Unit Rates

How do you compare prices, speeds, or other rates? Unit rates show the amount per ONE unit - making it easy to compare and make smart decisions!

---

What Is a Rate?

A rate compares two quantities with different units.

Examples:

• 120 miles in 2 hours
• $6 for 3 pounds
• 300 words in 5 minutes

Written as: quantity₁/quantity₂ or quantity₁ per quantity₂

Key: The units are DIFFERENT (miles and hours, dollars and pounds, etc.)

---

What Is a Unit Rate?

A unit rate is a rate where the second quantity is 1.

Examples:

• 60 miles per 1 hour (60 mph)
• $2 per 1 pound
• 60 words per 1 minute

The "per" means "for each one"

Unit rates make comparing easy!

---

Finding Unit Rates

Method: Divide the first quantity by the second quantity

Formula: Unit Rate = First Quantity ÷ Second Quantity

Example 1: 120 miles in 2 hours

Unit Rate = 120 miles ÷ 2 hours = 60 miles/hour

Answer: 60 miles per hour (60 mph)

Example 2: $15 for 3 books

Unit Rate = $15 ÷ 3 books =$5/book

Answer: $5 per book

---

Step-by-Step: Finding Unit Rates

Example: 180 students in 6 buses. Find students per bus.

Step 1: Identify quantities

• First quantity: 180 students
• Second quantity: 6 buses

Step 2: Divide 180 ÷ 6 = 30

Step 3: Write with units 30 students per bus

Answer: 30 students/bus

---

Unit Price

Unit price is a special unit rate used for shopping.

It shows price per ONE item (or per one unit like ounce, pound, etc.)

Example: $8 for 4 cans

Unit price = $8 ÷ 4 =$2 per can

Why useful? Compare different package sizes!

---

Comparing Unit Prices

Example: Which is the better buy?

• Option A: 3 bottles for $6
• Option B: 5 bottles for $9

Find unit price for each:

Option A: $6 ÷ 3 =$2 per bottle Option B: $9 ÷ 5 =$1.80 per bottle

Option B is better! ($1.80 <$2)

Rule: Lower unit price = better deal (for same product)

---

Speed as a Unit Rate

Speed is distance per unit of time.

Example 1: 240 miles in 4 hours

Speed = 240 miles ÷ 4 hours = 60 mph

Example 2: 150 meters in 25 seconds

Speed = 150 m ÷ 25 s = 6 m/s

Common speed units:

• mph (miles per hour)
• km/h (kilometers per hour)
• m/s (meters per second)
• ft/s (feet per second)

---

Unit Rate with Fractions

Sometimes you get fractional unit rates.

Example: 3 pizzas for 8 people

Unit rate = 3 ÷ 8 = 3/8 pizza per person

Each person gets 3/8 of a pizza

Or flip it: 8 ÷ 3 = 8/3 = 2⅔ people per pizza

Choose the perspective that makes sense for the problem!

---

Rate vs Unit Rate

Rate: Any comparison of quantities

• 120 miles in 2 hours
• $12 for 4 pounds

Unit Rate: Second quantity is 1

• 60 miles per 1 hour
• $3 per 1 pound

To convert rate to unit rate: DIVIDE!

---

Real-World Examples

Grocery Shopping:

• Brand A: 24 oz for $4.80 →$0.20 per oz
• Brand B: 16 oz for $3.84 →$0.24 per oz
• Brand A is better value!

Gas Mileage:

• Car travels 360 miles on 12 gallons
• 360 ÷ 12 = 30 miles per gallon (mpg)

Typing Speed:

• 450 words in 10 minutes
• 450 ÷ 10 = 45 words per minute (wpm)

Heart Rate:

• 150 beats in 2 minutes
• 150 ÷ 2 = 75 beats per minute (bpm)

Wages:

• Earned $84 in 7 hours
• $84 ÷ 7 =$12 per hour

---

Complex Comparisons

Example: Which copy shop is cheaper?

• Shop A: 75 copies for $3.75
• Shop B: 120 copies for $5.40

Find unit prices: Shop A: $3.75 ÷ 75 =$0.05 per copy Shop B: $5.40 ÷ 120 =$0.045 per copy

Shop B is cheaper! (by half a cent per copy)

Even small differences add up with large quantities!

---

Unit Rates in Recipes

Scaling recipes uses unit rates!

Example: Recipe uses 6 cups flour for 4 loaves

Unit rate = 6 ÷ 4 = 1.5 cups per loaf

To make 7 loaves: 7 × 1.5 = 10.5 cups of flour

Unit rate helps scale up or down!

---

Different Rate Formats

All these mean the same:

• 60 miles per hour
• 60 mph
• 60 miles/hour
• 60 mi/hr

The "/" means "per"

Example formats:

• $5 per pound =$5/lb
• 30 miles per gallon = 30 mpg
• 65 words per minute = 65 wpm

---

When Unit Rate Has Decimals

Example: 5 pounds for $7

Unit price = $7 ÷ 5 =$1.40 per pound

Money is usually rounded to 2 decimal places (cents)

Example: 3 items for $5

Unit price = $5 ÷ 3 =$1.666...

Round: $1.67 per item (round up for money)

---

Population Density

Population density is people per square mile (or km).

Example: City has 450,000 people in 150 square miles

Density = 450,000 ÷ 150 = 3,000 people per square mile

Higher density = more crowded!

---

Efficiency and Productivity

Work rate: Tasks per hour Fuel efficiency: Miles per gallon Production rate: Items per day

Example: Factory makes 1,200 widgets in 8 hours

Rate = 1,200 ÷ 8 = 150 widgets per hour

Unit rates help measure efficiency!

---

Common Mistakes to Avoid

❌ Mistake 1: Dividing in wrong order

• Wrong: $2 per 8 oz → 8 ÷ 2 = 4
• Right: $2 per 8 oz → 2 ÷ 8 =$0.25 per oz

❌ Mistake 2: Comparing without unit rates

• Can't directly compare "3 for $6" vs "5 for$9"
• Must find unit price first!

❌ Mistake 3: Forgetting units

• Always include units in your answer!
• Not "60" but "60 mph"

❌ Mistake 4: Rounding too early

• Keep extra decimals during calculation
• Round only in final answer

❌ Mistake 5: Choosing higher price

• Lower unit price = better deal
• Don't just look at total cost!

---

Using Unit Rates to Solve Problems

Problem: If 4 pizzas cost $36, how much do 7 pizzas cost?

Step 1: Find unit rate $36 ÷ 4 =$9 per pizza

Step 2: Multiply by new quantity 7 × $9 =$63

Answer: $63 for 7 pizzas

---

Problem-Solving Strategy

To find unit rate:

1. Identify the two quantities
2. Divide first by second
3. Write answer with correct units
4. Check: Does it make sense?

To compare options:

1. Find unit rate for each option
2. Compare the unit rates
3. Choose lower for price, higher for value
4. Consider quality too!

To use unit rate:

1. Find the unit rate
2. Multiply by desired quantity
3. Calculate final answer

---

Quick Reference

Unit Rate Formula: First Quantity ÷ Second Quantity

Common Unit Rates:

• Speed: distance ÷ time
• Unit price: price ÷ quantity
• Wage: money ÷ hours
• Mileage: miles ÷ gallons
• Typing: words ÷ minutes

Comparing:

• Find unit rate for each
• Lower unit price = better buy
• Higher unit value = better performance

Remember: Always include UNITS!

---

Practice Tips

Tip 1: Set up division carefully

• What do you want "per one"?
• That goes in denominator

Tip 2: Check reasonableness

• If car goes 240 miles in 4 hours
• Should be around 60 mph (reasonable)
• Not 960 mph (unreasonable!)

Tip 3: Use calculator for complex division

• Especially with money
• Round money to 2 decimal places

Tip 4: Compare apples to apples

• Same units (both $/oz or both$/lb)
• Same quality if possible
• Consider other factors (freshness, brand, etc.)

---

Summary

Unit rate shows amount per ONE unit:

• Makes comparisons easy
• Found by dividing
• Always include units

Finding unit rates: Unit Rate = First Quantity ÷ Second Quantity

Common uses:

• Shopping (unit price)
• Travel (speed, mileage)
• Work (productivity, wages)
• Sports (stats per game)

Benefits:

• Compare different options
• Make informed decisions
• Calculate scaled quantities
• Understand efficiency

Key skill: Unit rates help you make smart choices in everyday life - from shopping to planning trips to budgeting time!

Mastering unit rates empowers you to analyze and compare rates in the real world!