TACHS: Constant rates: find the amount for one unit of time
Use a unit rate to find distances, times and amounts, including trips with stops, machines working together and average speed.
How do you use a constant speed or rate to answer distance and time questions?
Short answer
Divide amount by time to get the unit rate, then multiply by a new time or divide a new amount by the rate. For average speed, divide total distance by total time.
Know first: Multiplying and dividing whole numbers, 60 minutes = 1 hour
Step 1 of 6
A constant rate is the same amount every minute
A constant rate means equal amounts in equal times. If a drone flies 1,200 meters in 8 minutes at a constant speed, it flies 1,200 ÷ 8 = 150 meters every minute. That number is the unit rate, and it answers every follow-up question.
Write units on every number. Meters per minute times minutes gives meters. Meters divided by meters per minute gives minutes.
| You want | Use | Example at 150 m per minute |
|---|---|---|
| Distance | rate × time | 150 × 4 = 600 m |
| Time | distance ÷ rate | 1,050 ÷ 150 = 7 minutes |
| Trip time with a stop | moving time + stopped time | 7 + 2 = 9 minutes |
| Two machines together | add their rates | 1/3 + 1/6 = 1/2 tank per hour |
| Average speed | total distance ÷ total time | Never the average of two speeds |
Stopped time adds minutes but no distance.
Step 2 of 6
Unit rate first, then scale
Worked example: A drone at constant speed
A drone flies 1,200 meters in 8 minutes at a constant speed. How far does it fly in 20 minutes?
- A150
- B1,220
- C2,400
- D3,000
- 1
Find the unit rate
1,200 ÷ 8 = 150 meters per minute.
- 2
Multiply by the new time
150 × 20 = 3,000 meters.
- 3
Check another way
20 minutes is 20/8 = 2.5 times as long as 8 minutes, and 2.5 × 1,200 = 3,000.
- 4
See why the other choices are there
150 is the rate, not the distance. 1,220 adds 20 to 1,200. 2,400 doubles the distance, but 20 minutes is more than twice 8 minutes.
Answer
The drone flies 3,000 meters.
Wrong: A runner goes 4 km at 12 km per hour, then 2 km at 3 km per hour. Average speed: (12 + 3) ÷ 2 = 7.5 km per hour.
Right: Total distance 4 + 2 = 6 km. Total time 20 + 40 = 60 minutes, or 1 hour. Average speed: 6 ÷ 1 = 6 km per hour.
The runner spent twice as long going slowly, so the slow part pulls the average down. Only total distance ÷ total time accounts for that.
Step 3 of 6
Stops and teamwork change the time
A stop adds time but no work. A copier makes 240 copies in 4 minutes, so 60 copies per minute. If it runs 5 minutes, stops 3 minutes, and runs 2 more, it runs 7 minutes in a 10-minute period and makes 7 × 60 = 420 copies.
When two machines work together, add their rates. If hose A fills 1/3 of a pool each hour and hose B fills 1/6, together they fill 1/3 + 1/6 = 1/2 of the pool each hour, so the pool takes 2 hours.
Step 4 of 6
Rate, then the question
Check yourself · Question 1
A cyclist rides 54 km in 3 hours at a constant speed. How far does she ride in 5 hours at that speed?
Answer: C
Unit rate: 54 ÷ 3 = 18 km per hour. Distance: 18 × 5 = 90 km.
Check yourself · Question 2
Maya bikes 6 km in 20 minutes, then walks 4 km in 40 minutes. What is her average speed for the whole trip, in kilometers per hour?
Answer: B
Total distance: 6 + 4 = 10 km. Total time: 20 + 40 = 60 minutes, which is 1 hour. Average speed: 10 ÷ 1 = 10 km per hour.
Trap answer: 12
Averaging the speeds
- Why it looks right
- Two speeds appear, and averaging two numbers is the usual way to find an average.
- Why it's wrong
- Maya walked for 40 minutes but biked for only 20. An average of the two speeds counts them equally, which overstates the fast part.
- The right answer
- 10 km per hour: 10 km in 1 hour.
Check yourself · Question 3
A robot moves at 80 meters per minute. It must travel 1,000 meters and stops twice along the way, for 2 minutes each time. How many minutes does the whole trip take?
Answer: C
Moving time: 1,000 ÷ 80 = 12.5 minutes. Stopped time: 2 × 2 = 4 minutes. Total: 12.5 + 4 = 16.5 minutes.
Common mistake
"The drone's rate is minutes ÷ meters, so it is 8 ÷ 1,200."
- Why it's tempting
- Both numbers are in the question, and it is easy to divide them in either order.
- Do this instead
- Say the rate in words: meters per minute means meters ÷ minutes. Units on every number catch an upside-down rate.
Step 5 of 6
What to remember
Remember
- Find the unit rate first, with units: amount ÷ time.
- Rate × time gives an amount; amount ÷ rate gives a time. Stopped time adds minutes but no distance.
- Average speed is total distance ÷ total time, and machines working together add their rates.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what is the unit rate, and which wrong choice would you get by adding instead of multiplying?