How to work with ratios, unit rates and rate of change
TEAS 7 outline: Solve real-world ratio and rate-of-change problems
Also covers HESI A2 Math (Ratios and proportions)
- A ratio compares two amounts in the order the words name them.
- A rate compares different units, like miles per hour.
- Unit rate: divide so the bottom is 1, like 125 mL per hour.
- Rate of change is change in output over change in input.
- Convert minutes to hours by dividing by 60: 45 minutes is 0.75 hour.
Key facts
- A ratio compares two quantities in the order named; in 'blue to red', blue goes first (on top).
- Simplify a ratio like a fraction by dividing both parts by their greatest common factor; leave an improper ratio such as 3/2 as it is.
- A rate compares quantities with different units, such as miles per hour or milliliters per hour.
Show all 7 key factsShow fewer
- A unit rate has 1 in the denominator: divide the first quantity by the second (1,000 mL ÷ 8 h = 125 mL per hour).
- Distance = rate × time, and the time must be in the same unit as the rate (convert minutes to hours for miles per hour).
- Rate of change = change in output ÷ change in input, in output units per input unit; on a graph it is the slope of the line.
- A negative rate of change means the quantity is going down.
Quick check1 question · 2 more at the end
A nursing study group has 14 first-year students and 21 second-year students. What is the ratio of first-year students to second-year students in simplest form?
Correct answer: C.
Write the ratio in the order named, first-year to second-year: 14:21, or 14/21. Both numbers divide by their greatest common factor, 7, giving 2:3. Watch the order and make sure you compare the two groups, not one group to the total.
A This reverses the order, giving second-year to first-year (21:14).
B This compares first-year students to the whole group (14:35), a part-to-whole ratio.
C Correct: 14:21 divided by 7 is 2:3.
D This compares second-year students to the whole group (21:35).
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: A ratio compares two quantities in the order the words name them, written a:b or a/b, simplified like a fraction. A rate compares different units, and a unit rate divides so the second amount is 1, such as 125 mL per hour. Rate of change, or slope, is the change in one quantity divided by the change in the other.
1 What is the difference between a ratio and a rate?A ratio compares two amounts of the same kind, while a rate compares two different units. 1 min
A ratio compares two amounts of the same kind, while a rate compares two different units.
| Term | What it compares | Example |
|---|---|---|
| Ratio | Same kind of thing, in the order named | 12 blue tubes to 28 red tubes = 3:7 |
| Rate | Different units | 90 miles in 1.5 hours |
| Unit rate | Different units, per 1 | 60 miles per hour |
| Rate of change | How much one quantity changes per unit of another | −8% battery per hour |
The word per means "for each one" and tells you which number goes on the bottom.
2 Worked example 1: How do you write and simplify a ratio?Write the ratio in the order the words give, then simplify it like a fraction. 1 min
Write the ratio in the order the words give, then simplify it like a fraction.
A lab tray holds 12 blue-capped tubes and 28 red-capped tubes. Find the ratio of blue to red.
- Blue first: 12/28.
- Both divide by 4: 12 ÷ 4 = 3 and 28 ÷ 4 = 7.
- Answer: 3:7 (or 3/7).
Wrong turn: writing 7:3 (reversed) or 12:40 = 3:10 (blue to all tubes, a part-to-whole ratio the question did not ask for).
3 Worked example 2: How do you find a unit rate, like a flow rate or speed?To find a unit rate, divide the amount by the number of units that come after the word "per". 1 min
To find a unit rate, divide the amount by the number of units that come after the word "per".
Flow rate: 1,000 mL of fluid runs over 8 hours. mL per hour = 1,000 ÷ 8 = 125 mL per hour. Mental math works: half of 1,000 is 500, half again is 250, half again is 125.
Speed: a shuttle drives 90 miles in 1 hour 30 minutes.
- Convert the time: 30 minutes = 30 ÷ 60 = 0.5 hour, so the time is 1.5 hours.
- Divide: 90 ÷ 1.5 = 60 miles per hour (use the calculator).
- Check with distance = rate × time: 60 × 1.5 = 90 miles.
Wrong turn: dividing backward (8 ÷ 1,000) or treating 1 hour 30 minutes as 1.3 hours.
4 Worked example 3: How do you find a rate of change?Rate of change is the change in the output divided by the change in the input. 1 min
Rate of change is the change in the output divided by the change in the input.
A phone battery shows 90% at 1:00 and 66% at 4:00. If it keeps draining at the same rate, when will it reach 10%?
- Change in battery: 66 − 90 = −24 percentage points.
- Change in time: 4 − 1 = 3 hours.
- Rate of change: −24 ÷ 3 = −8 percentage points per hour. The minus sign means it is going down.
- From 66% to 10% is 56 points: 56 ÷ 8 = 7 hours after 4:00, which is 11:00.
On a graph of battery versus time, this rate of change is the slope of the line. Wrong turn: subtracting 90 − 66 on top but 4 − 1 on the bottom. That gives +8, as if the battery were charging. Another slip is counting 7 hours from 1:00 instead of 4:00.
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: Keep the order the question names: 'A to B' is A:B, not B:A.
Watch out: Part to part is not part to whole: 12 blue to 28 red is 12:28, not 12:40.
Watch out: Divide in the order of 'per': miles per gallon is miles ÷ gallons, not gallons ÷ miles.
Watch out: Convert minutes to hours by dividing by 60: 45 minutes is 0.75 hour, not 0.45 hour.
Watch out: Subtract in the same order on the top and bottom of a rate of change, or the sign comes out wrong.
How to remember it
One phrase covers almost every rate problem on this topic.
Memory hook: "Per means divide, and the word after per goes on the bottom."
Self-check:
- Write the ratio of 15 apples to 25 oranges in simplest form.
- Find the speed of a runner who covers 6 miles in 48 minutes.
- A plant is 10 cm tall on day 2 and 22 cm tall on day 6. What is its rate of change?
Show answers
- 3:5.
- Apples first: 15/25.
- Both divide by 5: 15 ÷ 5 = 3 and 25 ÷ 5 = 5.
- 7.5 miles per hour.
- Convert the time: 48 ÷ 60 = 0.8 hour.
- Divide: 6 ÷ 0.8 = 7.5 miles per hour.
- Check: 7.5 × 0.8 = 6 miles.
- 3 cm per day.
- Change in height: 22 − 10 = 12 cm.
- Change in time: 6 − 2 = 4 days.
- Rate of change: 12 ÷ 4 = 3 cm per day. It is positive, so the plant is growing.