Rates
By the end of this lesson, you’ll be able to:
- Understand rates as ratios comparing different units.
- Solve problems involving speed, density, and unit rates.
- Convert between units using dimensional analysis.
Key Ideas
A rate compares quantities with different units:
- speed: miles/hour (mph)
- cost per unit: dollars/pound
- density: grams/cm³
- speed: miles/hour (mph)
A unit rate is a rate with a denominator of 1.
Example:
\[ \frac{150\text{ miles}}{3\text{ hours}} = 50\text{ mph} \]Work-rate formula often used on SAT/ACT:
\[ \text{rate} = \frac{\text{work}}{\text{time}} \]
More generally:
\[ \text{rate} = \frac{\text{quantity}}{\text{time}} \]
which can be rearranged as
\[ \text{quantity} = \text{rate} \times \text{time} \]
and
\[ \text{time} = \frac{\text{quantity}}{\text{rate}}. \]
- Use dimensional analysis to convert units cleanly: \[ \frac{\text{miles}}{\text{hour}} \times \frac{1\text{ hr}}{60\text{ min}} \]
Track units carefully.
Rates problems often test whether you know how to cancel and convert units correctly.
Common Problem Types
Speed / Distance / Time
Example: A car travels 180 miles in 3 hours.
Rate = \(180/3 = 60\) mph.
Cost per Unit
Example: 12 lb of apples cost $18 → $18/12 = $1.50 per lb.
Density
Example: Density = 3 g/cm³; volume = 5 cm³
\[ \text{mass} = \text{density} \times \text{volume} = 3 \times 5 = 15\text{ g} \]
Unit Conversions
Example: Convert 90 km/hr to m/s:
\(90 \times \frac{1000}{1} \times \frac{1}{3600} = 25\) m/s.
Combined Work (Light Intro)
Example: Rate(A) = 4 units/hr, Rate(B) = 6 units/hr → together: 10 units/hr.
Multi-Step Conversions
Example: Convert 60 miles per hour to feet per second.
\[ 60\frac{\text{mi}}{\text{hr}} \times \frac{5280\text{ ft}}{1\text{ mi}} \times \frac{1\text{ hr}}{3600\text{ s}} \]
Cancel miles and hours:
\[ 60 \times \frac{5280}{3600} = 88 \]
Answer:
\[ 88\frac{\text{ft}}{\text{s}} \]
Strategies
- Write rates as fractions with units attached.
- Cancel units visually to avoid mistakes.
- Convert to a unit rate first if unsure.
- Use a quick table for multi-step conversions or scaling.
Worked Examples
Example 1
Question: A cyclist travels 24 miles in 1.5 hours. What is her speed?
Solution: \(24/1.5 = 16\) mph.
Example 2
Question: A machine produces 300 parts in 4 hours. How many parts per hour?
Solution: \(300/4 = 75\) per hour.
Example 3
Question: Convert 240 minutes to hours using dimensional analysis.
Solution:
\[ 240\text{ min} \times \frac{1\text{ hr}}{60\text{ min}} = 4\text{ hr} \]
Example 4
Question: Convert 45 miles per hour to feet per second.
Solution:
\[ 45\frac{\text{mi}}{\text{hr}} \times \frac{5280\text{ ft}}{1\text{ mi}} \times \frac{1\text{ hr}}{3600\text{ s}} \]
\[ 45 \times \frac{5280}{3600} = 66 \]
Answer: \(66\) ft/s
- Ignoring units or failing to cancel units when converting.
- Mixing up reciprocals in rate conversions (e.g., mph → min/mile).
- Converting numbers but not units.
- Combining multiple conversions into one messy step instead of chaining them cleanly.
- Forgetting to convert minutes ↔︎ hours before computing speed or rate.
Practice Problems
- Find the unit rate: 72 miles in 1.5 hours.
- Cost per ounce: $4.80 for 12 ounces.
- Density: 4 g/cm³, volume 7 cm³ → mass?
- Convert 2.5 hours to minutes.
1.
Unit rate = \(72 \div 1.5 = 48\) mph.
Answer: 48 mph
2.
Cost per ounce = \(4.80 \div 12 = 0.40\).
Answer: $0.40/oz
3.
Mass = \(4 \times 7 = 28\) grams.
Answer: 28 g
4.
Convert hours → minutes: \(2.5 \times 60 = 150\).
Answer: 150 minutes
Summary
- Rates compare quantities with different units; unit rates standardize them.
- Speed, cost/unit, density, and work-rate are common SAT contexts.
- Dimensional analysis is the cleanest way to convert units.
- Units must match before comparing or combining rates.
- Write the rate as a fraction with units on top and bottom.
- Always cancel units—never assume they disappear on their own.
- Convert to a unit rate first if the situation feels messy.
- For conversions, multiply by fractions that equal 1 (like \(\frac{60\text{ min}}{1\text{ hr}}\)).