Revision Notes – Time, Work & Distance
(Tailored for JKSSB Social Forestry Worker – Basic Mathematics)
1. Core Idea
All three topics share a single proportional relationship:
\[
\text{Quantity} = \text{Rate} \times \text{Time}
\]
- Distance – Rate = Speed, Quantity = Distance.
- Work – Rate = Work‑done per unit time (efficiency), Quantity = Total work (usually taken as 1 unit).
- Time – The common variable that links the two rates.
Understanding this equivalence lets you translate any problem into the same algebraic form and solve it quickly.
2. Fundamental Formulas
| Concept | Symbol | Formula | Typical Units | ||
|---|---|---|---|---|---|
| Speed | \(S\) | \(S = \dfrac{D}{T}\) | km/h, m/s | ||
| Distance | \(D\) | \(D = S \times T\) | km, m | ||
| Time | \(T\) | \(T = \dfrac{D}{S}\) | h, min, s | ||
| Work Rate (Efficiency) | \(E\) | \(E = \dfrac{W}{T}\) | work / h (often “unit work”) | ||
| Total Work | \(W\) | \(W = E \times T\) | 1 (whole job) or any given amount | ||
| Combined Efficiency | \(E_{comb}\) | \(E_{comb}=E_1+E_2+\dots\) (if working together) | work / h | ||
| Time for Combined Work | \(T_{comb}\) | \(T_{comb}= \dfrac{W}{E_{comb}}\) | h | ||
| Relative Speed (Same Direction) | \(S_{rel}\) | \(S_{rel}= | S_1-S_2 | \) | km/h |
| Relative Speed (Opposite Direction) | \(S_{rel}\) | \(S_{rel}=S_1+S_2\) | km/h | ||
| Average Speed (Equal Distances) | \(S_{avg}\) | \(S_{avg}= \dfrac{2S_1S_2}{S_1+S_2}\) | km/h | ||
| Average Speed (Equal Times) | \(S_{avg}\) | \(S_{avg}= \dfrac{S_1+S_2}{2}\) | km/h |
Mnemonic: “D = S × T” → Distance Speed Time – think of a road sign that says “DST”.
For work, replace S with E (Efficiency) and D with W (Work): W = E × T → “WET”.
3. Work & Efficiency – Key Points
- Unit Work Convention: Treat the whole job as 1 unit unless otherwise stated.
- Efficiency (E): If a person can finish a job in x days, his daily efficiency = \(1/x\).
- Inverse Relation: More workers ⇒ less time (provided they work at same efficiency).
- Formula Shortcut:
\[
\text{Time taken by } n \text{ workers} = \frac{\text{Time taken by 1 worker}}{n}
\]
(Only when all have identical efficiency.)
- Mixed Efficiencies:
- Compute each person’s daily work = \(1/(\text{individual time})\).
- Add them → combined daily work.
- Required days = \(1 / (\text{combined daily work})\).
- Example Trick: If A can do a job in 6 h and B in 9 h, combined time =
\[
T = \frac{1}{\frac{1}{6}+\frac{1}{9}} = \frac{1}{\frac{5}{18}} = \frac{18}{5}=3.6\text{ h}=3\text{ h }36\text{ min}
\]
4. Speed, Distance & Time – Core Techniques
4.1 Uniform Motion
- Direct proportion: Distance ∝ Speed (when time fixed).
- Inverse proportion: Time ∝ 1/Speed (when distance fixed).
4.2 Problems with Two Segments
| Situation | Formula to Use |
|---|---|
| Different speeds for equal distances | Use harmonic mean: \(S_{avg}= \frac{2S_1S_2}{S_1+S_2}\) |
| Different speeds for equal times | Use arithmetic mean: \(S_{avg}= \frac{S_1+S_2}{2}\) |
| One part known, other unknown (total distance/time given) | Set up equations: \(d_1 = s_1 t_1\), \(d_2 = s_2 t_2\); use \(d_1+d_2 = D\) or \(t_1+t_2 = T\). |
4.3 Relative Speed (Moving Objects)
- Same direction: Subtract speeds.
- Opposite direction: Add speeds.
- Meeting time: \(t = \dfrac{\text{initial separation}}{\text{relative speed}}\).
- Overtaking time (same direction): Same formula, using the faster minus slower speed.
4.4 Boats & Streams (Optional but often asked)
- Upstream speed: \(S_u = S_b – S_c\)
- Downstream speed: \(S_d = S_b + S_c\) – Where \(S_b\) = speed of boat in still water, \(S_c\) = speed of current.
- Shortcut:
\[
S_b = \frac{S_d+S_u}{2},\qquad S_c = \frac{S_d-S_u}{2}
\]
5. Mnemonics & Memory Aids
| Topic | Mnemonic | Explanation |
|---|---|---|
| D‑S‑T | “Driving Slowly Takes Time” | Helps recall \(D = S \times T\). |
| Work | “Work Equals Time” (WET) | Work = Efficiency × Time. |
| Average Speed (Equal Distance) | “Harmonic Mean for Distance” | Use harmonic mean. |
| Average Speed (Equal Time) | “Arithmetic Mean for Time” | Use arithmetic mean. |
| Relative Speed (Opposite) | “Opposite Add” | Add speeds. |
| Relative Speed (Same) | “Same Subtract” | Subtract speeds. |
| Boat Stream | “Up Slow, Down Fast” | Upstream slower, downstream faster. |
| Combined Work | “Add Efficiencies, Time D**ivides” | \(T_{comb}=W/(E_1+E_2+…)\). |
6. Problem‑Solving Strategy (Step‑by‑Step)
- Identify the Quantity – Is it distance, work, or time?
- Write the Basic Equation – \(Q = R \times T\).
- Extract Given Data – Convert all units to a common system (km/h ↔ m/s, days ↔ hours).
- Set Up Unknowns – Usually one variable (speed, efficiency, or time).
- Form Equation(s) – Use relative speed, average speed, or combined efficiency as needed.
- Solve Algebraically – Keep fractions; avoid premature decimal conversion.
- Check Units & Reasonableness – Does the answer make sense? (e.g., a person cannot finish a job in negative time).
- Answer in Required Format – Often the exam asks for hours/minutes or days; convert back if needed.
7. Worked Examples (Illustrative)
Example 1 – Pure Work
A can finish a trench in 12 days. B can finish the same trench in 8 days. If they work together, how long will it take?
- A’s daily work = \(1/12\).
- B’s daily work = \(1/8\).
- Combined daily work = \(1/12 + 1/8 = (2+3)/24 = 5/24\).
- Days required = \(1 ÷ (5/24) = 24/5 = 4.8\) days = 4 days 19 h 12 min.
Example 2 – Speed & Distance (Two‑Part Journey)
A train travels 150 km at 50 km/h and the next 150 km at 75 km/h. Find average speed for the whole trip.
- Equal distances → harmonic mean:
\[
S_{avg}= \frac{2 \times 50 \times 75}{50+75}= \frac{7500}{125}=60\text{ km/h}.
\]
Example 3 – Relative Speed (Opposite Direction)
Two cars start from points A and B, 300 km apart, and move towards each other at 40 km/h and 60 km/h. When will they meet?
- Relative speed = \(40+60=100\) km/h.
- Time = \(300/100 = 3\) h.
Example 4 – Boat & Stream
A boat goes 24 km upstream in 6 hours and the same distance downstream in 4 hours. Find speed of boat in still water and speed of current.
- Upstream speed = \(24/6 = 4\) km/h.
- Downstream speed = \(24/4 = 6\) km/h. – Boat speed \(S_b = (4+6)/2 = 5\) km/h.
- Current speed \(S_c = (6-4)/2 = 1\) km/h.
Example 5 – Mixed Work (Different Efficiencies)
Three workers P, Q, R can complete a task in 10, 15, and 30 days respectively. How long will they take if they work together?
- Daily work: \(P=1/10\), \(Q=1/15\), \(R=1/30\).
- Sum = \( \frac{3+2+1}{30}= \frac{6}{30}= \frac{1}{5}\).
- Time = \(1 ÷ (1/5) = 5\) days.
8. Quick Reference Tables
8.1 Unit Conversions (Frequently Needed)
| From | To | Factor |
|---|---|---|
| km/h | m/s | Multiply by \( \frac{5}{18}\) |
| m/s | km/h | Multiply by \( \frac{18}{5}\) |
| days | hours | × 24 |
| hours | minutes | × 60 |
| minutes | seconds | × 60 |
| km | metres | × 1000 |
| metres | km | ÷ 1000 |
8.2 Common Fractions for Work Problems
| Individual Time (days) | Daily Work (Fraction) |
|---|---|
| 2 | 1/2 |
| 3 | 1/3 |
| 4 | 1/4 |
| 5 | 1/5 |
| 6 | 1/6 |
| 8 | 1/8 |
| 10 | 1/10 |
| 12 | 1/12 |
| 15 | 1/15 |
| 20 | 1/20 |
| 24 | 1/24 |
| 30 | 1/30 |
Use these to add efficiencies quickly.
8.3 Average Speed Formulas at a Glance | Condition | Formula |
| ———– | ——— |
| Equal distances | \( \displaystyle S_{avg}= \frac{n}{\sum_{i=1}^{n}\frac{1}{S_i}} \) (harmonic mean) |
| Equal times | \( \displaystyle S_{avg}= \frac{\sum_{i=1}^{n} S_i}{n} \) (arithmetic mean) |
| Two speeds, different distances \(d_1,d_2\) | \( S_{avg}= \frac{d_1+d_2}{\frac{d_1}{S_1}+\frac{d_2}{S_2}} \) |
| Two speeds, different times \(t_1,t_2\) | \( S_{avg}= \frac{S_1 t_1+S_2 t_2}{t_1+t_2} \) |
9. Typical Exam‑Style Questions & How to Crack Them
| Question Type | Key Insight | Shortcut |
|---|---|---|
| “If A does a job in x days and B in y days, how long together?” | Add efficiencies. | \(T = \frac{xy}{x+y}\) (derived from \(1/x+1/y = (x+y)/(xy)\)). |
| “A works for a few days then leaves; B finishes. Find total days.” | Work done by A + work done by B = 1. | Compute A’s contribution = (days A worked)/x, then solve for B’s days. |
| “Two trains start from stations … meet after … find speed of one.” | Use relative speed and distance = speed × time. | Distance covered until meeting = relative speed × time; subtract known distance to get unknown speed. |
| “A person walks part of the way and cycles the rest; total time given. Find walking speed.” | Set up two time equations: \(t_w = d_w/S_w\), \(t_c = d_c/S_c\). | Use total distance and total time to solve simultaneous equations (often one variable cancels). |
| “Boat goes upstream … downstream … find speed of stream.” | Use upstream/downstream formulas. | \(S_c = (S_d – S_u)/2\). |
| “Average speed for a round trip with different speeds each way.” | Harmonic mean (equal distances). | \(S_{avg}= \frac{2S_1S_2}{S_1+S_2}\). |
10. Revision Checklist (Before the Exam)
- [ ] Know the three core equations: D = S × T, W = E × T, T = D / S (and the work version).
- [ ] Be fluent in unit conversions (especially km/h ↔ m/s). – [ ] Memorise the two average‑speed formulas (harmonic for equal distance, arithmetic for equal time).
- [ ] Practice adding fractions for work problems – use the “individual time → daily work” table.
- [ ] Visualise relative speed: same direction → subtract, opposite → add.
- [ ] For boat‑stream questions, write down upstream and downstream speeds first, then solve for boat speed and current.
- [ ] Check that your answer is in the units asked (hours/minutes, days, km/h, etc.).
- [ ] If time permits, do a quick sanity check: Does a higher speed give less time? Does more workers reduce days? —
11. Final Thought All problems in Time, Work & Distance boil down to a simple proportionality: output = rate × time. By converting every scenario into this template, you eliminate confusion and can solve even the trickiest mixed questions with confidence. Keep the mnemonics handy, practice a few sets each day, and you’ll walk into the exam hall ready to score full marks on this topic.
—
Good luck, and revise smart!
