Rates

TipLearning Objectives

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

  • 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: \[ \text{rate} = \frac{\text{work}}{\text{time}} \]

  • Use dimensional analysis to convert units cleanly: \[ \frac{\text{miles}}{\text{hour}} \times \frac{1\text{ hr}}{60\text{ min}} \]

Important

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 → mass = \(3 \times 5 = 15\) 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.

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:

\[ 240\text{ min} \times \frac{1\text{ hr}}{60\text{ min}} = 4\text{ hr} \]

WarningCommon Mistakes
  • 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

  1. Find the unit rate: 72 miles in 1.5 hours.
  2. Cost per ounce: $4.80 for 12 ounces.
  3. Density: 4 g/cm³, volume 7 cm³ → mass?
  4. 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 \(\frac{60\text{ min}}{1\text{ hr}}\)).