Ratios & Percentages
A 45-Minute Interactive Lesson
By M.RajaRao, MSc, MEd
(0:00 - 0:05) 🚀 Hook: Where's the Math?
Let's start with a quick game. Look at the image from your textbook. It mentions:
- A cycle vs. a scooter
- Students who like math
- A football team's wins
Ratios and percentages are everywhere! They help us compare things (like speeds), understand parts of a whole (like discounts in a store), and track success (like a team's win record).
Today's Goal: Learn to speak this language of comparison.
(0:05 - 0:15) 👨🏫 I Do: Watch Me Solve It
I'll model two key skills using problems from your sheet. Pay close attention to the steps.
Concept 1: Simplifying Ratios
Problem (1a): Ratio of cycle speed (15 km/hr) to scooter speed (30 km/hr).
- Write the ratio: `15 : 30`
- Think of it as a fraction: 1530
- Simplify: Divide both numbers by their greatest common divisor (which is 15).
- `15 ÷ 15 = 1`
- `30 ÷ 15 = 2`
- Final Ratio: 1 : 2
This means for every 1 km/hr the cycle goes, the scooter goes 2 km/hr. The units (km/hr) are the same, so we can compare them directly!
Concept 2: Ratios to Percentages
Problem (2a): Convert the ratio `3 : 4` to a percentage.
- Write as a fraction: 34
- Goal: "Percent" means "per 100". We need to make the denominator (bottom number) 100.
- Convert: (34) * 100%
- Calculate: (3 * 100) / 4 = 3004 = 75
- Final Answer: 75%
So, `3:4` is the same as 75100, which is 75%.
(0:15 - 0:25) 🤝 We Do: Let's Solve Together
Let's try two more. These have small tricks. We'll do them step-by-step.
Problem (1c): Ratio of 50 paise to ₹5
Teacher: What's the trap here? Look at the units.
Student: One is "paise" and one is "rupees". They're different!
Teacher: Exactly! We MUST convert to the same unit. Which is easier?
Student: Let's turn rupees into paise. ₹1 = 100 paise.
Teacher: Great. So, ₹5 = ?
Student: 5 * 100 = 500 paise.
Teacher: Perfect. Now what's our ratio?
Student: `50 : 500`
Teacher: How do we simplify it?
Student: Divide both by 10... `5 : 50`. Then divide both by 5... `1 : 10`.
Final Answer: 1 : 10
Problem (3): 72% of 25 students like math. How many do NOT?
Teacher: This is a 2-step question. What's the first step?
Student: Find out how many students *like* math. So, 72% of 25.
Teacher: How do we write "72% of 25" as math?
Student: (72100) * 25
Teacher: Good. Let's solve it. (72 * 25) / 100 = 1800100 = 18. So, 18 students like math. Are we done?
Student: No, the question asks how many do *NOT* like math.
Teacher: Right! So what's the final step?
Student: Total students (25) minus the ones who like math (18). `25 - 18 = 7`.
Final Answer: 7 students
(0:25 - 0:40) 🧠 You Do: Your Turn!
Time to practice on your own. Try these problems from the sheet. Click the "Show Answer" toggle to check your work when you're done.
Problem 1 (from 2b): Convert the ratio `2 : 3` to a percentage.
Hint: This one will involve a decimal or a fraction!
Show Answer
- Fraction: 23
- Calculate: (23) * 100% = 2003
- Answer: `66.66...%` or 66.7% (rounded) or 6623 %.
Problem 2 (from 4): A team won 10 matches. This was 40% of their total matches. How many matches did they play in all?
Hint: We know the part (10) and the percent (40%). We need to find the whole (total).
Show Answer
- Let 'T' be the Total Matches.
- Equation: `40% of T = 10`
- Math: (40100) * T = 10 or
0.4 * T = 10 - Solve for T: `T = 10 / 0.4`
- Answer: `T = 25`. They played 25 matches in total.
🌟 Challenge Problem (from 1b): Find the ratio of 5 m to 10 km.
Hint: Remember the "We Do" problem. Check your units! (1 km = 1000 m)
Show Answer
- Convert: `10 km = 10 * 1000 m = 10,000 m`
- Ratio: `5 : 10,000`
- Simplify (Divide by 5): `1 : 2000`
- Answer: 1 : 2000
(0:40 - 0:43) 🏁 Conclusion: Key Takeaways
Great work today! Let's review the two big ideas:
- Ratios are Comparisons: Always make sure the units are the same (like paise-to-paise or m-to-m) before you compare and simplify.
- Percentages are Ratios out of 100: To turn *any* fraction or ratio into a percent, just multiply it by 100%.
(0:43 - 0:45) 👇 Self-Assessment: Check Your Understanding
How are you feeling about these topics right now? This helps me know what to review next time. (Just think about your answer!)
1: Confused
"I'm still not sure what a ratio is or how to simplify."
2: Getting It
"I can simplify ratios, but the percentage word problems are tricky."
3: Confident
"I can simplify ratios *and* solve percentage problems."