Calculating unit rate: A step-by-step guide for Grade 7

📚 What is Unit Rate?

A unit rate is a ratio that compares a quantity to one unit of another quantity. Think of it as figuring out the cost of one item or the distance traveled in one hour. It simplifies comparisons, making it easy to see which option offers the best value.

📜 A Brief History

While the concept of ratios and proportions dates back to ancient civilizations like the Egyptians and Babylonians, the explicit use of 'unit rates' as a defined term likely emerged with the growth of commerce and trade. As societies developed more complex economic systems, the need for standardized comparisons of value became increasingly important.

🌟 Key Principles of Calculating Unit Rate

• ➗ Division is Key: The core operation in finding a unit rate is division. You'll divide the quantity you want to measure (like cost or distance) by the number of units.
• 🏷️ Identify the Quantities: Clearly identify what you're comparing. Are you looking at cost per item, distance per hour, or something else?
• 🎯 Set up the Ratio: Express the relationship as a ratio. For example, if something costs $10 for 5 items, the ratio is $10/5 items.
• ➗ Perform the Division: Divide the first quantity (e.g., cost) by the second quantity (e.g., number of items). This will give you the unit rate.
• 🧮 Include Units: Always include the units in your answer. For instance, "$2 per item" or "60 miles per hour".

🧭 Step-by-Step Calculation

1. Step 1: Identify the two quantities you are comparing. For instance, cost and quantity, or distance and time.
2. Step 2: Write the quantities as a ratio (a fraction). Make sure the quantity you want to find the unit rate for is in the numerator (top).
3. Step 3: Divide the numerator by the denominator. This will give you the value for one unit.
4. Step 4: Write your answer with the appropriate units.

🌍 Real-World Examples

Example 1: Grocery Shopping 🛒

A box of 10 apples costs $5. What is the unit price per apple?

Solution:

Unit Price = $\frac{\text{Total Cost}}{\text{Number of Apples}} = \frac{$5}{10} = $0.50$ per apple.

Example 2: Traveling 🚗

You drive 150 miles in 3 hours. What is your average speed (unit rate of miles per hour)?

Solution:

Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{150 \text{ miles}}{3 \text{ hours}} = 50$ miles per hour.

Example 3: Comparing Prices 💰

Brand A offers 6 bottles of juice for $12, while Brand B offers 8 bottles for $15. Which brand offers the better deal?

Conclusion: Brand B offers a slightly better deal at approximately $1.88 per bottle compared to Brand A's $2 per bottle.

📝 Practice Quiz

Calculate the unit rate for the following problems:

1. You earn $48 for working 6 hours. What is your hourly wage?
2. A package of 20 cookies costs $4. What is the cost per cookie?
3. You can type 350 words in 5 minutes. What is your typing speed in words per minute?

Answers:

1. $8 per hour
2. $0.20 per cookie
3. 70 words per minute

✅ Conclusion

Understanding and calculating unit rates is a valuable skill that extends beyond the classroom. From making informed purchasing decisions to understanding speeds and efficiency, the ability to determine unit rates empowers you to make smart choices in everyday life. Keep practicing, and you'll master this essential mathematical concept in no time!