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

The unit rate for ratio (with ) is:

Compute by dividing: → 5 dollars per hamburger

Note: — a unit rate per zero items describes nothing

Same Symbol — Two Operations

Same notation. Different operation.

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 (with , ) produces 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 quantities; time not required
✓ Unit rate = amount of per one ; compute by
✓ Rate language = value + unit of + per + unit of
✓ Every ratio gives two unit rates — ask "per one of WHAT?"

Unlike quantities is the test — not whether time is involved
as a ratio reads "75 to 15"; as a unit rate it equals 5
"5" alone is not a unit rate — "5 dollars per hamburger" is

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