Time, Work and Distance

Overview

Time, Work and Distance is one of the most application-oriented topics in the MP TET Mathematics section. These problems test a candidate's ability to apply arithmetic reasoning to real-life situations—workers completing tasks, taps filling tanks, vehicles covering distances, and boats moving in streams. The topic draws heavily on concepts of ratio, proportion, and the unitary method.

For MP TET, expect 2–4 questions from this combined topic across Varg-1, Varg-2, and Varg-3 papers. Questions range from straightforward calculations to multi-step word problems. Mastery requires understanding the underlying relationships (work ∝ 1/time, distance = speed × time) and developing the ability to quickly set up equations from worded scenarios.

The key to success is recognising problem types instantly and applying the correct formula without confusion. Most errors stem from mixing up concepts or misreading the question—skills that improve with systematic practice.

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Key Concepts

• **Work and Time are inversely related**: If A completes a job in 10 days, A's one day work = 1/10. More efficient workers take less time.

• **Combined Work**: When A and B work together, their combined one day work = (1/a) + (1/b), where a and b are their individual completion times.

• **Efficiency Ratio**: If A is twice as efficient as B, then A takes half the time B takes. Efficiency ∝ 1/Time.

• **Distance-Speed-Time Triangle**: Distance = Speed × Time. Rearrange as needed: Speed = Distance/Time, Time = Distance/Speed.

• **Relative Speed**: When two objects move in the same direction, relative speed = difference of speeds. When moving towards each other, relative speed = sum of speeds.

• **Average Speed**: For a journey with two different speeds, Average Speed = (2 × S₁ × S₂)/(S₁ + S₂) when distances are equal. Do NOT simply average the speeds.

• **Upstream and Downstream (Boats/Streams)**: Downstream speed = Boat speed + Stream speed. Upstream speed = Boat speed − Stream speed.

• **Pipes and Cisterns**: Inlet pipes do positive work (fill); outlet pipes do negative work (empty). Combine as algebraic sum.

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Formulas / Key Facts

| Concept | Formula | |---------|---------| | One day's work | If total work done in n days, one day work = 1/n | | Combined work (A and B together) | 1/T = 1/a + 1/b, so T = (a × b)/(a + b) | | Work with efficiency | Work = Efficiency × Time | | Distance formula | D = S × T | | Relative speed (same direction) | S_rel = S₁ − S₂ | | Relative speed (opposite direction) | S_rel = S₁ + S₂ | | Average speed (equal distances) | S_avg = 2S₁S₂/(S₁ + S₂) | | Downstream speed | S_down = B + R (B = boat, R = river/stream) | | Upstream speed | S_up = B − R | | Speed of boat in still water | B = (S_down + S_up)/2 | | Speed of stream | R = (S_down − S_up)/2 |

**Unit Conversions**:

• 1 km/hr = 5/18 m/s
• 1 m/s = 18/5 km/hr

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Worked Examples

### Example 1: Time and Work (Combined Work) **Problem**: A can complete a task in 12 days, B can complete it in 18 days. In how many days will they finish the work together?

**Solution**:

• A's one day work = 1/12
• B's one day work = 1/18
• Combined one day work = 1/12 + 1/18 = (3 + 2)/36 = 5/36
• Time to complete = 36/5 = **7.2 days or 7 days and 4.8 hours**

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### Example 2: Time and Distance (Relative Speed) **Problem**: Two trains 150 m and 100 m long are moving in opposite directions at 40 km/hr and 32 km/hr. In what time will they cross each other?

**Solution**:

• Total distance to cover = 150 + 100 = 250 m
• Relative speed = 40 + 32 = 72 km/hr = 72 × (5/18) = 20 m/s
• Time = Distance/Speed = 250/20 = **12.5 seconds**

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### Example 3: Boats and Streams **Problem**: A boat goes 24 km downstream in 3 hours and returns upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream.

**Solution**:

• Downstream speed = 24/3 = 8 km/hr
• Upstream speed = 24/4 = 6 km/hr
• Speed of boat (B) = (8 + 6)/2 = **7 km/hr**
• Speed of stream (R) = (8 − 6)/2 = **1 km/hr**

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### Example 4: Pipes and Cisterns **Problem**: Pipe A fills a tank in 6 hours, Pipe B empties it in 8 hours. If both are opened together, in how many hours will the tank be filled?

**Solution**:

• A's work per hour = +1/6 (fills)
• B's work per hour = −1/8 (empties)
• Net work per hour = 1/6 − 1/8 = (4 − 3)/24 = 1/24
• Time to fill = **24 hours**

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Adding individual times to get combined time (12 + 18 = 30 days) | Add work rates (1/12 + 1/18), then invert to find time | | Averaging speeds directly for average speed calculation | Use harmonic mean formula: 2S₁S₂/(S₁ + S₂) for equal distances | | Forgetting to convert km/hr to m/s when distance is in metres | Always check units; multiply by 5/18 to convert km/hr to m/s | | Using same-direction formula for opposite-direction problems | Opposite direction → add speeds; same direction → subtract speeds | | Treating outlet pipes as positive work | Outlets empty the tank, so their work rate is negative | | Ignoring that upstream speed < downstream speed | Remember: stream opposes upstream motion, assists downstream |

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Quick Reference

• **Combined work formula**: T = (a × b)/(a + b) days when A takes 'a' days and B takes 'b' days.

• **Distance = Speed × Time** — the fundamental relation; rearrange as needed.

• **Opposite direction = Add speeds; Same direction = Subtract speeds.**

• **Average speed ≠ arithmetic mean** — use 2S₁S₂/(S₁ + S₂) for equal distances.

• **Boat speed = (Downstream + Upstream)/2; Stream speed = (Downstream − Upstream)/2.**

• **1 km/hr = 5/18 m/s** — memorise this conversion factor.