SHSAT: Unit rate: find the amount for one, then scale
Find a unit rate from one pair of numbers, use it to scale to a new amount, and spot when a warm-up or starting amount breaks the proportion.
If something happens at a constant rate, how do you get from one known pair to any other amount?
Short answer
Divide to find the amount for one unit, then multiply by the new number of units. This works only when the rate is constant and the count starts at zero.
Know first: Dividing and multiplying whole numbers, Simplifying fractions
Step 1 of 5
A constant rate turns one division into any answer
A unit rate is the amount that goes with exactly one unit of something else: liters per minute, pages per minute, dollars per pound. Divide the amount by the units it took, and you have the rate.
Once you have the amount for one, multiply by any number of units. This works only under one condition: the rate stays constant, and the count starts from zero. Then doubling the time doubles the output.
| You divide | You get | Use it to find |
|---|---|---|
| liters ÷ minutes | liters per minute | how many liters in some number of minutes |
| minutes ÷ liters | minutes per liter | how many minutes for some number of liters |
| new time ÷ old time | a scale factor | the new amount, by multiplying the old amount |
Write the units next to every number. The units tell you which division you need.
Step 2 of 5
Find the rate, then scale it
Worked example: A pump at a constant rate
A pump moves 135 liters of water in 9 minutes at a constant rate. How many liters does it move in 16 minutes?
- A16/15
- B142
- C240
- D2160
- 1
Write the pair with units
135 liters in 9 minutes.
- 2
Divide for the rate
135 ÷ 9 = 15 liters per minute.
- 3
Scale to the new time
15 × 16 = 240 liters. The units check: (liters per minute) × minutes leaves liters.
- 4
See why the other choices are there
16/15 divides 9 ÷ 135, which is minutes per liter, then multiplies by 16. 142 adds the 7 extra minutes to 135 liters, mixing units. 2160 multiplies 135 by 16 and never divides by 9.
Answer
240 liters
When the rate is not a whole number
A printer prints 52 pages in 8 minutes. How many pages in 20 minutes?
Unit rate: 52 ÷ 8 = 6.5 pages per minute, and 6.5 × 20 = 130
Scale factor: 20 ÷ 8 = 5/2, and 52 × 5/2 = 26 × 5 = 130
Without a calculator, pick the route that keeps the numbers whole.
Find the wrong step
A machine is switched on. It warms up for 2 minutes and makes nothing, then makes 9 bolts per minute. How many bolts has it made 10 minutes after it was switched on?
- 1
The rate while it runs is 9 bolts per minute.
- 2
It runs for 10 minutes, so it makes 9 × 10 = 90 bolts.
What went wrong
The first 2 minutes were warm-up. The output is not proportional to the full 10 minutes.
- 3
The answer is 90 bolts.
The fix
Only 10 − 2 = 8 minutes were spent making bolts, so the machine made 9 × 8 = 72 bolts.
Step 3 of 5
Set up the rate yourself
Check yourself · Question 1
A faucet fills 63 cups in 7 minutes at a constant rate. How many cups does it fill in 12 minutes?
Answer: C
The rate is 63 ÷ 7 = 9 cups per minute. In 12 minutes the faucet fills 9 × 12 = 108 cups.
Trap answer: 68
Adding instead of scaling
- Why it looks right
- 12 minutes is 5 more than 7, so adding 5 feels like a small, safe adjustment.
- Why it's wrong
- Each extra minute adds 9 cups, not 1 cup. Adding 5 to 63 treats minutes as if they were cups.
- The right answer
- 108, because 9 cups per minute × 12 minutes = 108 cups.
Check yourself · Question 2
A robot paints 20 tiles in 8 minutes at a constant rate. How many tiles does it paint in 30 minutes?
Answer: C
The rate is 20 ÷ 8 = 5/2 tiles per minute, and 5/2 × 30 = 75. The scale factor route agrees: 30 ÷ 8 = 15/4, and 20 × 15/4 = 75.
Common mistake
I divide whichever way gives a whole number, since there's no calculator.
- Why it's tempting
- A clean whole number feels like a sign that you chose the right division.
- Do this instead
- Let the units decide. Tiles ÷ minutes gives tiles per minute, even when it's 5/2. If you want to avoid fractions, use the scale factor route.
Step 4 of 5
What to remember
Remember
- A unit rate is the amount divided by the units it took. Write the units so you divide in the right order.
- Multiply the rate by the new number of units, or multiply the old amount by a scale factor.
- The method needs a constant rate that starts from zero. Remove any warm-up time or starting amount first.
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, which two numbers did you divide, and what units did the result have?