🚀 Numerical Ability – NA07AFCAT Level★ High Priority
📌 AFCAT Focus 2022–2026: TSD gives 2–3 questions per paper — the highest in NA.
Tested sub-topics: train crossing (platform/pole/another train),
boats & streams (upstream/downstream), and relative speed.
Master the unit conversion (×5/18 for km/h→m/s) and the three train-crossing cases —
they account for over 60% of AFCAT TSD questions.
1. Speed, Distance & Time — Basics
Fig. 1.1 — The SDT Triangle: Derive Any Variable Instantly
TRAINS
2. Train Problems — Three Cases
Fig. 2.1 — Train Crossing: What Distance Does the Train Cover?
Same direction: Relative speed = |S₁ − S₂|. Use when overtaking, gap closing
Opposite direction: Relative speed = S₁ + S₂. Use for head-on meeting
Time to meet: = Distance / Relative speed
Circular track: Same dir: meet when faster covers 1 extra lap; Opp dir: meet when combined covers 1 full lap
🏁 Race Problems
"A beats B by x metres": When A finishes the full distance, B is still x m short of the finish line
"A beats B by t seconds": B takes t seconds longer than A to complete the race
"A gives B a start of x metres": B begins from x m ahead of the starting line; A starts from 0
"A gives B a start of t seconds": B starts t seconds before A; both finish race of same distance
Dead heat: Both finish together — same time, same position
Speed ratio = Distance ratio (when time is the same for both)
Winning margin: In a race of L m, if A beats B by x m: when A covers L, B covers (L−x) m
Circular track: Same dir: meet when relative gain = 1 lap (L/|S₁−S₂|). Opposite dir: meet every L/(S₁+S₂)
✎ Worked Example — Train Crossing Platform
A train 240 m long passes a platform 160 m long in 20 seconds. Find the speed of the train.
Distance = Train + Platform = 240 + 160 = 400 m
Speed = Distance / Time = 400 / 20 = 20 m/s
In km/h: 20 × 18/5 = 72 km/h
✔ Speed = 72 km/h
✎ Worked Example — Boats & Streams
A boat goes 30 km upstream in 3 hours and 30 km downstream in 2 hours. Find the speed of the stream.
Upstream speed U = 30/3 = 10 km/h
Downstream speed D = 30/2 = 15 km/h
Speed of stream = (D − U)/2 = (15 − 10)/2 = 2.5 km/h
✔ Stream speed = 2.5 km/h
✎ Worked Example — Race Problem
In a 100 m race, A beats B by 10 m and C by 15 m. By how much does B beat C in the same race?
When A runs 100 m → B runs 90 m, C runs 85 m.
So when B runs 90 m, C runs 85 m.
When B runs 100 m, C runs: 85 × 100/90 = 94.44 m
B beats C by: 100 − 94.44 = 5.56 m ≈ 5⁵⁄₉ m
✔ B beats C by 5⁵⁄₉ metres
📐 Formula Sheet — NA07
Core SDT
S = D/T D = S×T T = D/S km/h → m/s: ×5/18 m/s → km/h: ×18/5 Avg speed (equal dist): 2s₁s₂/(s₁+s₂)
Trains
Past pole: dist = L train Past platform: dist = L train + L platform Two trains: dist = L₁ + L₂ Opp dir: speed = S₁+S₂ | Same: |S₁−S₂|
Boats & Streams
Downstream: D = b + s Upstream: U = b − s Boat speed: b = (D+U)/2 Stream speed: s = (D−U)/2
Relative Speed
Same direction: |S₁ − S₂| Opposite direction: S₁ + S₂ Circular (same dir): meet every L/|S₁−S₂| sec Circular (opp dir): meet every L/(S₁+S₂) sec
Race Shortcuts
"A beats B by x m": B is x m short "A gives B start of x": B starts x m ahead Speed ratio = Distance ratio (same time) Time ratio = Distance/Speed
Quick Numbers
36 km/h = 10 m/s 54 km/h = 15 m/s 72 km/h = 20 m/s 90 km/h = 25 m/s
📝 Topic-Wise PYQs — NA07
Q1. A train 200 m long passes a pole in 10 seconds. What is the speed in km/h? AFCAT PYQ
(a) 60 km/h(b) 72 km/h(c) 80 km/h(d) 54 km/h
✔ Answer: (b) 72 km/h
Speed = 200/10 = 20 m/s = 20×18/5 = 72 km/h. Passing a pole → distance = train length only.
Q2. Two trains 120 m and 80 m long run in opposite directions at 50 and 40 km/h. Time to cross? AFCAT PYQ
(a) 6 sec(b) 8 sec(c) 9 sec(d) 10 sec
✔ Answer: (b) 8 sec
Relative speed = 50+40 = 90 km/h = 90×5/18 = 25 m/s. Distance = 120+80 = 200 m. Time = 200/25 = 8 sec.
Q3. Downstream speed = 18 km/h, upstream = 10 km/h. Find speed of boat in still water. AFCAT PYQ
(a) 14 km/h(b) 12 km/h(c) 4 km/h(d) 8 km/h
✔ Answer: (a) 14 km/h
Boat speed = (D+U)/2 = (18+10)/2 = 14 km/h. Stream = (18−10)/2 = 4 km/h.
Q4. A man covers equal distances at 30 km/h and 60 km/h. Average speed? ⚡ Tricky
(a) 45 km/h(b) 40 km/h(c) 42 km/h(d) 50 km/h
✔ Answer: (b) 40 km/h
Equal distance → harmonic mean: 2×30×60/(30+60) = 3600/90 = 40 km/h. (45 = arithmetic mean — wrong for equal distances!)
Q5. In a race of 1 km, A beats B by 100 m. B beats C by 100 m. By how much does A beat C? ⚡ Tricky
(a) 190 m(b) 200 m(c) 180 m(d) 210 m
✔ Answer: (a) 190 m
When A runs 1000 m, B runs 900 m. When B runs 1000 m, C runs 900 m. So when B runs 900 m, C runs 900×900/1000 = 810 m. A beats C by 1000−810 = 190 m.
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