## Time, Distance and Work

Theory

Constant Distance:

Let the distances travelled in each part of the journey be $$d_{1}, d_{2}, d_{3}$$ and so on till $$d_{n}$$ and the speeds in each part be $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.

If $$d_{1} = d_{2} = d_{3} =...= d_{n}$$= d, then the average speed is the harmonic mean of the speeds $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.

Constant Time:

Let the distances travelled in each part of the journey be $$d_{1}, d_{2}, d_{3}$$ and so on till $$d_{n}$$ and the time taken for each part be $$t_{1}, t_{2}, t_{3}$$ and so on till $$t_{n}$$.

If $$t_{1} = t_{2} = t_{3} =...= t_{n}$$= t, then the average speed is the arithmetic mean of the speeds $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.

Theory

The questions in this section can vary from being very easy to surprisingly difficult. This is a conceptual section (especially questions involving clocks) and some of the questions can consume a lot of time. While solving, write down the equations as far as possible to avoid mistakes. The few extra seconds can help you avoid silly mistakes. Also, check if the units of distance, speed and time match up. So if you see yourself adding a unit of distance like m to a unit of speed m/s, you would realize you have missed a term. Choose to apply the concept of relative speed wherever possible as it can greatly reduce the complexity of the problem. Like speed and distance, in time and work while working with terms ensure that you convert all terms to consistent units like man-hours.

Tip
• In a journey travelled with different speeds, if the distance covered in each stage is constant, the average speed is the harmonic mean of the different speeds.
• In a journey travelled with different speeds, if the time travelled in each stage is constant, the average speed is the arithmetic mean of the different speeds.
Formula

CLOCKS:

• In a well functioning clock, both hands meet after every 720 / 11 Mins.
• It is because the relative speed of the minute hand with respect to the hour hand = 11/2 degrees per minute.
Formula

PIPES & CISTERNS:

Inlet Pipe: A pipe which is used to fill the tank is known as Inlet Pipe.

Outlet Pipe: A pipe which can empty the tank is known as an Outlet Pipe.

• If a pipe can fill a tank in 'x' hours then the part filled per hour= 1/x
• If a pipe can empty a tank in 'y' hours, then the part emptied per hour= 1/y
• If pipe A can fill a tank 'x' hours and pipe can empty a tank in 'y' hours, if they are both active at the same time, then

The part filled per hour  =$$\frac{1}{x}-\frac{1}{y}$$(if y>x)

The part emptied per hour  =$$\frac{1}{y}-\frac{1}{x}$$(if x>y)

Formula

BOATS & STREAMS

• If the speed of water is 'W' and speed of a boat in still water is 'B'

1. Speed of the boat downstream is B+W
2. Speed of the boat upstream is B-W

The direction along the stream is called downstream. And, the direction against the stream is called upstream.

• If the speed of the boat downstream is x km/hr and the speed of the boat upstream is y km/hr, then

Speed of boat in still water= $$\frac{x+y}{2}$$km/hr
Rate of stream= $$\frac{x-y}{2}$$km/hr

• While converting the speed in m/s to km/hr, multiply it by 3.6(18/5).

1m/s = 3.6 km/h

• While converting km/hr into m/sec, we multiply by 5/18
Formula

Circular Tracks

If two people are running on a circular track with speeds in the ratio a:b where a and b are co-prime, then

• They will meet at a+b distinct points if they are running in the opposite directions.
• They will meet at |a-b| distinct points if they are running in the same direction.

If two people are running on a circular track having perimeter I, with speeds m and n,

• The time for their first meeting = $$\frac{I}{(m+n)}$$

(when they are running in opposite directions)

• The time for their first meeting = $$\frac{I}{(|m-n|)}$$

(when they are running in the same direction)

If a person P starts from A and heads towards B and another person Q starts from B and heads towards A and they meet after a time 't' then,

t = $$\sqrt{x*y}$$

where x = time taken (after the meeting) by P to reach B and y = time taken (after the meeting) by Q to reach A.

A and B started st a time towards each other. After crossing each other, they took $$T_{1}$$ hrs, $$T_{2}$$ hrs respectively to reach their destinations. If they travel at constant speeds $$S_{1}$$ and $$S_{2}$$ respectively all over the journey, Then
$$\frac{S_{1}}{S_{2}}$$=$$\sqrt{\frac{T_{2}}{T_{1}}}$$

Formula

Distance = Speed$$\times$$Time

Speed = $$\frac{Distance}{Time}$$

Time = $$\frac{Distance}{Speed}$$

While covering the Speed in m/s to km/hr. multiply it by 3.6. It is because 1m/s = 3.6 km/hr

If the ratio of the speeds of A and B is a : b, then

• The ratio of the times taken to cover the same distance is 1/a : 1/b or b : a.
• The ratio of distance travelled in equal time intervals is a : b

Average speed= $$\frac{Total Distance Travelled}{Total Time Taken}$$

If a part of a journey is travelled at speed $$S_{1}$$ km/hr in $$T_{1}$$ hours and the remaining part at speed $$S_{2}$$ km/hr in $$T_{2}$$ hours then.
Total distance travelled= $$S_{1}T_{1}$$+$$S_{2}T_{2}$$ km
Average speed=$$\frac{S_{1}T_{1}+S_{2}T_{2}}{T_{1}+T_{2}}$$ km/hr

If $$D_{1}$$ km is travelled at speed of $$S_{1}$$ km/hr, and $$D_{2}$$ km is travelled at speed of $$S_{2}$$ km/hr then

Average Speed= $$\frac{D_{1}+D_{2}}{\frac{D_{1}}{S_{1}}+\frac{D_{2}}{S_{2}}}$$ km/hr

• In a journey travelled at different speeds, if the distance covered in each stage is constant, the average speed is the harmonic mean of the different speeds.
• Suppose a man covers a certain distance st x km/hr and an equal distance at y km/hr

Then the average speed during the whole journey is $$\frac{2xy}{x+y}$$ km/hr

• In a journey travelled with different speeds, if the time travelled in each stage is constant, the average speed is the harmonic mean of the different speeds.
• If a man travelled for a certain time at the speed of x km/hr and travelled for an equal amount of time at the speed of y km/hr then

Then the average speed during the whole journey is $$\frac{x+y}{2}$$ km/hr

Formula

TRAINS:

Two trains of length $$L_1$$ and $$L_2$$ travelling at speeds of $$S_1$$ and $$S_2$$ cross each other in

• $$\frac{L_1+L_2}{S_1+S_2}$$ if they are going in opposite directions

• $$\frac{L_1+L_2}{S_1-S_2}$$ if they are going in the same direction
Formula
• If X can do a work in 'n' days, the fraction of work X does in a day is $$\frac{1}{n}$$
• If X can do a work in 'x' days, and Y can do a work in 'y' days, the number of days taken by both of them together is $$\frac{x*y}{x+y}$$
• If $$A_1$$ men can do $$B_1$$ work in $$C_1$$ days and $$A_2$$ men can do $$B_2$$ work in $$C_2$$ days, then $$\frac{A_1 C_1}{B_1}$$ =$$\frac{A_2 C_2}{B_2}$$
Formula
• While converting the speed in m/s to km/hr, multiply it by 3.6. It is because 1 m/s = 3.6 km/hr
Tip
• In a well functioning clock, both the hands meet after every $$\frac{720}{11}$$ mins. It is because relative speed of minute hand with respect to hour hand = $$\frac{11}{2}$$ degrees per minute.
Formula

Work:

• If X can do a work in 'n' days, the fraction of work X does in a day us 1/n
• If X can do a work in 'x' days, and Y can do a work in 'Y' days,

The number of days taken by both of them together is $$\frac{x*y}{x+y}$$

• If $$M_{1}$$ men work for $$H_{1}$$ hours per day and worked for $$D_{1}$$ days and completed $$W_{1}$$ work, and if $$M_{2}$$ men work for $$H_{2}$$ hours per day and worked for $$D_{2}$$ days and completed $$W_{1}$$ work, then

$$\frac{M_{1}H_{1}D_{1}}{W_{1}}$$=$$\frac{M_{2}H_{2}D_{2}}{W_{2}}$$