Hello students, welcome to today's mathematics lesson. I am so happy to be here with you to learn about something really interesting and useful – proportional reasoning. This is Chapter 7 of your Ganita Prakash textbook, and I promise you that by the end of this lesson, you will see proportional relationships everywhere in your daily life – in the food you cook, in the prices you see in the market, in the way buildings are constructed, and even in the pictures you click with your phone!
So students, let's begin by understanding what proportionality really means through something very familiar to all of us – digital images.
You must have noticed that when you take a photo on your phone, you can make it bigger or smaller. Sometimes you rotate it, sometimes you stretch it. Now, look at these five images of a tiger that we have here. Images A, C, and D look similar to each other, don't they? They look like the same picture, just resized. But images B and E look a bit strange – the tiger in image B looks stretched out, long and thin, while in image E it looks squashed, short and fat. Why does this happen? Let's measure these images to find out.
Here are the measurements. Image A is 60 millimetres wide and 40 millimetres tall. Image B is 40 millimetres wide and 20 millimetres tall. Image C is 30 millimetres wide and 20 millimetres tall. Image D is 90 millimetres wide and 60 millimetres tall. And image E is 60 millimetres wide and 60 millimetres tall – it's a square!
Now students, compare image A with image C. What do you notice? The width of C is 30, which is exactly half of 60. The height of C is 20, which is exactly half of 40. So both the width and the height have been multiplied by the same factor – 1/2 or half. That's why they look similar!
Now compare image A with image B. The width changed from 60 to 40 – that's a decrease of 20 millimetres. The height changed from 40 to 20 – that's also a decrease of 20 millimetres. The absolute change is the same, but look: the height became half (20 is half of 40), but the width did not become half (40 is not half of 60). So the width and height changed by different factors. That's why image B looks distorted!
Now what about image D compared to image A? Width changed from 60 to 90 – that's multiplied by 3/2 or 1.5. Height changed from 40 to 60 – that's also multiplied by 3/2 or 1.5. Same factor! So image D looks similar to A, just bigger.
And image E? It's 60 by 60. The width is the same as A, but the height is 60 instead of 40. So the factor for width is 1, but for height it's 1.5 – different factors! That's why it looks different.
So students, here's the key idea: when two quantities change by the same factor, we say the changes are proportional. This is the heart of proportional reasoning!
Now, let's talk about how we express this mathematically. We use something called a ratio. The ratio of width to height of image A is 60 : 60. Wait, no – it's 60 : 40, because width is 60 and height is 40. In a ratio like a : b, we say that for every 'a' units of the first quantity, there are 'b' units of the second quantity. So for image A, for every 60 millimetres of width, there are 40 millimetres of height.
The numbers 60 and 40 are called the terms of the ratio. Simple, isn't it?
Now, let's look at the ratios for our similar images. Image A is 60 : 40. Image C is 30 : 20. Image D is 90 : 60. Are these ratios related to each other? Yes! If you multiply both terms of 60 : 40 by 1/2, you get 30 : 20. If you multiply both terms of 60 : 40 by 3/2, you get 90 : 60. The terms changed by the same factor. That's what makes them proportional!
Now students, here's a very important point. We can check if two ratios are proportional by reducing them to their simplest form and comparing. The simplest form of a ratio is obtained by dividing both terms by their highest common factor, or HCF.
For image A, the ratio is 60 : 40. What is the HCF of 60 and 40? It's 20. Dividing both terms by 20, we get 3 : 2. That's the simplest form.
For image D, the ratio is 90 : 60. The HCF of 90 and 60 is 30. Dividing by 30, we get 3 : 2 again! So both simplify to 3 : 2. They are proportional!
What about image B? Its ratio is 40 : 20. The HCF is 20, so simplest form is 2 : 1. Image E is 60 : 60, which simplifies to 1 : 1. These are not the same as 3 : 2, so they are not proportional to images A, C, and D.
So students, here's the definition: when two ratios are the same in their simplest form, we say that the ratios are in proportion, or that the ratios are proportional. We use the symbol '::' to indicate this. So we write 60 : 40 :: 30 : 20, which means the ratio 60:40 is proportional to the ratio 30:20. Similarly, 60 : 40 :: 90 : 60.
Let me recap what we've learned so far: We started with the idea of similar images, where the width and height change by the same factor. Then we learned about ratios – a way to compare two quantities. And finally, we learned that when two ratios have the same simplest form, they are proportional. Great job!
Now let's practice this with some examples. This will help solidify your understanding.
Example 1: Are the ratios 3 : 4 and 72 : 96 proportional?
3 : 4 is already in its simplest form. For 72 : 96, we find the HCF of 72 and 96, which is 24. Dividing both terms by 24, we get 3 : 4. Since both simplify to 3 : 4, they are proportional. Simple, right?
Example 2: This is a practical problem about making lemonade. Kesang made 6 glasses of lemonade and added 10 spoons of sugar. Her father asked her to make 18 more glasses. How much sugar should she add to keep the same sweetness?
The key is that the ratio of glasses to sugar must remain the same. The ratio is 6 : 10. We need to find a ratio proportional to this where the first term is 18. How do we find the factor of change? We divide 18 by 6, which gives us 3. So the first term increased by a factor of 3. The second term must also increase by the same factor. 10 multiplied by 3 is 30. So the ratio becomes 18 : 30. Therefore, she should add 30 spoons of sugar. The ratio 6 : 10 :: 18 : 30 shows they are proportional.
Example 3: Nitin was building a 60-foot wall using 3 bags of cement. Hari was building a 40-foot wall using 2 bags of cement. Nitin worried that Hari's wall would be weaker because he used less cement. Is Nitin right?
We need to compare the ratio of length to cement used. For Nitin, it's 60 : 3, which simplifies to 20 : 1. For Hari, it's 40 : 2, which also simplifies to 20 : 1. Both ratios are the same in simplest form, so they are proportional! The walls are equally strong. Nitin need not worry at all!
Example 4 is an activity for you to try in your school – count the teachers and students, find the ratio, and compare it with the ratio in my school which is 5 : 170.
Example 5 asks you to measure your classroom blackboard and find its width to height ratio, then draw a similar rectangle in your notebook.
Example 6 is very interesting. When Neelima was 3 years old, her mother was 10 times her age, so 30 years old. The ratio is 3 : 30, which simplifies to 1 : 10. When Neelima is 12 (9 years later), her mother would be 39 (30 plus 9). The ratio becomes 12 : 39, which simplifies to 4 : 13. This is different from 1 : 10! So the ratio changes when we add the same number to both terms. Students, this is very important: adding or subtracting the same number from the terms of a ratio does not keep it proportional!
Example 7: We need to find missing numbers in ratios proportional to 14 : 21. For the first one, we have ____ : 42. Since 42 is 2 times 21, the first term should also be 2 times 14, which is 28. So it's 28 : 42. For the second, 6 : ____ . We need to find what factor multiplies 14 to give 6. That's 6/14 = 3/7. So we multiply 21 by 3/7, which gives 9. So it's 6 : 9. For the third, 2 : ____ . We can see that 14 divided by 7 gives 2, so we divide 21 by 7 to get 3. So it's 2 : 3.
Now students, I want to tell you about something really interesting – filter coffee! In South India, filter coffee is very popular. Manjunath, a coffee shop owner, usually mixes 15 millilitres of coffee decoction with 35 millilitres of milk. That's a ratio of 15 : 35. If he wants stronger coffee, he mixes 20 mL of decoction with 30 mL of milk – that's 20 : 30. Why is this stronger? Because there's more coffee relative to milk! For lighter coffee, he mixes 10 mL of coffee with 40 mL of milk – that's 10 : 40. Less coffee, more milk, so it's lighter!
The chapter also gives you a table to practice: for 300 mL coffee with 600 mL milk, is it regular, strong, or light? Remember, we compare with the regular ratio of 15 : 35, which simplifies to 3 : 7 or approximately 0.43. For 300 : 600, that's 1 : 2 or 0.5 – more milk relative to coffee, so lighter! You'll fill in the rest.
Now, there's a section called "Figure it Out" with practice problems. Let me explain a few of these.
For problem 1, we need to check which pairs of ratios are proportional using cross multiplication. Remember, for a : b :: c : d, the condition is a × d = b × c. Let's check (i) 4 : 7 :: 12 : 21. Is 4 × 21 = 7 × 12? 4 × 21 is 84, and 7 × 12 is also 84. So yes, they are proportional! You can check the others similarly.
For problem 2, we need to give three ratios proportional to 4 : 9. We can multiply both terms by any number. For example, 8 : 18 (multiply by 2), 12 : 27 (multiply by 3), 16 : 36 (multiply by 4). All these simplify back to 4 : 9.
Now students, I want to tell you about something fascinating from ancient Indian mathematics. The problems we've been solving – where we have three quantities and need to find the fourth – were known in ancient India as "Rule of Three" problems, or "Trairasika" in Sanskrit. The great mathematician Aryabhata, who lived around 499 CE, described how to solve these problems!
In the Rule of Three, there are three given numbers: the pramāṇa (measure or standard), the phala (fruit or result), and the icchhā (requisition or desired quantity). We need to find the icchhāphala (the desired result). Aryabhata's rule was: multiply the phala by the icchhā and divide by the pramāṇa. In our notation, if a : b :: c : d, then d = (b × c) / a. This is exactly what we do when we cross-multiply: a × d = b × c, so d = (b × c) / a.
This method was used in India for centuries to solve practical problems of trade, measurement, and everyday life. Isn't it wonderful that we are learning something that has been practiced in India for over 1500 years?
Let's see Example 9 with this understanding. A car travels 90 kilometres in 150 minutes. How far will it travel in 4 hours?
First, we need to convert 4 hours to minutes – that's 240 minutes. So we have 150 : 90 :: 240 : ? Using cross multiplication: 150 × x = 240 × 90, so x = (240 × 90) / 150 = 21600 / 150 = 144. So the car travels 144 km in 4 hours. Notice how we converted units first – that's very important!
Example 10 compares tea prices. In Himachal, 200 grams costs Rs 200, so the ratio is 200 : 200, which is 1 : 1. In Meghalaya, 1 kg (which is 1000 grams) costs Rs 800, so the ratio is 1000 : 800, which simplifies to 5 : 4. These are not proportional! Which tea is more expensive? To compare, we find the price per kg. In Himachal, 200g costs Rs 200, so 1kg costs Rs 1000. In Meghalaya, 1kg costs Rs 800. So the Himachal tea is more expensive!
Now there's a note about a problem that cannot be solved by the Rule of Three. Puneeth's father goes from Lucknow to Kanpur at 50 km/h and takes 2 hours. If he drives at 75 km/h, how long will it take? At first glance, you might think we can set up 50 : 2 :: 75 : x. But this doesn't work! When speed increases, time decreases. This is an example of inverse proportion, which you'll learn about in the next chapter. For now, remember that the Rule of Three applies only when the relationship is direct proportion.
Now let's learn about sharing in given ratios. This is very useful in real life, like splitting money or sharing things among friends.
Activity 3 shows us how to share 12 counters in different ratios. If we share equally, each gets 6, so the ratio is 6 : 6, which simplifies to 1 : 1. If one person gets 5, the other gets 7, so the ratio is 5 : 7.
Now, if we want to share in the ratio 3 : 1, how do we do it systematically? We need to divide 12 into parts that are in the ratio 3 : 1. The total number of parts is 3 + 1 = 4. Each part is 12 ÷ 4 = 3. So one person gets 3 × 3 = 9 counters, and the other gets 3 × 1 = 3 counters. That's 9 : 3, which is the same ratio as 3 : 1!
What if we have 42 counters and want to share in the ratio 4 : 3? Total parts = 4 + 3 = 7. Each part = 42 ÷ 7 = 6. So one person gets 4 × 6 = 24, and the other gets 3 × 6 = 18. That's 24 : 18, which is proportional to 4 : 3!
In general, students, if you want to divide a quantity x in the ratio m : n, then the first part is m × x / (m + n) and the second part is n × x / (m + n).
Let's see Example 11. Prashanti invested Rs 75,000 and Bhuvan invested Rs 25,000. They made a profit of Rs 4,000 and want to share it in the ratio of their investments. The investment ratio is 75000 : 25000, which simplifies to 3 : 1. Total parts = 4. Each part = 4000 ÷ 4 = 1000. So Prashanti gets 3 × 1000 = Rs 3,000 and Bhuvan gets 1 × 1000 = Rs 1,000.
Example 12 is trickier. A 40 kg mixture has sand and cement in the ratio 3 : 1. So sand = 3/4 × 40 = 30 kg, cement = 1/4 × 40 = 10 kg. We want to add some cement to make the ratio of sand to cement become 5 : 2. The sand stays at 30 kg. If the new ratio is 5 : 2, then for every 5 parts sand, there are 2 parts cement. So cement should be 2/5 × 30 = 12 kg. We already have 10 kg, so we need to add 2 kg more.
Now students, there's one more important topic – unit conversions. Proportional reasoning often requires converting between units. Here are some important ones:
For length, 1 metre equals 3.281 feet. For area, 1 square metre equals 10.764 square feet, 1 acre equals 43,560 square feet, 1 hectare equals 10,000 square metres or 2.471 acres. For volume, 1 millilitre equals 1 cubic centimetre, and 1 litre equals 1000 mL or 1000 cc. For temperature, Fahrenheit = 9/5 × Celsius + 32, and Celsius = 5/9 × (Fahrenheit – 32). For example, 25°C equals 77°F.
There are several practice problems at the end that ask you to use unit conversions and proportional reasoning together. Work through them carefully, making sure to convert units before setting up your proportion.
Now let's review everything we've learned in this chapter:
First, we learned about observing similarity in change – when two quantities change by the same factor, the change is proportional. We saw this with the images example.
Then we learned about ratios. A ratio a : b tells us that for every 'a' units of the first quantity, there are 'b' units of the second quantity. The numbers a and b are called the terms of the ratio.
We learned how to simplify ratios by dividing both terms by their HCF, and how to check if two ratios are proportional by comparing their simplest forms.
We learned the definition of proportion: when two ratios are the same in their simplest form, they are proportional. We use the symbol '::' to denote proportion.
We practiced solving many problems using proportional reasoning, including the lemonade problem, the wall-building problem, and the tea pricing problem.
We learned about the ancient Indian Rule of Three, or Trairasika, which was used to solve proportion problems, and we saw how Aryabhata described this method over 1500 years ago.
We learned how to share a quantity in a given ratio – by dividing the total into parts based on the ratio, finding the size of each part, and then multiplying.
And finally, we learned about unit conversions and how they're essential for solving proportional reasoning problems correctly.
Students, proportional reasoning is one of the most important mathematical skills you'll ever learn. It helps you compare quantities, solve real-world problems, and understand relationships in science, economics, and everyday life. Keep practicing, and you'll get better and better at it!
That's the end of today's lesson on Chapter 7: Proportional Reasoning-1. Thank you for listening attentively. Keep practicing the problems, and I'll see you in the next lesson!