Veda Skills Quantitative Aptitude Unit 11: Boats and Streams
- 2 topics
- 135 MCQs
- 8 short answers
- 2 long answers
- 27 flashcards
- 2 games
- ~100 min read
- English
Checked before publishing: 23 automated checks, including a plagiarism check against the textbook's own words. How we check our content
Upstream and Downstream
Turn the boat speed b and the stream speed s into the downstream speed b + s and the upstream speed b − s, go back from D and U to b and s, and use them for times, distances, round trips and time ratios.
What you will be able to do
- I can find the downstream and upstream speeds from b and s.
- I can find the speed in still water and the stream speed from D and U.
- I can find the time for each leg and for a round trip.
- I can use the time-ratio shortcut b : s = (k + 1) : (k − 1).
Key points
Still water means no current; b is the speed of the boat (or swimmer) there.
Downstream means with the current: D = b + s.
Upstream means against the current: U = b − s.
b is the average of D and U: b = (D + U) ÷ 2.
s is half the difference: s = (D − U) ÷ 2.
Time = distance ÷ the speed for that leg. Never use one speed for both legs.
If upstream takes k times as long as downstream over the same distance, b : s = (k + 1) : (k − 1).
The average speed of a round trip is (b² − s²) ÷ b, always less than b.
Full explanation
Raju's ferry at Varanasi
Raju rows tourists 2 km along the Ganga from Assi Ghat to Dashashwamedh Ghat. Going down the river takes him 20 minutes. Coming back up the same 2 km takes 30 minutes.
His arms work just as hard both ways. So how fast does Raju row, and how fast is the Ganga flowing? By the end of this topic you will answer that in one line.
The words used in exams
- Still water: water that does not flow, like a lake. The boat's speed there is $b$.
- Stream or current: the flowing river, with speed $s$.
- Downstream: moving with the current.
- Upstream: moving against the current.
Two speeds from one boat
With the current, the river carries the boat forward, so the speeds add. Against it, the river pulls back, so they subtract.
$$D = b + s \qquad U = b - s$$
Going back: b and s from D and U
Add the two equations: $D + U = 2b$. Subtract them: $D - U = 2s$. So
$$b = \frac{D + U}{2} \qquad s = \frac{D - U}{2}$$
Example: downstream 14 km/h, upstream 8 km/h. Then $b = 22 \div 2 = 11$ km/h and $s = 6 \div 2 = 3$ km/h.
Worked example: from distances and times
A boat goes 36 km downstream in 3 hours and 24 km upstream in 4 hours.
- $D = 36 \div 3 = 12$ km/h, $U = 24 \div 4 = 6$ km/h.
- $b = (12 + 6) \div 2 = 9$ km/h, $s = (12 - 6) \div 2 = 3$ km/h.
Trap: $(36 + 24) \div (3 + 4)$ is an average speed of the whole trip, not $b$.
Time for each leg
Each leg has its own speed, so find each time separately and then add.
Round trip: find the distance
A boat with $b = 10$ km/h goes to a village and back in 5 hours. The stream is 2 km/h. How far is the village?
Let the distance be $d$: $\frac{d}{12} + \frac{d}{8} = 5$, so $\frac{5d}{24} = 5$ and $d = 24$ km.
Shortcut: $d = \frac{T(b^2 - s^2)}{2b} = \frac{5 \times 96}{20} = 24$ km.
The time-ratio shortcut
If a man takes twice as long to row upstream as downstream over the same distance, his speeds are in the ratio $D : U = 2 : 1$. Then $b : s = (2 + 1) : (2 - 1) = 3 : 1$.
Average speed of a round trip
Going and coming back the same distance, the average speed is total distance ÷ total time:
$$\text{average} = \frac{2d}{\frac{d}{b+s} + \frac{d}{b-s}} = \frac{b^2 - s^2}{b}$$
With $b = 10$, $s = 2$: $96 \div 10 = 9.6$ km/h. It is less than $b$, because the slow leg lasts longer.
Units: minutes and metres
"1 km in 5 minutes" is $60 \div 5 = 12$ km/h. A speed in m/s becomes km/h on multiplying by $\frac{18}{5}$. Change everything to km and hours before you use the formulas.
Back to Raju
Downstream: 2 km in 20 min = 6 km/h. Upstream: 2 km in 30 min = 4 km/h.
So Raju rows at $b = (6 + 4) \div 2 = 5$ km/h and the Ganga flows at $s = (6 - 4) \div 2 = 1$ km/h. One line, as promised.
Concepts and terms
How fast the boat moves when the water is not flowing, written b. It is the boat's own effort.
Moving with the flow, the current adds its speed: D = b + s.
Moving against the flow, the current takes its speed away: U = b − s.
Add D and U and halve for b; subtract and halve for s. With D = 14 and U = 8: b = 11, s = 3.
Did you know
Fishermen on the Brahmaputra plan trips so that they go downstream when loaded and come back up empty, saving fuel and effort.
A log or a raft with no engine moves at exactly the speed of the stream, so it is a free speedometer for the river.
Memory tricks
Which sign? — Down adds, Up fights — Downstream the river pushes you (+ s); upstream it pushes back (− s).
b and s from D and U — Sum-half, difference-half — Half the sum is the boat, half the difference is the stream.
Practice MCQs with answers (76)
- A 9 km/h
- B 18 km/h
- C 15 km/h
- D 12 km/h
Show answer
Why:
Downstream the speeds add: 12 + 3 = 15 km/h.
- A 15 km/h
- B 10 km/h
- C 25 km/h
- D 20 km/h
Show answer
Why:
Downstream the speeds add: 15 + 5 = 20 km/h.
- A 10 km/h
- B 6 km/h
- C 12 km/h
- D 8 km/h
Show answer
Why:
Downstream the speeds add: 8 + 2 = 10 km/h.
- A 16 km/h
- B 24 km/h
- C 28 km/h
- D 20 km/h
Show answer
Why:
Downstream the speeds add: 20 + 4 = 24 km/h.
- A 13.5 km/h
- B 12 km/h
- C 15 km/h
- D 16.5 km/h
Show answer
Why:
Downstream the speeds add: 13.5 + 1.5 = 15 km/h.
- A 14 km/h
- B 18 km/h
- C 6 km/h
- D 10 km/h
Show answer
Why:
Upstream the current takes its speed away: 14 − 4 = 10 km/h.
- A 12 km/h
- B 18 km/h
- C 24 km/h
- D 6 km/h
Show answer
Why:
Upstream the current takes its speed away: 18 − 6 = 12 km/h.
- A 9 km/h
- B 6 km/h
- C 12 km/h
- D 3 km/h
Show answer
Why:
Upstream the current takes its speed away: 9 − 3 = 6 km/h.
- A 11 km/h
- B 25 km/h
- C 18 km/h
- D 32 km/h
Show answer
Why:
Upstream the current takes its speed away: 25 − 7 = 18 km/h.
- A 5.5 km/h
- B 10.5 km/h
- C 13 km/h
- D 8 km/h
Show answer
Why:
Upstream the current takes its speed away: 10.5 − 2.5 = 8 km/h.
- A 11 km/h
- B 6 km/h
- C 22 km/h
- D 3 km/h
Show answer
Why:
b = (D + U) ÷ 2 = (14 + 8) ÷ 2 = 11 km/h.
- A 4 km/h
- B 16 km/h
- C 8 km/h
- D 32 km/h
Show answer
Why:
b = (D + U) ÷ 2 = (20 + 12) ÷ 2 = 16 km/h.
- A 8 km/h
- B 28 km/h
- C 14 km/h
- D 4 km/h
Show answer
Why:
b = (D + U) ÷ 2 = (18 + 10) ÷ 2 = 14 km/h.
- A 10 km/h
- B 40 km/h
- C 5 km/h
- D 20 km/h
Show answer
Why:
b = (D + U) ÷ 2 = (25 + 15) ÷ 2 = 20 km/h.
- A 8 km/h
- B 3 km/h
- C 6 km/h
- D 16 km/h
Show answer
Why:
b = (D + U) ÷ 2 = (11 + 5) ÷ 2 = 8 km/h.
- A 2.5 km/h
- B 6.5 km/h
- C 5 km/h
- D 13 km/h
Show answer
Why:
b = (D + U) ÷ 2 = (9 + 4) ÷ 2 = 6.5 km/h.
- A 13 km/h
- B 4 km/h
- C 3 km/h
- D 6 km/h
Show answer
Why:
s = (D − U) ÷ 2 = (16 − 10) ÷ 2 = 3 km/h.
- A 8 km/h
- B 18 km/h
- C 5 km/h
- D 4 km/h
Show answer
Why:
s = (D − U) ÷ 2 = (22 − 14) ÷ 2 = 4 km/h.
- A 3 km/h
- B 6 km/h
- C 10 km/h
- D 4 km/h
Show answer
Why:
s = (D − U) ÷ 2 = (13 − 7) ÷ 2 = 3 km/h.
- A 5.5 km/h
- B 4.5 km/h
- C 9 km/h
- D 25.5 km/h
Show answer
Why:
s = (D − U) ÷ 2 = (30 − 21) ÷ 2 = 4.5 km/h.
- A 5 km/h
- B 15 km/h
- C 2.5 km/h
- D 3.5 km/h
Show answer
Why:
s = (D − U) ÷ 2 = (17.5 − 12.5) ÷ 2 = 2.5 km/h.
- A 3 km/h
- B 12 km/h
- C 60/7 km/h
- D 9 km/h
Show answer
Why:
D = 36 ÷ 3 = 12, U = 24 ÷ 4 = 6; b = (12 + 6) ÷ 2 = 9 km/h.
- A 2 km/h
- B 10 km/h
- C 3 km/h
- D 4 km/h
Show answer
Why:
D = 12, U = 8; s = (12 − 8) ÷ 2 = 2 km/h.
- A 14 km/h
- B 11 km/h
- C 10 km/h
- D 3 km/h
Show answer
Why:
D = 28 ÷ 2 = 14, U = 32 ÷ 4 = 8; b = (14 + 8) ÷ 2 = 11 km/h.
- A 8 km/h
- B 14 km/h
- C 4 km/h
- D 5 km/h
Show answer
Why:
D = 18, U = 10; s = (18 − 10) ÷ 2 = 4 km/h.
- A 9 km/h
- B 6.6 km/h
- C 2 km/h
- D 7 km/h
Show answer
Why:
D = 36 ÷ 4 = 9, U = 30 ÷ 6 = 5; b = (9 + 5) ÷ 2 = 7 km/h.
- A 4 km/h
- B 8 km/h
- C 16 km/h
- D 5 km/h
Show answer
Why:
D = 20, U = 12; s = (20 − 12) ÷ 2 = 4 km/h.
- A 2.5 km/h
- B 7.5 km/h
- C 10 km/h
- D 7 km/h
Show answer
Why:
D = 40 ÷ 4 = 10, U = 30 ÷ 6 = 5; b = (10 + 5) ÷ 2 = 7.5 km/h.
- A 13 km/h
- B 6 km/h
- C 5 km/h
- D 10 km/h
Show answer
Why:
D = 18, U = 8; s = (18 − 8) ÷ 2 = 5 km/h.
- A 4 hours
- B 4 h 30 min
- C 3 h 36 min
- D 3 hours
Show answer
Why:
Downstream speed 12 km/h, so time = 36 ÷ 12 = 3 hours.
- A 4 hours
- B 3 h 12 min
- C 5 hours
- D 2 h 40 min
Show answer
Why:
Upstream speed 12 km/h, so time = 48 ÷ 12 = 4 hours.
- A 6 hours
- B 8 hours
- C 10 hours
- D 7 h 30 min
Show answer
Why:
30 ÷ 10 + 30 ÷ 6 = 3 + 5 = 8 hours.
- A 5 hours
- B 3 h 20 min
- C 2 h 30 min
- D 3 h 30 min
Show answer
Why:
Downstream speed 16 km/h, so time = 40 ÷ 16 = 2 h 30 min.
- A 3 h 12 min
- B 5 hours
- C 4 h 30 min
- D 4 hours
Show answer
Why:
Upstream speed 8 km/h, so time = 32 ÷ 8 = 4 hours.
- A 7 hours
- B 6 hours
- C 8 hours
- D 4 hours
Show answer
Why:
48 ÷ 16 + 48 ÷ 12 = 3 + 4 = 7 hours.
- A 6 hours
- B 5 h 30 min
- C 3 hours
- D 5 hours
Show answer
Why:
30 ÷ 12 + 30 ÷ 10 = 2.5 + 3 = 5 h 30 min.
- A 30 km
- B 20 km
- C 24 km
- D 25 km
Show answer
Why:
d = T(b² − s²) ÷ 2b = 5 × 96 ÷ 20 = 24 km.
- A 40.5 km
- B 54 km
- C 27 km
- D 36 km
Show answer
Why:
d = T(b² − s²) ÷ 2b = 9 × 72 ÷ 18 = 36 km.
- A 30 km
- B 32 km
- C 40 km
- D 24 km
Show answer
Why:
d = T(b² − s²) ÷ 2b = 8 × 60 ÷ 16 = 30 km.
- A 30 km
- B 40 km
- C 45 km
- D 60 km
Show answer
Why:
d = T(b² − s²) ÷ 2b = 6 × 200 ÷ 30 = 40 km.
- A 35 km
- B 14 km
- C 20 km
- D 24.5 km
Show answer
Why:
d = T(b² − s²) ÷ 2b = 7 × 40 ÷ 14 = 20 km.
- A 1 km/h
- B 12 km/h
- C 6 km/h
- D 9 km/h
Show answer
Why:
b : s = (2 + 1) : (2 − 1) = 3 : 1, so b = 3 × 3 ÷ 1 = 9 km/h.
- A 8 km/h
- B 16 km/h
- C 2 km/h
- D 12 km/h
Show answer
Why:
b : s = (3 + 1) : (3 − 1) = 4 : 2, so b = 4 × 4 ÷ 2 = 8 km/h.
- A 20 km/h
- B 15 km/h
- C 10 km/h
- D 5/3 km/h
Show answer
Why:
b : s = (2 + 1) : (2 − 1) = 3 : 1, so b = 5 × 3 ÷ 1 = 15 km/h.
- A 4 : 1
- B 3 : 1
- C 2 : 1
- D 1 : 2
Show answer
Why:
(b + s) : (b − s) = 3 : 1, so b : s = (3 + 1) : (3 − 1) = 2 : 1.
- A 3 : 5
- B 5 : 1
- C 4 : 1
- D 5 : 3
Show answer
Why:
(b + s) : (b − s) = 4 : 1, so b : s = (4 + 1) : (4 − 1) = 5 : 3.
- A 3 km/h
- B 10 km/h
- C 12 km/h
- D 4/3 km/h
Show answer
Why:
b : s = (5 + 1) : (5 − 1) = 6 : 4, so b = 2 × 6 ÷ 4 = 3 km/h.
- A 4 km/h
- B 10 km/h
- C 3 km/h
- D 12 km/h
Show answer
Why:
(b + 2) × 4 = (b − 2) × 6 gives b = 2 × (6 + 4) ÷ (6 − 4) = 10 km/h.
- A 15 km/h
- B 6 km/h
- C 12 km/h
- D 5 km/h
Show answer
Why:
(b + 3) × 3 = (b − 3) × 5 gives b = 3 × (5 + 3) ÷ (5 − 3) = 12 km/h.
- A 50 km
- B 20 km
- C 16 km
- D 24 km
Show answer
Why:
b = 2 × 5 ÷ 1 = 10, so AB = (10 + 2) × 2 = 24 km.
- A 35 km
- B 25 km
- C 72 km
- D 30 km
Show answer
Why:
b = 1 × 12 ÷ 2 = 6, so AB = (6 + 1) × 5 = 35 km.
- A 10.4 km/h
- B 9.6 km/h
- C 10 km/h
- D 8 km/h
Show answer
Why:
Average = (b² − s²) ÷ b = (100 − 4) ÷ 10 = 9.6 km/h.
- A 6 km/h
- B 15 km/h
- C 9 km/h
- D 12 km/h
Show answer
Why:
Average = (b² − s²) ÷ b = (144 − 36) ÷ 12 = 9 km/h.
- A 20 km/h
- B 16 km/h
- C 20.8 km/h
- D 19.2 km/h
Show answer
Why:
Average = (b² − s²) ÷ b = (400 − 16) ÷ 20 = 19.2 km/h.
- A 12.5 km/h
- B 15 km/h
- C 5 km/h
- D 2.5 km/h
Show answer
Why:
D = 60 ÷ 4 = 15 km/h, U = 60 ÷ 6 = 10 km/h, so b = 12.5 km/h.
- A 2.5 km/h
- B 3 km/h
- C 6 km/h
- D 9 km/h
Show answer
Why:
D = 12 km/h, U = 6 km/h, so s = (12 − 6) ÷ 2 = 3 km/h.
- A 20 km/h
- B 12 km/h
- C 16 km/h
- D 4 km/h
Show answer
Why:
D = 60 ÷ 3 = 20 km/h, U = 60 ÷ 5 = 12 km/h, so b = 16 km/h.
- A 4/3 km/h
- B 10 km/h
- C 5 km/h
- D 8 km/h
Show answer
Why:
b − s = 60% of (b + s), so b × 40 = s × 160 and b = 2 × 160 ÷ 40 = 8 km/h.
- A 7 km/h
- B 4 km/h
- C 1/3 km/h
- D 8 km/h
Show answer
Why:
b − s = 75% of (b + s), so b × 25 = s × 175 and b = 1 × 175 ÷ 25 = 7 km/h.
- A 6 km/h
- B 9 km/h
- C 3 km/h
- D 12 km/h
Show answer
Why:
b − s = 50% of (b + s), so b × 50 = s × 150 and b = 3 × 150 ÷ 50 = 9 km/h.
- A a river in flood
- B the deepest part of a river
- C water with no current
- D water moving with the boat
Show answer
Why:
Still water does not flow, so only the boat's own speed b acts.
- A across the river
- B only in still water
- C against the current
- D in the direction of the current
Show answer
Why:
Downstream means with the flow of the river.
- A 2b
- B b + s
- C b
- D 2s
Show answer
Why:
(b + s) + (b − s) = 2b; the stream cancels.
- A move forward slowly
- B drift backwards
- C move at speed b
- D stay still
Show answer
Why:
b − s is negative, so the boat is carried back with the water.
- A zero
- B twice the stream speed
- C the speed of the stream
- D half the stream speed
Show answer
Why:
With b = 0, its downstream speed is 0 + s = s.
- A equal to s
- B equal to b
- C more than b
- D less than b
Show answer
Why:
(b² − s²) ÷ b = b − s²/b, which is below b whenever s > 0.
- A b = 15 km/h and s = 3 km/h
- B b = 15 km/h and s = 6 km/h
- C b = 30 km/h and s = 6 km/h
- D b = 12 km/h and s = 3 km/h
Show answer
Why:
b = 30 ÷ 2 = 15 and s = 6 ÷ 2 = 3. Forgetting to halve gives 6.
- A 6 km/h
- B 4 km/h
- C 15 km/h
- D 0.25 km/h
Show answer
Why:
1 km in a quarter hour is 4 km/h.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
(b + s) + (b − s) = 2b, so b is the average. R is exactly why.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
A is false: the average is (b² − s²) ÷ b, below b. R is true and is the reason the average drops.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
Downstream 12 km/h: 24 ÷ 12 = 2 h, true. Upstream 8 km/h: 24 ÷ 8 = 3 h, so 2 is false.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
(b + s) − (b − s) = 2s, so 1 is true. D ÷ U depends on b and s; for 12 and 8 it is 1.5. So 2 is false.
- A 7.5 km/h
- B 15 km/h
- C 9 km/h
- D 12 km/h
Show answer
Why:
The boat points against the flow, so U = 12 − 3 = 9 km/h.
- A 8 km/h
- B 26 km/h
- C 4 km/h
- D 13 km/h
Show answer
Why:
The faster arrow goes with the flow: D = 17, U = 9, so b = 26 ÷ 2 = 13 km/h.
- A 4 km/h
- B 2 km/h
- C 8 km/h
- D 13 km/h
Show answer
Why:
s = (17 − 9) ÷ 2 = 4 km/h.
- A 4 hours
- B 5 hours
- C 4 h 48 min
- D 6 hours
Show answer
Why:
P to Q is downstream: 36 ÷ 18 = 2 h. Q to P is upstream: 36 ÷ 12 = 3 h. Total 5 h.
Important short questions (4)
D = 15, U = 6, so b = 21 ÷ 2 = 10.5 km/h and s = 9 ÷ 2 = 4.5 km/h.
36 ÷ 12 + 36 ÷ 6 = 3 + 6 = 9 hours.
b : s = 4 : 2 = 2 : 1, so b = 4 km/h.
The boat spends longer on the slow upstream leg, so the slow speed counts for more of the time. The average is (b² − s²) ÷ b.
Long answer questions (1)
30/(15 + s) + 30/(15 − s) = 9/2. So 30 × 30 ÷ (225 − s²) = 9/2, giving 225 − s² = 200 and s² = 25, s = 5 km/h.
Check: 30 ÷ 20 = 1.5 h and 30 ÷ 10 = 3 h; total 4.5 h. Correct.
Flashcards (14)
Play this topic
- Quick river check
Mixed Boat Problems
Solve the exam's mixed boat questions: boats meeting, a floating hat, raft and boat timings, round trips with an unknown stream, extra time upstream, journeys with a midway stop, and a stream that changes speed.
What you will be able to do
- I can see when the stream cancels, as with boats moving towards each other.
- I can solve floating-object questions by thinking in the frame of the water.
- I can set up and solve round-trip and extra-time equations for b or s.
- I can handle journeys in several legs, such as A to B and back to a midpoint.
Key points
Two boats moving towards each other close the gap at b₁ + b₂; the stream cancels.
Two boats moving the same way on the same river separate at b₁ − b₂.
A dropped hat moves with the water, so a boat that turns back reaches it after the same time it spent moving away.
Round trip in time T: d/(b + s) + d/(b − s) = T, which simplifies to 2bd = T(b² − s²).
Extra time h upstream: d/(b − s) − d/(b + s) = h, which simplifies to 2sd = h(b² − s²).
If a raft takes T hours and a boat t hours downstream between two towns, the boat's upstream time is 1 ÷ (1/t − 2/T).
In a multi-leg trip, write the time of each leg with its own speed and add.
When the stream speeds up, downstream time falls and upstream time rises.
Full explanation
Anita's hat
Anita is rowing up the Narmada when the wind blows her hat into the water. She does not notice and keeps rowing for 15 minutes. Then she turns round to fetch it.
How long will she take to reach the hat? You might reach for b + s and b − s. There is a far quicker way, and this topic shows it, along with the other mixed questions exams love.
Think from the water
Imagine you sit on a raft. To you the water is still. A boat rows away from you at $b$ and comes back at $b$. The river only carries the whole picture along the bank; it does not change who reaches whom, or when.
Boats coming towards each other
A boat from A goes downstream at $b_1 + s$. A boat from B comes upstream at $b_2 - s$. The gap closes at
$$(b_1 + s) + (b_2 - s) = b_1 + b_2$$
So two towns 60 km apart, with boats of 12 and 8 km/h, meet after $60 \div 20 = 3$ hours, whatever the stream.
The floating hat
In the water frame the hat stays still and Anita rows away at $b$ for 15 minutes, then back at $b$. So she needs 15 minutes to return.
The hat floated for $15 + 15 = 30$ minutes. If the Narmada flows at 4 km/h, she picks it up $4 \times \frac{1}{2} = 2$ km downstream of where it fell.
Raft and boat
A raft drifts from A to B in 12 hours. A boat goes from A to B in 3 hours. How long does the boat take from B to A?
Per hour: stream $= \frac{1}{12}$ of AB, downstream $= \frac{1}{3}$, so $b = \frac{1}{3} - \frac{1}{12} = \frac{1}{4}$ and upstream $= \frac{1}{4} - \frac{1}{12} = \frac{1}{6}$. The boat takes 6 hours back.
Round trip with an unknown stream
A boat with $b = 10$ km/h goes 21 km downstream and back in 5 hours. Find $s$.
$2bd = T(b^2 - s^2)$: $2 \times 10 \times 21 = 5(100 - s^2)$, so $100 - s^2 = 84$, $s^2 = 16$ and $s = 4$ km/h.
Check: $21 \div 14 + 21 \div 6 = 1.5 + 3.5 = 5$. Correct.
"It takes h hours more upstream"
A boat with $b = 12$ km/h takes 2 hours more to go 45 km upstream than downstream. Then
$$\frac{45}{12 - s} - \frac{45}{12 + s} = 2 \Rightarrow 90s = 2(144 - s^2)$$
so $s^2 + 45s - 144 = 0$, $(s + 48)(s - 3) = 0$ and $s = 3$ km/h. In the exam, try the options: $45 \div 9 - 45 \div 15 = 5 - 3 = 2$.
Journeys with a stop in the middle
A boat goes from A down to B, 48 km, and then back up to C, the midpoint of AB. Treat each leg separately.
When the stream changes
A boat with $b = 12$ km/h goes 40 km upstream. Normally $s = 2$, so it takes $40 \div 10 = 4$ h. After rain $s = 4$, so it takes $40 \div 8 = 5$ h, 1 hour more. Find each time with its own stream speed, then compare.
Back to Anita
She rowed away from the hat for 15 minutes in the water frame, so she reaches it 15 minutes after turning. Her own speed and the river's speed never entered the answer. That is the power of choosing the right frame.
Concepts and terms
One boat gains s and the other loses s, so the gap closes at b₁ + b₂ whatever the stream.
Seen from a raft, the water is still and the boat moves at b both ways. Floating things never move in this frame.
d/(b + s) + d/(b − s) = T becomes 2bd = T(b² − s²). Given three of b, s, d, T you can find the fourth.
If a raft needs T hours for a stretch, the stream covers 1/T of it each hour. Boats then work like pipes in a tank.
Did you know
The hat puzzle is often credited to Einstein's way of thinking: choose the frame in which the problem is easiest.
Memory tricks
Meeting boats — Face to face, forget the flow — Towards each other the stream cancels; use b₁ + b₂.
Floating hat — Away ten, back ten — If you noticed after 10 minutes, you get the hat back 10 minutes after turning.
Practice MCQs with answers (59)
- A 3 hours
- B 15 hours
- C 3 h 30 min
- D 6 hours
Show answer
Why:
Closing speed = (12 + 3) + (8 − 3) = 20 km/h; time = 60 ÷ 20 = 3 hours.
- A 9 hours
- B 3 hours
- C 3 h 30 min
- D 6 hours
Show answer
Why:
Closing speed = (20 + 4) + (10 − 4) = 30 km/h; time = 90 ÷ 30 = 3 hours.
- A 15 hours
- B 3 h 30 min
- C 3 hours
- D 6 hours
Show answer
Why:
Closing speed = (9 + 2) + (6 − 2) = 15 km/h; time = 45 ÷ 15 = 3 hours.
- A 3 h 30 min
- B 6 hours
- C 18 hours
- D 3 hours
Show answer
Why:
Closing speed = (14 + 5) + (10 − 5) = 24 km/h; time = 72 ÷ 24 = 3 hours.
- A 45 km
- B 15 km
- C 36 km
- D 27 km
Show answer
Why:
They meet after 60 ÷ 20 = 3 h. The upper boat goes at 15 km/h, so it covers 15 × 3 = 45 km.
- A 18 km
- B 72 km
- C 60 km
- D 48 km
Show answer
Why:
They meet after 90 ÷ 30 = 3 h. The upper boat goes at 24 km/h, so it covers 24 × 3 = 72 km.
- A 30 km
- B 24 km
- C 36 km
- D 12 km
Show answer
Why:
They meet after 48 ÷ 16 = 3 h. The upper boat goes at 12 km/h, so it covers 12 × 3 = 36 km.
- A 30 min
- B 7.5 min
- C It depends on the speed of the boat
- D 15 min
Show answer
Why:
Relative to the water the ball is still and the boat moves at b both ways, so returning takes the same 15 minutes.
- A 2 km
- B 4 km
- C 3 km
- D 1 km
Show answer
Why:
The ball floats for 20 + 20 = 40 minutes at 3 km/h: 3 × 40/60 = 2 km.
- A 4 km
- B 3 km
- C 1.5 km
- D 6 km
Show answer
Why:
The ball floats for 45 + 45 = 90 minutes at 2 km/h: 2 × 90/60 = 3 km.
- A 3 km
- B 1 km
- C 2 km
- D 4 km
Show answer
Why:
The ball floats for 10 + 10 = 20 minutes at 6 km/h: 6 × 20/60 = 2 km.
- A 7 hours
- B 4 hours
- C 9 hours
- D 6 hours
Show answer
Why:
Per hour: stream = 1/12, downstream = 1/3, so upstream = 1/3 − 2/12 = 1/6 of the stretch. Time = 6 hours.
- A 6 hours
- B 8 hours
- C 4 h 48 min
- D 20 hours
Show answer
Why:
Per hour: stream = 1/24, downstream = 1/4, so upstream = 1/4 − 2/24 = 1/6 of the stretch. Time = 6 hours.
- A 8 hours
- B 3 h 20 min
- C 4 hours
- D 2 h 30 min
Show answer
Why:
Per hour: stream = 1/10, downstream = 1/2, so upstream = 1/2 − 2/10 = 0.3 of the stretch. Time = 3 h 20 min.
- A 16 hours
- B 8 hours
- C 6 h 40 min
- D 5 hours
Show answer
Why:
Per hour: stream = 1/20, downstream = 1/4, so upstream = 1/4 − 2/20 = 0.15 of the stretch. Time = 6 h 40 min.
- A 5 km/h
- B 3 km/h
- C 6 km/h
- D 4 km/h
Show answer
Why:
2bd = T(b² − s²): 420 = 5(100 − s²), so s² = 16 and s = 4 km/h.
- A 2 km/h
- B 4 km/h
- C 3 km/h
- D 1 km/h
Show answer
Why:
2bd = T(b² − s²): 192 = 6(36 − s²), so s² = 4 and s = 2 km/h.
- A 6 km/h
- B 4 km/h
- C 5 km/h
- D 3 km/h
Show answer
Why:
2bd = T(b² − s²): 768 = 6(144 − s²), so s² = 16 and s = 4 km/h.
- A 2 km/h
- B 3 km/h
- C 1 km/h
- D 0.5 km/h
Show answer
Why:
2bd = T(b² − s²): 120 = 5(25 − s²), so s² = 1 and s = 1 km/h.
- A 12 km/h
- B 11 km/h
- C 9 km/h
- D 10 km/h
Show answer
Why:
Try b = 10: 48 ÷ 12 + 48 ÷ 8 = 4 + 6 = 10 hours. It fits.
- A 9 km/h
- B 10 km/h
- C 8 km/h
- D 11 km/h
Show answer
Why:
Try b = 9: 24 ÷ 12 + 24 ÷ 6 = 2 + 4 = 6 hours. It fits.
- A 9 km/h
- B 8 km/h
- C 7 km/h
- D 10 km/h
Show answer
Why:
Try b = 8: 15 ÷ 10 + 15 ÷ 6 = 1.5 + 2.5 = 4 hours. It fits.
- A 6 hours
- B 9 hours
- C 8 hours
- D 12 hours
Show answer
Why:
A to B: 48 ÷ 12 = 4 h. B to C: 24 ÷ 6 = 4 h. Total 8 hours.
- A 3 hours
- B 5 hours
- C 6 hours
- D 4 hours
Show answer
Why:
A to B: 40 ÷ 20 = 2 h. B to C: 20 ÷ 10 = 2 h. Total 4 hours.
- A 36 km
- B 54 km
- C 18 km
- D 53.2 km
Show answer
Why:
d ÷ 18 + (d/2) ÷ 10 = 3.8 gives d = 36 km.
- A 30 km
- B 60 km
- C 88 km
- D 72 km
Show answer
Why:
d ÷ 12 + (d/2) ÷ 10 = 8 gives d = 60 km.
- A 2 hours
- B 1 h 30 min
- C 1 hour
- D 30 min
Show answer
Why:
Before: 40 ÷ 10 = 4 h. After: 40 ÷ 8 = 5 h. Extra 1 hour.
- A 3 hours
- B 4 hours
- C 2 h 30 min
- D 2 hours
Show answer
Why:
Before: 96 ÷ 16 = 6 h. After: 96 ÷ 12 = 8 h. Extra 2 hours.
- A 1 hour
- B 32 min
- C 2 hours
- D 20 min
Show answer
Why:
Before: 60 ÷ 12 = 5 h. After: 60 ÷ 10 = 6 h. Extra 1 hour.
- A 18 km
- B 24 km
- C 30 km
- D 32 km
Show answer
Why:
D = 36 ÷ 3 = 12, so U = 12 − 2 × 2 = 8 km/h, and 8 × 3 = 24 km.
- A 16 km
- B 24 km
- C 22 km
- D 28 km
Show answer
Why:
D = 34 ÷ 2 = 17, so U = 17 − 2 × 3 = 11 km/h, and 11 × 2 = 22 km.
- A 39 km
- B 24 km
- C 36 km
- D 30 km
Show answer
Why:
D = 42 ÷ 4 = 10.5, so U = 10.5 − 2 × 1.5 = 7.5 km/h, and 7.5 × 4 = 30 km.
- A 4 km/h
- B 11 km/h
- C 5 km/h
- D 6 km/h
Show answer
Why:
(b + s)1 = (b − s)2 gives s = b(2 − 1) ÷ (2 + 1) = 4 km/h.
- A 5 km/h
- B 4 km/h
- C 6.4 km/h
- D 8 km/h
Show answer
Why:
(b + s)3 = (b − s)5 gives s = b(5 − 3) ÷ (5 + 3) = 4 km/h.
- A 12 km/h
- B 9 km/h
- C 6 km/h
- D 7 km/h
Show answer
Why:
(b + s)2 = (b − s)4 gives s = b(4 − 2) ÷ (4 + 2) = 6 km/h.
- A 8
- B 5
- C 3
- D 4
Show answer
Why:
One round trip = 16 ÷ 16 + 16 ÷ 8 = 3 h, so 12 ÷ 3 gives 4 complete trips.
- A 5
- B 4
- C 10
- D 6
Show answer
Why:
One round trip = 36 ÷ 18 + 36 ÷ 12 = 5 h, so 26 ÷ 5 gives 5 complete trips.
- A 1 km/h
- B 2 km/h
- C 4 km/h
- D 3 km/h
Show answer
Why:
Check s = 2: 24 ÷ 8 − 24 ÷ 12 = 3 − 2 = 1 h. It fits 2sd = h(b² − s²).
- A 1 km/h
- B 4 km/h
- C 2 km/h
- D 3 km/h
Show answer
Why:
Check s = 2: 15 ÷ 6 − 15 ÷ 10 = 2.5 − 1.5 = 1 h. It fits 2sd = h(b² − s²).
- A 5 km/h
- B 4 km/h
- C 2 km/h
- D 3 km/h
Show answer
Why:
Check s = 3: 45 ÷ 9 − 45 ÷ 15 = 5 − 3 = 2 h. It fits 2sd = h(b² − s²).
- A 5 km/h
- B 6 km/h
- C 4 km/h
- D 7 km/h
Show answer
Why:
Check s = 5: 40 ÷ 10 − 40 ÷ 20 = 4 − 2 = 2 h. It fits 2sd = h(b² − s²).
- A 4 km/h
- B 2 km/h
- C 4.8 km/h
- D 3 km/h
Show answer
Why:
D = 5k, U = 3k, b = 4k = 8, so k = 2 and s = 1k = 2 km/h.
- A 8 km/h
- B 30/7 km/h
- C 4 km/h
- D 2.5 km/h
Show answer
Why:
D = 7k, U = 3k, b = 5k = 10, so k = 2 and s = 2k = 4 km/h.
- A 10 km/h
- B 9 km/h
- C 6 km/h
- D 3 km/h
Show answer
Why:
D = 3k, U = 2k, b = 2.5k = 15, so k = 6 and s = 0.5k = 3 km/h.
- A 12 km
- B 21 km
- C 72 km
- D 30 km
Show answer
Why:
Both get +3, so the gap grows at 14 − 10 = 4 km/h: 4 × 3 = 12 km.
- A 20 km
- B 12 km
- C 60 km
- D 28 km
Show answer
Why:
Both get +4, so the gap grows at 18 − 12 = 6 km/h: 6 × 2 = 12 km.
- A only the stream speed
- B the stream and the boats equally
- C only their still-water speeds
- D which boat is upstream
Show answer
Why:
(b₁ + s) + (b₂ − s) = b₁ + b₂.
- A it depends on the current
- B 3 minutes
- C 12 minutes
- D 6 minutes
Show answer
Why:
Direction does not matter: in the water frame the bottle is still and he moves at b both ways.
- A increases
- B stays the same
- C becomes zero
- D decreases
Show answer
Why:
Round-trip time = 2bd ÷ (b² − s²); a larger s makes the bottom smaller and the time larger.
- A 8 km/h
- B 4 km/h
- C −4 km/h
- D 16 km/h
Show answer
Why:
Speeds are positive, so take s = 4 and drop −4.
- A all of AB, with the current
- B all of AB, against the current
- C half of AB, against the current
- D half of AB, with the current
Show answer
Why:
B to C is half the stretch and goes back up the river.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
Closing speed is b₁ + b₂ either way, and R gives the reason.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
A is true (water frame). R is false: b + s and b − s differ. Equality holds only relative to the water.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
In both cases the stream cancels: b₁ − b₂ and b₁ + b₂.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
45 ÷ 9 − 45 ÷ 15 = 5 − 3 = 2, true. With s = b, upstream speed is 0, true.
- A 10 hours
- B 7 h 12 min
- C 6 hours
- D 7 hours
Show answer
Why:
A to B: 48 ÷ 12 = 4 h. B to C: 24 ÷ 8 = 3 h. Total 7 hours.
- A 3 hours
- B 13 h 30 min
- C 2 h 15 min
- D 2 h 42 min
Show answer
Why:
Closing speed = 11 + 7 = 18 km/h (the 2 km/h cancels), so 54 ÷ 18 = 3 h.
- A 15 km
- B 39 km
- C 33 km
- D 27 km
Show answer
Why:
They meet after 3 h. X moves at 11 + 2 = 13 km/h, so it covers 13 × 3 = 39 km.
- A 3 km
- B 1.25 km
- C 2.5 km
- D 5 km
Show answer
Why:
Back also takes 25 min, so the cap floats 50 min at 3 km/h: 3 × 50/60 = 2.5 km.
Important short questions (4)
Closing speed 16 + 12 = 28 km/h, so 84 ÷ 28 = 3 hours.
Upstream rate = 1/3 − 2/15 = 3/15 = 1/5, so 5 hours.
Relative to the water the hat is at rest and the boat moves at b away and b back. Equal speeds over equal distances give equal times.
d = T(b² − s²) ÷ 2b = 6 × 32 ÷ 12 = 16 km.
Long answer questions (1)
(a) 45/(12 − s) − 45/(12 + s) = 2.
(b) 90s = 2(144 − s²), so s² + 45s − 144 = 0, (s + 48)(s − 3) = 0, s = 3 km/h.
(c) 45 ÷ 15 + 45 ÷ 9 = 3 + 5 = 8 hours.
Flashcards (13)
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Veda Skills Quantitative Aptitude Unit 11 — frequently asked questions
D = 15, U = 6, so b = 21 ÷ 2 = 10.5 km/h and s = 9 ÷ 2 = 4.5 km/h.
36 ÷ 12 + 36 ÷ 6 = 3 + 6 = 9 hours.
b : s = 4 : 2 = 2 : 1, so b = 4 km/h.
Closing speed 16 + 12 = 28 km/h, so 84 ÷ 28 = 3 hours.
Upstream rate = 1/3 − 2/15 = 3/15 = 1/5, so 5 hours.
Relative to the water the hat is at rest and the boat moves at b away and b back. Equal speeds over equal distances give equal times.
All Veda Skills Quantitative Aptitude units
All Quantitative Aptitude units · All Skills & Exams courses
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