Finding Price Per Item

A pack of 8 water bottles costs 12 dollars. What is the price per bottle?

Identify the ratio: 12 dollars for 8 bottles = $\frac{12 \text{ dollars}}{8 \text{ bottles}}$

Divide to find unit rate: $12 \div 8 = 1.5$ = 1.50 dollars

Write the unit rate: 1.50 dollars per 1 bottle = 1.50 dollars per bottle

Answer: The unit rate is 1.50 dollars per bottle.

Calculating Speed

A train travels 420 kilometers in 6 hours. What is its average speed?

Set up the ratio: Distance : Time = 420 km : 6 hours = $\frac{420 \text{ km}}{6 \text{ hours}}$

Divide to find unit rate: $420 \div 6 = 70$ = $70$

Express as speed: 70 kilometers per 1 hour = $70$ km/h

Answer: The train's average speed is $70$ km/h.

Comparing Unit Rates

Store A sells 5 notebooks for 8 dollars. Store B sells 3 notebooks for 5 dollars. Which store has the better deal?

Find Store A's unit rate: $8 \div 5 = 1.60$ = 1.60 dollars/notebook

Find Store B's unit rate: $5 \div 3 \approx 1.67$ = 1.67 dollars/notebook

Compare unit rates: $1.60 < 1.67$ = Store A is cheaper

State the answer: Lower price per notebook = better deal = Store A

Answer: Store A has the better deal at 1.60 dollars per notebook (vs 1.67 dollars at Store B).

Using Unit Rates to Solve Problems

A factory produces 840 toys in 12 hours. How many toys will it produce in 5 hours?

Find the unit rate: $840 \div 12 = 70$ = $70$ toys/hour

Multiply by desired time: $70 \times 5 = 350$ = $350$ toys

Verify the answer: $350 \div 5 = 70$ toys/hour (matches!) = Correct

Answer: The factory will produce $350$ toys in 5 hours.

Mistake: Dividing in the wrong order

Why: Students sometimes divide 8 by 12 instead of 12 by 8 when finding price per item.

Correct: Always divide the quantity you want to find (price) by the number of units (items). 12 dollars / 8 items = 1.50 dollars per item.

Mistake: Forgetting the units

Why: Writing just '70' instead of '70 km/h' loses important information.

Correct: Always include both units in your answer: km/h, dollars/item, pages/day, etc.

Mistake: Comparing rates with different units

Why: You cannot directly compare 2 dollars/bottle with 50 cents/ounce.

Correct: Convert to the same units first, then compare the unit rates.

Smart Shopping

Stores often show unit prices on shelf labels, but you can calculate them yourself when they are missing.

A 750ml bottle costs 3 dollars. Unit rate: $3 \div 0.75 = 4$ dollars per liter.

Travel Planning

Speed is a unit rate that helps you plan trips and estimate arrival times.

If you drive at an average speed of 80 km/h, a 320 km trip takes $320 \div 80 = 4$ hours.

Q: What's the difference between a rate and a unit rate?

A: A rate compares two quantities with different units (like 150 km in 3 hours). A unit rate is a special rate where the second quantity is exactly 1 (like 50 km per 1 hour, or 50 km/h).

Q: Why do we use unit rates instead of regular rates?

A: Unit rates make comparison easy. It's hard to compare '120 km in 2 hours' with '200 km in 3 hours', but comparing '60 km/h' with '66.7 km/h' is simple!

Q: How do I know which number to divide by?

A: Divide by the number that you want to make equal to 1. For 'price per item', divide by the number of items. For 'speed per hour', divide by the number of hours.