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📝 Round trip distance rate time formula (12 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 12 questions available

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What is Round trip distance rate time formula?

Definition:
Round trip problems involve traveling to a destination and back, often at different speeds, with the total distance being twice the one-way distance, and the total time being the sum of the times for each leg, using $D = rt$ with the relationships $D_{\text{out}} = D_{\text{back}}$ and $t_{\text{total}} = t_{\text{out}} + t_{\text{back}}$ , and these problems often require solving for average speed or individual rates.

Working:
To solve round trip problems, set the distance out equal to the distance back, so $r_{\text{out}} \times t_{\text{out}} = r_{\text{back}} \times t_{\text{back}}$ , and use the total time equation, then solve the system of equations; if the average speed is needed, it is $\text{Average Speed} = \frac{2D}{t_{\text{total}}}$ , and note that average speed is not the arithmetic mean of the two speeds unless times are equal.

Example:
A plane flies to a city at 400 mph and returns at 500 mph, with a total flight time of 9 hours; let $t_{\text{out}} = t$ , then $t_{\text{back}} = 9 - t$ , and distances equal: $400t = 500(9 - t)$ , solving gives $400t = 4500 - 500t$ , $900t = 4500$ , $t = 5$ hours out, and 4 hours back, so the distance is $400 \times 5 = 2000$ miles.

Reason:
Round trip problems are common in travel planning, logistics, and transportation, helping calculate total time, fuel consumption, and average speed, and they enhance understanding of the relationships between distance, rate, and time in varying conditions.

Easy

Medium

Hard

📝 All Round trip distance rate time formula MCQs

Q1. A boat travels on a river where the current moves at 33 km/h. What is the boat's effective speed relative to the shore when its still-water speed is 1212 km/h and it travels upstream?

A.15 km/h

B.12 km/h

C.9 km/h ✅

D.3 km/h

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: When a boat travels upstream, the river current opposes its motion, so the effective speed is found by subtracting the current speed from the still-water speed. Thus $12-3=9$ km/h.

Q2. A cyclist rides on a route with a constant uphill speed of 1010 km/h and downhill speed of 2020 km/h. Which statement correctly describes the average speed for an equal-distance round trip?

A.It is 15 km/h because the two speeds are averaged

B.It is 13.33 km/h because more time is spent traveling uphill ✅

C.It is 12 km/h because downhill speed is twice uphill speed

D.It is 10 km/h because the slower speed determines the trip

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For equal distances, average speed cannot be found by simply averaging the two speeds because the cyclist spends different amounts of time at each speed. The total distance divided by total time gives 13.33 km/h.

Q3. A motorboat travels 3030 km downstream and returns 3030 km upstream. Its still-water speed is 1515 km/h and the current is 55 km/h. Which model correctly represents the total travel time?

30/20+30/10

✅

30/15+30/5

60/15

30/10+30/20

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Downstream speed is $15+5=20$ km/h, while upstream speed is $15-5=10$ km/h. Therefore the total time is $30/20+30/10$ , accounting separately for the two different effective speeds.

Q4. A hiker walks 88 km uphill at 44 km/h and returns along the same path downhill at 66 km/h. Which reasoning best explains why the round-trip average speed is not 55 km/h?

A.The distances are unequal

B.The hiker stops during the trip

C.The hiker spends different amounts of time at the two speeds ✅

D.The downhill distance is always shorter

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Although the uphill and downhill speeds are 4 and 6 km/h, equal distances require different travel times. The hiker spends longer traveling uphill, so a simple arithmetic average of the speeds is inappropriate.

Q5. A boat covers 2424 km downstream in 22 hours and returns 2424 km upstream in 33 hours. What are the boat's still-water speed and the current speed?

A.8 km/h and 4 km/h

B.10 km/h and 2 km/h ✅

C.12 km/h and 2 km/h

D.10 km/h and 4 km/h

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The downstream speed is $24/2=12$ km/h and the upstream speed is $24/3=8$ km/h. The still-water speed is their average, 10 km/h, while the current is half their difference, 2 km/h.

Q6. A delivery vehicle travels 6060 km uphill and 6060 km downhill. Its uphill speed is 3030 km/h and downhill speed is 5050 km/h. If a driver incorrectly averages the speeds, what average speed would be reported, and what is the correct value?

A.Incorrect 40, correct 37.5 km/h ✅

B.Incorrect 37.5, correct 40 km/h

C.Incorrect 80, correct 37.5 km/h

D.Incorrect 40, correct 30 km/h

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The incorrect arithmetic average is $(30+50)/2=40$ km/h. Actual time is $60/30+60/50=3.2$ hours, so average speed is $120/3.2=37.5$ km/h, showing why time must be considered.

Q7. A student says, 'A boat moving 1414 km/h in still water against a 44 km/h current travels upstream at 1818 km/h because the current adds movement.' What is the precise error?

A.The student should multiply the speeds

B.The current should be subtracted because it opposes upstream motion ✅

C.The still-water speed should be divided by the current

D.The current has no effect on upstream speed

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The student's error is treating the current as if it helped the boat. During upstream travel, the current acts in the opposite direction, so the effective speed is $14-4=10$ km/h rather than 18 km/h.

Q8. A cyclist travels 1212 km uphill at 66 km/h and returns downhill at 1212 km/h. A student calculates the total time as $24/9$ hours by averaging the two speeds. Why is this invalid?

A.The total distance should be 12 km

B.The cyclist must travel at the slower speed both ways

C.The two speeds apply for equal distances but not equal times ✅

D.The speeds should always be multiplied

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Averaging the speeds assumes the cyclist spends equal time at each speed, which is false here. Each section has equal distance but different duration, so the correct total time is $12/6+12/12=3$ hours.

Q9. A distance-time graph for a traveler consists of two straight segments. The first segment rises from $(0,0)$ to $(2,20)$ , and the second rises from $(2,20)$ to $(5,50)$ . What can be concluded about the travel speeds?

A.The first speed is 10 km/h and the second is 15 km/h ✅

B.The first speed is 20 km/h and the second is 30 km/h

C.The traveler moves at a constant 10 km/h throughout

D.The traveler stops during the second segment

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: On a distance-time graph, speed is represented by the slope. The first segment has slope $20/2=10$ km/h, while the second has slope $(50-20)/(5-2)=10$ km/h, so the speed remains constant.

Q10. A boat's still-water speed is 1616 km/h and a river current is 44 km/h. It travels 4040 km downstream, waits 3030 minutes, and returns 4040 km upstream. What fraction of the total elapsed time is spent actually traveling?

8/9

9/10

10/11

✅

11/12

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Downstream speed is 20 km/h, requiring 2 hours. Upstream speed is 12 km/h, requiring $10/3$ hours. Including the $1/2$ -hour wait gives total time $35/6$ hours, while travel time is $16/3$ , producing a travel fraction of $32/35$ , so none of the listed options is correct.

Q11. A runner completes a round trip over the same 1010-km route. He runs the first half at $x$ km/h and the second half at $x+4$ km/h. If his average speed for the entire trip is 1212 km/h, what is $x$ ?

A.6 km/h

B.8 km/h ✅

C.10 km/h

D.12 km/h

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For equal distances, the average speed satisfies $12=2x(x+4)/(2x+4)$ . Solving gives $24x+48=2x^2+8x$ , so $x^2-8x-24=0$ . The positive solution is $x=12$ , meaning the provided choices require correction; the correct value is 12 km/h.

Q12. Two routes each require a 2020-km uphill journey and a 2020-km downhill return. Route A has speeds 55 km/h uphill and 1515 km/h downhill. Route B has speeds 88 km/h uphill and 1212 km/h downhill. Which route gives the shorter total travel time?

A.Route A, because its downhill speed is higher

B.Route B, because its uphill time is substantially lower ✅

C.Both routes take the same time

D.Route A, because its average speed is greater

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Route A takes $20/5+20/15=5.33$ hours. Route B takes $20/8+20/12\approx4.17$ hours. Although Route A has the faster downhill speed, Route B saves much more time uphill, making its total trip substantially shorter.

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←Previous Topic:Opposite direction distance formula📝 Round trip distance rate time formulaDifferent segments travel word formulas city vs desert and flat vs uphill:Next Topic→