SHSAT: Rates, unit conversions and average speed
Scale a rate to a new amount, convert minutes to hours, combine rates for teamwork and chases, and find average speed correctly.
How do you solve rate problems when units change, two things work at once, or a trip has several parts?
Short answer
Write amount = rate × time with units on every number. Convert units before combining, add or subtract rates when things act at the same time, and find average speed from total distance and total time.
Know first: Unit rates, Dividing by a fraction
Step 1 of 5
Rates add, subtract and convert
A rate compares two quantities with different units: parts per minute, kilometers per hour. Every rate problem comes back to one relationship: amount = rate × time.
When two things work at once, their rates combine. Two pumps filling the same tank add their rates. Water flowing in and out subtracts. A car chasing a bus closes the gap at the difference of their speeds.
| Situation | Combine like this | Example |
|---|---|---|
| two workers on one job | add rates | 1/6 + 1/3 = 1/2 job per hour, so 2 hours |
| inflow and outflow | subtract rates | 7 in, 4 out: net 3 liters per minute |
| one mover catching another | subtract speeds | closing speed 60 − 40 = 20 km/h |
| average speed for a trip | total distance ÷ total time | include rest time if the question says so |
Convert minutes to hours by dividing by 60 before using a speed in kilometers per hour.
Step 2 of 5
Use the rate that matches the question
Worked example: Time for more jars
A machine fills 36 jars in 15 minutes. At the same rate, how many minutes does it need to fill 60 jars?
- A10
- B25
- C39
- D144
- 1
Choose the useful rate
The question asks for minutes, so find minutes per jar: 15 ÷ 36 = 5/12 minute per jar.
- 2
Multiply
60 × 5/12 = 25 minutes.
- 3
Check
36 jars in 15 minutes is 12/5 jars per minute. In 25 minutes: 25 × 12/5 = 60 jars.
- 4
See why the other choices are there
10 is the time for only the extra 24 jars. 39 adds 24 jars to 15 minutes. 144 multiplies 60 by jars per minute, which gives a number with the wrong units.
Answer
25 minutes
Changing minutes to hours
A train travels 50 kilometers in 40 minutes. Its speed in kilometers per hour:
40 minutes = 40/60 = 2/3 hour
50 ÷ 2/3 = 50 × 3/2 = 75 kilometers per hour
Wrong: A cyclist rides 12 km at 4 km/h and returns 12 km at 6 km/h. Average speed = (4 + 6) ÷ 2 = 5 km/h.
Right: Times: 12 ÷ 4 = 3 hours and 12 ÷ 6 = 2 hours. Average speed = 24 km ÷ 5 hours = 4.8 km/h.
The cyclist spends more time at the slower speed, so the average is pulled below 5. Always divide total distance by total time.
Step 3 of 5
Combine rates carefully
Check yourself · Question 1
One hose fills a pool in 6 hours. A second hose fills it in 3 hours. Working together, how many hours do they take?
Answer: B
Rates: 1/6 and 1/3 pool per hour. Together: 1/6 + 2/6 = 3/6 = 1/2 pool per hour, so one pool takes 2 hours.
Check yourself · Question 2
A cyclist leaves a park at 9:00 riding 10 kilometers per hour. A second cyclist leaves the same park at 9:30 on the same route, riding 15 kilometers per hour. How many minutes after 9:30 does the second cyclist catch up?
Answer: C
By 9:30 the first cyclist is 10 × 1/2 = 5 kilometers ahead. The gap closes at 15 − 10 = 5 kilometers per hour, so it takes 1 hour, or 60 minutes.
Trap answer: 20
Ignoring that the leader keeps moving
- Why it looks right
- 5 kilometers at 15 kilometers per hour is 1/3 hour, and the arithmetic is clean.
- Why it's wrong
- That would be right if the first cyclist stopped. Instead, each hour the gap shrinks by only 15 − 10 = 5 kilometers.
- The right answer
- 60 minutes, from 5 km ÷ 5 km/h.
Check yourself · Question 3
A hiker walks 10 kilometers uphill at 5 kilometers per hour, rests for 1 hour, then walks back 10 kilometers at 2 kilometers per hour. What is the average speed over the whole trip, including the rest, in kilometers per hour?
Answer: A
Times: 10 ÷ 5 = 2 hours, rest 1 hour, 10 ÷ 2 = 5 hours, so 8 hours in all. Distance is 20 kilometers. Average speed = 20 ÷ 8 = 5/2.
Common mistake
Writing 36 minutes as 0.36 hour.
- Why it's tempting
- Decimal points and minutes both look like parts of a whole, so the digits seem to carry over.
- Do this instead
- An hour has 60 minutes, not 100. Divide by 60: 36 minutes = 36/60 = 3/5 hour.
Step 4 of 5
What to remember
Remember
- Amount = rate × time. Choose minutes per item or items per minute based on what the question asks.
- Add rates for teamwork, subtract for inflow and outflow or for one mover chasing another.
- Average speed is total distance ÷ total time, with minutes converted to hours.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, did you use parts per minute or minutes per part, and how did the units tell you which one to use?