Rates and Unit Rates

In this lesson:

• Distinguish a rate from a same-unit ratio
• Find the unit rate from any ratio
• Write unit rates using rate language

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Distinguish a rate (ratio of unlike quantities) from a same-unit ratio
2. Define the unit rate as the value of per one of
3. Compute a unit rate from a ratio using a ratio table or division
4. Read and write a unit rate in rate language: "___ per ___"
5. Recognize that every ratio yields two unit rates — and
6. Choose the unit rate that fits a question using: "per one of WHAT?"

Which Is the Better Deal?

A hamburger stand charges 75 dollars for 15 burgers.

A hot dog cart charges 24 dollars for 6 hot dogs.

Which one gives you more food for your dollar — and how would you figure it out?

Ratios You Know vs. Something New

Familiar ratios from last lesson:

• 5 red marbles : 3 blue marbles
• 18 boys : 12 girls (both quantities share the same unit)

New phrases:

• "$3 a pound" → "$0.79 per ounce" → "55 miles per hour"

What is different about the new phrases?

Rates Are Ratios of Unlike Units

A rate is a ratio whose two quantities are measured in different units.

• Marbles to marbles → ratio (same units)
• Dollars to hamburgers → rate (unlike units)
• Miles to hours → rate (unlike units)

The notation stays the same: , "a to b", or

Unlike Units — Time Is Not Required

• "$3 per pound" — no time
• "$0.79 per ounce" — no time
• "30 students per classroom" — no time
• "55 miles per hour" — this one has time

Quick Check: Rate or Ratio?

Classify each:

1. "40 students on 2 buses"
2. "5 red marbles to 3 blue marbles"
3. "60 miles per hour"

Think before the next slide — which are rates and which are ratios?

Rate or Ratio — Answers Revealed

1. "40 students on 2 buses" → rate (unlike units)
2. "5 red to 3 blue marbles" → ratio (same units)
3. "60 miles per hour" → rate (unlike units)

Every rate can collapse to a single per-one number — that's next.

Watch How the Unit Rate Appears

When the hamburger column reads 1, the dollar column reads the unit rate.

The Unit Rate Is Value Per One

Compute by dividing: → 5 dollars per hamburger

Same Symbol — Two Operations

Finding the Unit Rate: Hamburger Stand

Given: 75 dollars for 15 hamburgers.

Step 1: dollars, hamburgers

Step 2:

Step 3: 5 dollars per hamburger

Worked Example: Recipe (Fractional Rate)

Given: 3 cups of flour for 4 cups of sugar.

Find the unit rate of flour per cup of sugar.

Step 1: cups flour, cups sugar

Step 2:

Step 3: cup flour per cup of sugar

Try It: Cookies and Cost

Given: 24 cookies for 6 dollars. Find the cost per one cookie.

1. Identify: ___ dollars, ___ cookies
2. Divide: ___
3. Write with units: ___ per ___

Scale a table if it helps: 24 cookies → 12 → 6 → 1

Cookie Answer: 25 Cents Per Cookie

Unit rate: 25 cents per cookie

Unit rates can be fractions — that is expected and correct.

The Value Needs a Story

You can compute a unit rate:

But 5 by itself isn't the full answer.

A unit rate is a number with a story — and the story lives in the unit label.

Next: how to read and write that story precisely.

Rate Language Has Three Parts

• "5 dollars per hamburger"
• "3/4 cup flour for each cup of sugar"

Rate Language Runs in Both Directions

Numeric → sentence:

• 5 dollars/hamburger → "5 dollars per hamburger"
• 60 mi/hr → "60 miles per hour"

Sentence → numeric:

• "55 miles per hour" →
• "0.79 dollars per ounce" →

Unit Label Is Part of the Answer

Wrong: The unit price is 5.

Right: The unit price is 5 dollars per hamburger.

"5" alone could mean anything. "5 dollars per hamburger" is a unit rate.

Template: value + unit of + per + unit of

Practice: Write and Compute Unit Rates

1. cup flour per cup of sugar → write as a sentence
2. 30 students per classroom → write as a sentence
3. "180 miles in 3 hours" → compute unit rate in mi/hr
4. "48 dollars for 6 hours" → compute unit rate

One Ratio Can Answer Two Questions

75 dollars for 15 hamburgers

Question A: "Cost per one hamburger?"
→ Per one of WHAT? → hamburger → dollars/hamburger

Question B: "Hamburgers per one dollar?"
→ Per one of WHAT? → dollar → hamburger/dollar

Every Ratio Produces Two Unit Rates

• Every ratio () yields two unit rates

• dollars per hamburger
• hamburger per dollar

"Per One of WHAT?" Picks the Direction

To choose which unit rate to compute:

1. Read the question carefully
2. Ask: "Per one of WHAT?"
3. That quantity is the denominator
4. The other quantity is the numerator

Example: "How much does one hamburger cost?"
→ Per one of WHAT? → hamburger → denominator = hamburgers

Predict: Which Direction for Flour?

Recipe ratio: 3 cups flour : 4 cups sugar

"Find the unit rate of flour per cup of sugar."

Predict first — which is it?

• A. cup flour per cup of sugar
• B. cups sugar per cup of flour

Choose A or B before advancing.

Recipe: Both Unit Rates Derived

Ratio: 3 cups flour : 4 cups sugar

Both valid — the question picks which.

Find and Fix the Error

A student wrote:

"75 dollars for 15 hamburgers — the unit rate is dollars per hamburger."

What went wrong? Fix it.

Apply "per one of WHAT?" — what is the question actually asking?

Find Both Unit Rates: Lawn Mowing

A landscaper mows 4 lawns in 7 hours.

Find two unit rates. For each one:

1. State the question it answers
2. Compute the unit rate
3. Write the answer in rate language

No scaffolding — complete the full procedure.

What You Now Know How to Do

✓ Rate = ratio of unlike units; time not required
✓ Unit rate = value per one; compute by a ÷ b
✓ Rate language: value + "per" + unit name
✓ Two unit rates per ratio — ask "per one of WHAT?"

Next: Unit Rates Power Problem-Solving

In 6.RP.A.3, you'll use unit rates to solve problems:

• "15 hamburgers cost 75 dollars — how much do 8 cost?"
  → 5 dollars/hamburger × 8 = 40 dollars
• "240 miles in 4 hours — how far in 7 hours?"
  → Find the rate, then multiply