Question 1

What is the basic formula for work-time problems?

Accepted Answer

Work = Rate × Time is the fundamental formula. If A completes work in 10 days, A's work rate = 1/10 per day (one-tenth of total work daily). For combined work (two people working together), add their individual rates: A+B together = 1/10 + 1/15 = (3+2)/30 = 1/6 per day, so they complete work in 6 days together. Key insight: work rates ADD when working together. Formula derivation: If A takes 'a' days and B takes 'b' days alone, together they take (a×b)/(a+b) days. Example: A does job in 12 days, B in 18 days. Together = (12×18)/(12+18) = 216/30 = 7.2 days. This tool visualizes these calculations step-by-step with animated characters showing work progress bars filling up over time, making abstract formulas concrete and understandable for exam preparation.

Question 2

How do I solve relative speed train crossing problems?

Accepted Answer

Relative speed depends on direction of motion: Same direction = |Speed₁ - Speed₂|, Opposite direction = Speed₁ + Speed₂. Train crossing problems have 4 types: Train crosses stationary object (pole, man): Time = Train length / Train speed. Train crosses platform/bridge: Time = (Train length + Platform length) / Train speed. Two trains cross each other (opposite direction): Time = (Train₁ length + Train₂ length) / (Speed₁ + Speed₂). Two trains cross (same direction): Time = (Train₁ length + Train₂ length) / |Speed₁ - Speed₂|. Example: 150m train at 60 km/h crosses 250m bridge. Convert speed: 60 km/h = 60×(5/18) = 16.67 m/s. Total distance = 150+250 = 400m. Time = 400/16.67 = 24 seconds. Pro tip: always convert km/h to m/s by multiplying by 5/18, or m/s to km/h by multiplying by 18/5. This tool's animated diagrams show trains moving toward/away from each other, helping visualize when to add vs subtract speeds—crucial for avoiding common exam mistakes.

Question 3

What's the trick for boats in stream problems?

Accepted Answer

Boat problems involve three speeds: boat's speed in still water (B), stream/current speed (S), downstream speed (D), upstream speed (U). Key relationships: Downstream = B + S (current helps, speeds add), Upstream = B - S (current opposes, subtract), Boat speed B = (D + U) / 2 (average of downstream and upstream), Stream speed S = (D - U) / 2 (half the difference). Example: Boat goes 30 km downstream in 2 hours, returns upstream in 3 hours. Downstream speed D = 30/2 = 15 km/h. Upstream speed U = 30/3 = 10 km/h. Boat speed B = (15+10)/2 = 12.5 km/h. Stream speed S = (15-10)/2 = 2.5 km/h. Verification: Downstream = 12.5+2.5 = 15✓, Upstream = 12.5-2.5 = 10✓. Common mistakes: forgetting to account for stream when calculating time (use downstream/upstream speed, NOT boat speed alone), mixing up D and U formulas. This tool has story mode featuring animated boats rowing with/against current, showing visually why downstream is faster—helps remember formulas intuitively rather than memorization.

Question 4

How do I handle problems with multiple workers joining/leaving?

Accepted Answer

Break complex work problems into time segments: calculate work done in each segment separately, then sum. Method: Step 1 - Find each person's work rate (rate = 1/days_to_complete). Step 2 - Identify time segments when different people work. Step 3 - Calculate work done per segment (work = rate × time). Step 4 - Sum all segments to get total work. Example: A completes job in 15 days, B in 20 days. A works alone for 5 days, then B joins. How many more days? A's rate = 1/15, B's rate = 1/20. Segment 1 (first 5 days, A alone): Work = 1/15 × 5 = 1/3. Remaining work = 1 - 1/3 = 2/3. Segment 2 (A+B together): Combined rate = 1/15 + 1/20 = (4+3)/60 = 7/60. Time = Work / Rate = (2/3) / (7/60) = (2/3) × (60/7) = 120/21 = 5.71 days. Total time = 5 + 5.71 = 10.71 days. Variations: workers leaving (calculate work until they leave, then remaining people continue), alternating days (A works day 1, B works day 2, repeat—calculate 2-day cycle work), efficiency changes (different rates at different times—segment by time period). This tool's calculator mode has segment-by-segment breakdown showing exact work completed each phase with progress bars for visual tracking.

Question 5

What are common shortcuts for time-distance calculations in exams?

Accepted Answer

Essential shortcuts saving 30-60 seconds per question: Speed conversion (memorize): 1 m/s = 18/5 km/h = 3.6 km/h, 1 km/h = 5/18 m/s. Quick method: multiply km/h by 5/18 to get m/s (60 km/h = 60×5/18 = 16.67 m/s). Average speed (harmless mean): If traveling distance D at speed S₁, returning same distance D at speed S₂, average speed ≠ (S₁+S₂)/2. Correct formula: Average = 2S₁S₂/(S₁+S₂). Example: Go at 40 km/h, return at 60 km/h. Average = 2×40×60/(40+60) = 4800/100 = 48 km/h (NOT 50). Meeting point: Two people start toward each other from distance D apart. Time to meet = D / (Speed₁ + Speed₂). They meet at point dividing distance in ratio Speed₁:Speed₂. Ratio method: If speeds are in ratio 3:4, distances covered in same time are also 3:4. Time ratio (inverse): If covering same distance, time ratio = inverse of speed ratio. Speed 3:4 means time 4:3. Percentage method: Speed increases 20% means time decreases by 20/120 = 1/6 = 16.67% (not 20%). Formula: Time change = -Speed change / (100 + Speed change). Mental math: 25% = 1/4, 33.33% = 1/3, 50% = 1/2, 66.67% = 2/3—use fractions for faster calculation. This tool's quiz mode highlights which shortcut applies to each problem, teaching pattern recognition for quick formula selection during timed exams.

🚄 Work, Time & Distance Calculator

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