Hello there, young mathematicians! Welcome to today's lesson on Ratio and Proportion. I'm so glad you've joined me. Today, we're going to explore how we compare quantities, how to work with ratios, what proportion means, and how to use something called the unitary method to solve real-life problems. Let's dive right in!
Let us begin with a simple question. How do we compare things in everyday life? Sometimes we say one thing is greater than another, or less than another. Sometimes we say something is double, or half. There are two main ways we compare quantities.
The first way is by subtraction. If Peter is 15 years old and Devgan is 10 years old, we can say Peter is 15 minus 10, which is 5 years older. This tells us the difference between them.
The second way is much more powerful. It's called comparison by division. Imagine a boy weighs 22 kilograms, and his father weighs 66 kilograms. If we divide 66 by 22, we get 3. This tells us the father's weight is 3 times the son's weight. This comparison by division is what we call a ratio.
Here is the precise definition. The ratio of two quantities, both of the same kind and in the same unit, obtained by dividing one quantity by the other, is called their ratio.
We write the ratio of x to y as x : y, or as x/y. We read this as "x is to y."
Now, two very important conditions must be satisfied. First, both quantities must be of the same kind. You cannot find the ratio of length to mass. That would be meaningless. Second, both quantities must be in the same unit. If one length is 60 centimetres and another is 1.5 metres, you must convert both to centimetres or both to metres before finding their ratio.
Let me show you an example. Find the ratio of 5 grams to 15 kilograms. First, convert 15 kilograms to grams. That gives us 15,000 grams. So the ratio becomes 5 to 15,000, which simplifies to 1 : 3000 after dividing both terms by 5.
Here's another one. The ratio of 800 centimetres to 1.2 metres. Convert 1.2 metres to centimetres, giving 120 centimetres. So we have 800 to 120, which simplifies to 20 : 3 after dividing both terms by 40. We divide 800 by 40 to get 20, and 120 by 40 to get 3.
Let me introduce some important terms. In any ratio, the first number is called the antecedent, and the second number is called the consequent. In the ratio 1 to 4, 1 is the antecedent and 4 is the consequent.
Here are four key properties you must remember. First, if you multiply both terms of a ratio by the same non-zero number, the ratio stays the same. Second, a ratio must always be expressed in its lowest terms. Third, and this is crucial, a ratio has no unit. The ratio of 15 kilometres to 20 kilometres is 3 to 4, not 3 to 4 kilometres. Fourth, the order matters. The ratio of 5 to 8 is completely different from the ratio of 8 to 5.
Now, how do we simplify ratios that involve fractions or mixed numbers? There are two methods.
Method one: divide the first term by the second term. For 3 and a half to 2 and a third, write this as 7 halves to 7 thirds. Dividing 7 halves by 7 thirds gives 7 halves times 3 sevenths, which equals 21 fourteenths, or 3 halves, or 3 : 2 after simplifying.
Method two: multiply both terms by the LCM of their denominators. The LCM of 2 and 3 is 6. So 7 halves times 6 gives 21, and 7 thirds times 6 gives 14. The ratio 21 is to 14 simplifies to 3 : 2.
For ratios with three terms, like 2 thirds to 4 fifths to 1 half, multiply all terms by the LCM of 3, 5, and 2, which is 30. This gives 20 to 24 to 15. These numbers have no common factor, so this is the simplest form.
Sometimes we need to divide a quantity according to a given ratio. Suppose 12 sweets are to be divided between A and B in the ratio 1 is to 3. The total parts equal 1 plus 3, which is 4. So A gets 1 fourth of 12, which is 3 sweets. B gets 3 fourths of 12, which is 9 sweets.
Here's a more advanced example. A pole of 165 centimetres is divided in the ratio 7 is to 8. Total parts equal 15. The shorter part is 7 fifteenths of 165, which equals 7 times 11, or 77 centimetres. The longer part is 8 fifteenths of 165, which equals 8 times 11, or 88 centimetres.
We can also use algebra. Let the parts be 7x and 8x. Then 7x plus 8x equals 165, so 15x equals 165, and x equals 11. The parts are 77 and 88 centimetres.
Now let us move to proportion. When two ratios are equal, we say they form a proportion.
Consider this. 96 boys to 144 girls gives a ratio of 2 is to 3. 180 rupees to 270 rupees also gives a ratio of 2 is to 3. Since these ratios are equal, we write 96 is to 144 equals 180 is to 270. This equality of two ratios is called a proportion.
We read this as "96 is to 144 as 180 is to 270." The symbol for proportion is two colons, written as 96 : 144 :: 180 : 270.
In any proportion, the first and fourth terms are called the extremes. The second and third terms are called the means. A fundamental property is that the product of the extremes equals the product of the means.
For example, if 8, x, 9, and 36 are in proportion, meaning 8 is to x as 9 is to 36, then by the property of proportion, 8 times 36 equals x times 9. So x equals 8 times 36 divided by 9, which equals 8 times 4, or 32.
Finally, let us explore the unitary method. This is a powerful technique where we first find the value of one unit, then use it to find any required quantity.
Suppose 12 pens cost 960 rupees. We need to find the cost of 5 pens. First, find the cost of one pen: 960 divided by 12 equals 80 rupees. Then, the cost of 5 pens is 80 times 5, which equals 400 rupees.
Here is another situation. Six identical articles weigh 2.4 kilograms. We need to find the weight of 8 such articles. One article weighs 2.4 divided by 6, which is 0.4 kilograms. So 8 articles weigh 0.4 times 8, which is 3.2 kilograms.
This method works because of direct variation. When two quantities increase or decrease together, they are in direct variation. More articles means more cost. Fewer articles means less cost.
Let me show you a practical application. A car covers 192 kilometres in 3 hours. How far will it go in 5 hours at the same speed? First, the distance in one hour is 192 divided by 3, which equals 64 kilometres. Then, in 5 hours, it covers 64 times 5, which is 320 kilometres.
Let me quickly recap the key takeaways from today's lesson.
First, a ratio compares two quantities of the same kind by division, and both quantities must be in the same unit.
Second, a ratio has no unit, must be in lowest terms, and the order of terms matters.
Third, a proportion is an equality of two ratios, and in any proportion, the product of extremes equals the product of means.
Fourth, the unitary method involves finding the value of one unit first, then scaling to find the required quantity.
Fifth, quantities in direct variation increase or decrease together.
And sixth, always convert quantities to the same unit before finding ratios or applying the unitary method.
You've done wonderfully following along with ratio and proportion today. These concepts appear everywhere in mathematics and in real life, from cooking recipes to speed calculations to financial planning. Keep practising, stay curious, and remember that mathematics is all about understanding relationships between quantities. Until next time, happy learning!