeting at the start
  → Back at the start together