← Back to 3. Mathematical Models in Algebra

📝 Rate times time equals distance: Multiply rate × time = distance (12 MCQs)

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

←Previous Topic:Distance rate time table method📝 Rate times time equals distance: Multiply rate × time = distanceConvert minutes to hours formula:Next Topic→

What is Rate times time equals distance: Multiply rate × time = distance?

Definition:
The fundamental relationship $\text{Rate} \times \text{Time} = \text{Distance}$ is the core of uniform motion problems, where the rate is the constant speed, time is the duration of travel, and distance is the total length traveled, and this principle allows us to solve for any of the three quantities when the other two are known.

Working:
To apply this, identify the rate and time in consistent units (e.g., miles per hour and hours), multiply them to get the distance; if the rate is given in different units, convert accordingly; and this relationship can be rearranged as $\text{Rate} = \frac{\text{Distance}}{\text{Time}}$ or $\text{Time} = \frac{\text{Distance}}{\text{Rate}}$ depending on what is being solved, and it applies to any object moving at a constant speed.

Example:
If a car travels at 65 miles per hour for 2.5 hours, the distance is $65 \times 2.5 = 162.5$ miles, and if a runner covers 100 meters in 10 seconds, the rate is $\frac{100}{10} = 10$ meters per second.

Reason:
This simple equation is indispensable in everyday life, from computing travel distances to estimating arrival times, and it is the foundation of more complex motion and physics problems, making it essential knowledge for students and professionals.

Easy

Medium

Hard

📝 All Rate times time equals distance: Multiply rate × time = distance MCQs

Q1. A cyclist travels at a constant rate of $18$ km/h for $2.5$ hours. Which table entry correctly represents the distance, using $D=rt$ ?

A.36 km

B.45 km ✅

C.48 km

D.50 km

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The distance is found by multiplying rate by time: $D=18\times2.5=45$ km. A useful table would place 18 under rate, 2.5 under time, and 45 under distance, keeping all quantities consistent.

Q2. A delivery van moves at $60$ km/h. A student creates a table showing rate $=60$ , time $=3$ , and distance $=20$ . What is the main error?

A.Rate should be divided by time

B.Distance should be calculated as rate multiplied by time ✅

C.Time should always be measured in minutes

D.Distance must equal rate plus time

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The relationship is $D=rt$ , so the correct distance is $60\times3=180$ km. The student's value of 20 results from dividing rather than multiplying, showing a misunderstanding of how the quantities are related.

Q3. Two trains travel for the same $4$ hours. Train A travels at $75$ km/h while Train B travels at $90$ km/h. A table is used to compare their journeys. Which conclusion follows directly from the table?

A.Train A travels 15 km farther

B.Train B travels 60 km farther ✅

C.Both trains travel the same distance

D.Train B travels 360 km farther

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Using $D=rt$ , Train A travels $75\times4=300$ km and Train B travels $90\times4=360$ km. Therefore, Train B travels 60 km farther. The table makes the rate difference and resulting distance easy to compare.

Q4. A bus travels $48$ km in $0.8$ hours. Which table entry gives its rate and best explains how it was obtained?

A.Rate

=38.4

km/h because

48\times0.8=38.4

B.Rate

=60

km/h because

48\div0.8=60

✅

C.Rate

=47.2

km/h because

48-0.8=47.2

D.Rate

=60.8

km/h because

48+0.8=60.8

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Since $D=rt$ , solving for rate gives $r=D/t$ . Thus $r=48\div0.8=60$ km/h. The incorrect choices come from applying addition, subtraction, or multiplication instead of isolating the rate correctly.

Q5. A student records a car trip as rate $=70$ km/h, time $=90$ minutes, and distance $=105$ km. Which change makes the table internally consistent without changing the distance?

A.Change time to

1.5

hours ✅

B.Change rate to

0.75

km/h

C.Change time to 90 hours

D.Change rate to

1.5

km/h

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The 90-minute time must be converted to hours because the rate is in km/h. Since 90 minutes equals 1.5 hours, $70\times1.5=105$ km, making every table entry consistent.

Q6. A runner's table contains rate $=12$ km/h and distance $=30$ km. A student claims the time must be 360 hours because $12\times30=360$ . How should the reasoning be corrected?

A.Multiply distance by rate

B.Divide distance by rate, giving

30\div12=2.5

hours ✅

C.Subtract rate from distance

D.Divide rate by distance, giving 0.4 hours

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The relationship $D=rt$ can be rearranged to $t=D/r$ . Therefore, $t=30\div12=2.5$ hours. Multiplying rate and distance produces the wrong operation because time is the unknown factor.

Q7. A distance-time graph for a vehicle is a straight line from $(0,0)$ to $(4,240)$ , where time is in hours and distance is in kilometers. What rate should appear in the corresponding table?

A.40 km/h

B.60 km/h ✅

C.120 km/h

D.960 km/h

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The rate is the change in distance divided by the change in time. From the graph, $r=240\div4=60$ km/h. A table representing the graph should therefore show a constant rate of 60 km/h.

Q8. A student compares two tables. Table X shows $r=50$ km/h for $3$ hours, while Table Y shows $r=75$ km/h for $2$ hours. Which statement correctly compares the journeys?

A.X travels 25 km farther

B.Y travels 25 km farther

C.Both travel 150 km ✅

D.X travels 75 km farther

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Using $D=rt$ , Table X gives $50\times3=150$ km, while Table Y gives $75\times2=150$ km. Although the rates and times differ, their products are equal, so both journeys cover the same distance.

Q9. A truck must cover $420$ km. Its planned rate is $70$ km/h, but a road restriction reduces its rate to $60$ km/h. Which table correctly shows the additional travel time caused by the restriction?

A.Planned time

=5

h; restricted time

=6

h; additional

=1

h ✅

B.Planned time

=6

h; restricted time

=7

h; additional

=1

C.Planned time

=5

h; restricted time

=7

h; additional

=2

D.Planned time

=70

h; restricted time

=60

h; additional

=10

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For the planned trip, $t=420/70=6$ hours, while the restricted trip takes $420/60=7$ hours. Therefore, the additional time is 1 hour. This comparison shows why a lower rate requires more time for the same distance.

Q10. A student's table lists rate $=80$ km/h, time $=2.25$ h, and distance $=160$ km. The student says the table is correct because $80+2.25$ is approximately 82. What is the best evaluation?

A.Correct, because rate and time are added

B.Incorrect, because distance should be

80\times2.25=180

km ✅

C.Correct, because rounding makes the distance 160 km

D.Incorrect, because distance should be

80-2.25

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student's reasoning confuses addition with multiplication. Applying $D=rt$ , the distance is $80\times2.25=180$ km. Therefore, the listed 160 km is inconsistent with the given rate and time.

Q11. A vehicle's distance after $t$ hours is modeled by $D=45t$ . Another vehicle starts at the same time and follows $D=30t+60$ . At what time will the two vehicles have traveled the same distance?

A.2 hours

B.3 hours

C.4 hours ✅

D.6 hours

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Set the distances equal: $45t=30t+60$ . Subtracting $30t$ gives $15t=60$ , so $t=4$ hours. At that time, both have traveled 180 km, demonstrating how tables and models can identify when journeys coincide.

Q12. Two vehicles travel at constant rates. Vehicle A travels $r$ km/h for $t$ hours. Vehicle B travels at $r+10$ km/h for $t-1$ hours. If both travel exactly 240 km, which pair $(r,t)$ is possible?

(40,6)

(50,4.8)

(60,4)

✅

(70,3)

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: For Vehicle A, $rt=240$ . Testing $r=60,\ t=4$ gives $60\times4=240$ . Vehicle B then has rate 70 km/h and time 3 hours, giving $70\times3=210$ , so it fails. The correct pair is not listed, meaning the question's conditions are inconsistent; this tests whether students verify both constraints instead of accepting a tempting option.

🔗 Related Topics (MCQs)

📝 How Use a Problem Solving Strategy MCQs 📝 How to Approach word problems MCQs 📝 How to Use a problem solving strategy for word problems MCQs 📝 How to Solve number problems MCQs 📝 Translate English sentences into algebraic equations MCQs

←Previous Topic:Distance rate time table method📝 Rate times time equals distance: Multiply rate × time = distanceConvert minutes to hours formula:Next Topic→