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📝 Opposite direction distance formula (14 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 14 questions available

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What is Opposite direction distance formula?

Definition:
Opposite direction distance problems involve two objects moving away from each other in opposite directions, and the total distance between them is the sum of the distances each travels, using the formula $D_{\text{total}} = r_1 t + r_2 t$ , where $r_1$ and $r_2$ are their speeds, and $t$ is the time traveled, as they are moving directly apart, so their distances add.

Working:
To solve these problems, create a table with the two objects, note their rates and times (often the same time if they start simultaneously), and set the sum of their distances equal to the given total distance; if they start at different times, account for that in the time expressions, and solve for the unknown time or rate using the equation $(r_1 + r_2)t = D_{\text{total}}$ .

Example:
Two trains leave the same station at the same time in opposite directions, one at 60 mph and the other at 80 mph; to find when they are 420 miles apart, set $60t + 80t = 420$ , so $140t = 420$ , and $t = 3$ hours, meaning after 3 hours they are 420 miles apart.

Reason:
These problems are useful for understanding relative motion, navigation, and logistics, and they apply to real-world situations like cars leaving a point in opposite directions, aircraft separation, or ships at sea, making them practical for various fields.

Easy

Medium

Hard

📝 All Opposite direction distance formula MCQs

Q1. Two cyclists start from the same point and travel in opposite directions at constant speeds. Cyclist A travels at 1818 km/h and Cyclist B at 2222 km/h. What equation correctly represents their separation $d$ after $t$ hours?

d=4t

d=18t+22t

✅

d=22t-18t

d=40+t

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: When two objects move in opposite directions from the same starting point, their distances from the starting point increase on opposite sides. Therefore, their separation equals the sum of their individual distances: $18t+22t=40t$ .

Q2. A pair of runners move away from the same starting line in opposite directions at constant speeds. Which quantity determines how quickly the distance between them increases?

A.The difference between their speeds

B.The sum of their speeds ✅

C.The slower runner's speed only

D.The time already traveled

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For objects moving apart in opposite directions, each runner contributes to the increasing separation. Thus, the separation grows at the sum of their speeds. Using only the difference would incorrectly model objects traveling in the same direction.

Q3. Two cars leave an intersection simultaneously in opposite directions. Car A travels 6060 km/h while Car B travels 7575 km/h. After 2.42.4 hours, what is their separation?

A.144 km

B.180 km

C.324 km ✅

D.135 km

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Car A covers $60(2.4)=144$ km and Car B covers $75(2.4)=180$ km. Since they move in opposite directions, their separation is $144+180=324$ km.

Q4. A student claims that if two buses move in opposite directions at 5050 km/h and 7070 km/h, their separation increases at 2020 km/h. What is wrong with the reasoning?

A.The buses should be considered stationary

B.The speeds must be multiplied

C.Opposite directions require adding the speeds ✅

D.Only the faster bus determines separation

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The student has used the speed difference, which applies when comparing motion in the same direction under appropriate conditions. For opposite directions, both distances increase away from the starting point, so their speeds add.

Q5. Two delivery vans leave the same warehouse at the same time in opposite directions. One travels 4848 km/h and the other 5252 km/h. A manager wants them to be 250250 km apart. How long must they travel?

A.2.0 hours

B.2.5 hours ✅

C.5.0 hours

D.1.25 hours

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Their combined rate of separation is $48+52=100$ km/h. Setting the separation equal to 250 gives $100t=250$ , so $t=2.5$ hours. The key modeling step is adding the opposite-direction speeds.

Q6. Two trains begin at the same station and move in opposite directions. Train A travels 9090 km/h. Train B travels at an unknown constant speed. After 33 hours, they are 510510 km apart. What is Train B's speed?

A.60 km/h

B.70 km/h ✅

C.80 km/h

D.90 km/h

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Together, the trains increase their separation by $510/3=170$ km/h. Since Train A contributes 90 km/h, Train B must contribute $170-90=80$ km/h. Therefore, the correct speed is 80 km/h.

Q7. A graph of separation versus time is a straight line passing through $(0,0)$ and $(4,280)$ , where time is measured in hours and separation in kilometers. What does the slope represent?

A.The initial distance between the objects

B.The average speed of only the faster object

C.The rate at which separation increases ✅

D.The time required for the objects to meet

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The slope of a separation-versus-time graph gives the change in separation divided by the change in time. Here it is $280/4=70$ km/h, representing the combined rate at which the objects move apart.

Q8. Two boats leave a dock simultaneously in opposite directions. Boat A travels 3030 km/h and Boat B travels 4545 km/h. After 22 hours, Boat B increases its speed to 6060 km/h while Boat A continues at 3030 km/h. How far apart are they after 55 hours?

A.285 km

B.300 km

C.315 km ✅

D.345 km

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: During the first 2 hours, their separation increases at $30+45=75$ km/h, giving 150 km. During the next 3 hours, it increases at $30+60=90$ km/h, giving 270 km. Total separation is 420 km, so none of the listed values is correct.

Q9. Two people start at the same park and walk in opposite directions. Person A walks 55 km/h for 4040 minutes, while Person B walks 44 km/h for 5555 minutes. How far apart are they when Person B stops?

A.5.67 km

B.7.67 km ✅

C.8.67 km

D.9.67 km

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Convert the times carefully. Person A walks $2/3$ hour and covers $5(2/3)=3.33$ km. Person B walks $55/60=11/12$ hour and covers $4(11/12)=3.67$ km. Their separation is $3.33+3.67=7$ km, so the listed choices contain an error.

Q10. A position-time graph shows two straight lines starting from the same point. One line has slope $+35$ and the other has slope $-25$ , with position measured in kilometers and time in hours. What does the graph imply about their separation after 44 hours?

A.They are 40 km apart

B.They are 60 km apart

C.They are 100 km apart ✅

D.They are 240 km apart

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The positive and negative slopes indicate motion in opposite directions. After 4 hours, their positions relative to the origin are 140 km and $-100$ km. The distance between these positions is $140-(-100)=240$ km, not 100 km.

Q11. Two vehicles move away from the same checkpoint in opposite directions. Vehicle A travels 7272 km/h and Vehicle B travels 5858 km/h. A worker reasons that after $t$ hours their separation is $72t-58t$ . Which interpretation best evaluates this model?

A.It is correct because separation always uses subtraction

B.It is correct only when

t=1

C.It is incorrect because opposite-direction distances must be added ✅

D.It is incorrect because time should be divided by the speeds

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The model $72t-58t$ gives the difference in distances traveled, which does not represent separation when the vehicles move in opposite directions. Their positions lie on opposite sides of the checkpoint, so the distances must be added.

Q12. A rescue team sends two vehicles from a base in opposite directions. Vehicle A travels 8080 km/h and Vehicle B 6565 km/h. They must reach a combined separation of 435435 km. However, Vehicle A leaves 3030 minutes after Vehicle B. How long after Vehicle A starts will the required separation occur?

A.2 hours

B.2.5 hours

C.3 hours ✅

D.3.5 hours

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Let $t$ be hours after Vehicle A starts. Vehicle A travels $80t$ , while Vehicle B has traveled $65(t+0.5)$ . Their separation is $80t+65(t+0.5)=435$ . Solving $145t+32.5=435$ gives $t\approx2.78$ hours, so the closest listed choice is 3 hours.

Q13. Two objects move in opposite directions from the same point at constant speeds. One travels 4040 km/h and the other 6060 km/h. A model predicts their separation after 66 hours is 120120 km. Which correction produces the correct result and explains the error?

A.Use

60-40

, giving 120 km; subtraction is appropriate

B.Use

60+40

, giving 600 km; the speeds represent increasing separation ✅

C.Use

60\times40

, giving 2400 km; multiplication combines the speeds

D.Use

6/100

, giving 0.06 km; time must be divided

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The correct separation rate is $40+60=100$ km/h because both objects move away from each other. Over 6 hours, separation becomes $100(6)=600$ km. The original model incorrectly subtracts speeds, a common same-direction misconception.

Q14. A runner and cyclist leave the same location at the same time in opposite directions. The runner travels 1212 km/h and the cyclist 2828 km/h. At the same time, a second cyclist leaves the runner's direction at 88 km/h. Which statement correctly compares the runner's separation from the first cyclist with the second cyclist's separation from the runner after 33 hours?

A.They are equal at 120 km each

B.The first separation is 120 km and the second is 12 km ✅

C.The first separation is 84 km and the second is 24 km

D.The first separation is 36 km and the second is 60 km

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The runner and first cyclist move in opposite directions, so their separation rate is $12+28=40$ km/h, producing 120 km after 3 hours. The second cyclist travels in the runner's direction, so that separation is $12-8=4$ km/h, or 12 km.

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