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Arithmetic II

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Time & Speed
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CAT 2025 Lesson : Time & Speed - Concepts & Cheatsheet

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Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

   12. Cheatsheet

1) When a distance \space D  \spaceis covered at a speed of \space S  \spaceat time \space T, then

 \space D = S×T, S = DTD \space = \space S \times T, \space S \space = \space \dfrac{D}{T} and T = DST \space = \space \dfrac{D}{S}

      \space \space \space \space \space \space1.1) Distance and Speed have a direct relationship, i.e. D∝SD \propto S

      \space \space \space \space \space \space1.2) Distance and Time have a direct relationship, i.e. D∝TD \propto T

      \space \space \space \space \space \space1.3) Speed and Time, however, have an inverse relationship, i.e. S∝1TS \propto \dfrac{1}{T}

          \space \space \space \space \space \space \space \space \space \space1.3.1) If ratio of speed is S1:S2:S3S_1 : S_2 : S_3, then ratio of time taken is 1S1:1S2:1S3 \dfrac{1}{S_1} : \dfrac{1}{S_2} : \dfrac{1}{S_3}

2) 1 km/hr=518 m/s1 \space km/hr = \dfrac{5}{18} \space m/s and 1 m/s=185 km/hr1 \space m/s = \dfrac{18}{5} \space km/hr

3) Average Speed = Total Distance CoveredTotal Time Taken\dfrac{Total \space Distance \space Covered}{Total \space Time \space Taken}

      \space \space\space \space \space \space3.1) When Time Taken are equal, then the average speed is the Arithmetic Mean of the Speeds.

      \space \space \space \space \space \space3.2) When Distances are equal, then the average speed is the Harmonic Mean of the Speeds.

4) When Distance is constant,

      ∙ \space \space \space \space \space \space\bullet if one of time or speed increases by ab \dfrac{a}{b} of itself, then the other decreases by
       ab+a \space \space \space \space \space \space \space \dfrac{a}{b + a} of itself

      ∙ \space \space \space \space \space \space\bullet if one of time or speed decreases by ab \dfrac{a}{b} of itself, then the other increases by
       ab−a \space \space \space \space \space \space \space \dfrac{a}{b - a} of itself

5) Where S1S_1 and S2S_2 are two speeds with S1 > S2S_1 \space > \space S_2

      ∙ \space \space \space \space \space \space\bullet in opposite directions, Time taken = DistanceRelative Speed = DS1+S2=\space \dfrac{Distance}{Relative \space Speed} \space = \space \dfrac{D}{S_1 + S_2}

      ∙ \space \space \space \space \space \space\bullet in same direction, Time taken = DistanceRelative Speed = DS1−S2=\space \dfrac{Distance}{Relative \space Speed} \space = \space \dfrac{D}{S_1 - S_2}

6) If a train of length l takes time t to pass an

      ∙ \space \space \space \space \space \space\bullet object of negligible length at a speed s, then l=s×tl = s \times t

      ∙ \space \space \space \space \space \space\bullet object of length l2l_2 at a speed ss, then l1+l2=s×tl_1 + l_2 = s \times t

7) If a train of length ll moving at speed s1s_11 takes time tt to pass an

      ∙ \space \space \space \space \space \space\bullet object of negligible length moving at speed s2s_2, moving in opposite direction,
       l=(s1+s2)×t \space \space \space \space \space \space \space l = (s_1 + s_2) \times t

      ∙ \space \space \space \space \space \space\bullet object of negligible length moving at speed s2s_2, moving in same direction, then
       l=(s1−s2)×t \space \space \space \space \space \space \space l = (s_1 - s_2) \times t

8) If two trains of length l1l_1 and l2l_2 moving at speeds of s1s_1 and s2s_2 respectively, take time tt to cross each other

      ∙ \space \space \space \space \space \space\bullet in opposite directions, then l1+l2=(s1+s2)×tl_1 + l_2 = (s_1 + s_2) \times t

      ∙ \space \space \space \space \space \space\bullet in same direction with s1>s2s_1 > s_2, then l1+l2=(s1−s2)×tl_1 + l_2 = (s_1 - s_2) \times t

9) Two people walking to and fro from two ends of a corridor of length ll, when meeting for the nthn^{th} time, together cover a distance of (2n−1)l(2n - 1)l

10) Two people running around circular track at speeds of s1s_1 and s2(s1>s2)s_2(s_1 > s_2) meet for nthn^{th} time,

      ∙ \space \space \space \space \space \space \bullet when running in opposite directions, time taken =nls1+s2 = \dfrac{nl}{s_1 + s_2}

      ∙ \space \space \space \space \space \space \bullet when running in same directions, time taken =nls1−s2 = \dfrac{nl}{s_1 - s_2}

11) Where ss is the speed of the boat in still water and aa is the speed of the stream, the relative speed while

      ∙ \space \space \space \space \space \space \bullet travelling downstream =s+a= s + a

      ∙ \space \space \space \space \space \space \bullet travelling upstream =s−a= s - a

12) Where xx and yy are downstream and upstream speeds, boat speed =x+y2= \dfrac{x + y}{2} and stream speed =x−y2= \dfrac{x - y}{2}

13)  Downstream TimeUpstream Time = Upstream SpeedDownstream Speed\space \dfrac{Downstream \space Time}{Upstream \space Time} \space = \space \dfrac{Upstream \space Speed}{Downstream \space Speed}

14) The angles covered by the minute and hour hand for 1 hour and 1 minute are provided below.

           \space \space \space \space \space \space \space \space \space \space \space Minute Hand Hour Hand
1 Hour 3600360^0 30030^0
1 Minute 606^0 0.500.5^0

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