Hello, in this lesson we are going to study ratio and proportion .
First, let’s start with the “Ratio and Proportion”. While we are learning this topic, we need to tell that ratio and proportion are different from each other. Let’s start with the Ratio and say that we are going to study proportion later on in this lesson.
So, what is the meaning of the word “Ratio”? To explain the word “Ratio” let’s remember the sets, and mention that sets have elements. As for that ratio means to compare the elements of two sets. We use the ratio to compare their greatness, quantities, and amounts. Now let’s clarify ratio with an example. lets say I have two apples and my sister has 3 apples. If we compare my apples to my sister’s apples, we will have the ratio of two to three. We can think ratio as a fraction. So, my apples over my sister’s apples, 2 over 3, gives us the ratio of two to three.
While we are showing ratio as a writing, we write A colon B or A fraction line B. If we go through our example, 2 colon 3, or 2 division line 3. We read this expression as the ratio of A to B. Again if we go through our example, we read as the ratio of 2 to 3.
Now, let’s study how we get the ratio of 2 to 3. For example; why we get the ratio as 2 to 3 instead of 3 to 2? Because in the question of our example it’s written that “if we compare my apples to my sister’s apples.” If we compare my apples to my sister’s the first number needs to be the number of my apples, and the second number needs to be the number of my sister’s apples. If our example said us to compare my sister’s apples to my apples first we would write the number of my sisters apples then we write the number of my apples, and our answer would be the ratio of 3 to 2. So, the answers would be different from each other.
We call the two numbers, which we use while writing the ratio, terms. For example the numbers 2 and 3 are the terms, which we used here as number of apples. When we say the ratio of A to B. A and B are the terms of the ratio. Terms are separated as first term and second term. If we talk about the ratio of A to B, the first term is A, which is the first quantity. And, B is the second term, which is the second quantity. While we are comparing my apples to my sister’s apples; number of my apples is the first term and the number of my sister’s apples is the second term. Like we said before comparing my apples to my sister’s apples and comparing my sister’s apples to my apples are going to give different results. So, we need to be careful about which quantity is which term.
Ratios are separated into two: “unit rate” and “dimensionless ratio”. Unit Rate is the ratio that we get when we compare two quantities, which have different units. For example; one person walks 10 meters in 1 minute, if we compare the length of the road that this person can walk with the time, we will get the result as the ratio of 10 meters to one minute, we won’t find it as the ratio of 10 to 1. If we think this process as a fraction: 10 over 1 equal 10, so our result is 10. To get the unit of our result; we divide the first quantity’s unit by the second quantity’s unit. So, the answer of our example is 10 meters/minutes. And, this means 10 meters per minute. So, that person can walk 10 meters per minute.
On the other hand dimensionless ratio is the ratio that we get when we compare two quantities, which have the same unit. For Example; John has six pencils, and Emily has 12 pencils. If we compare John’s pencils with Emily’s pencils, our result will be the ratio of 6 to 12 or 6 over 12. To get the unit of our result we divide the first quantity’s unit with the second quantity’s unit. But, because of pencil to pencil is the same unit; we won’t have any units. So, our result will be dimensionless ratio without any unit.
So, let’s talk about the properties of the ratio. Ratio can be simplified and extended. While we are simplifying we divide the both terms by the same number. To divide the both terms by the same number we need to find their greatest common divisor. If we go through our last example, in other words if we want to simplify the ratio of 6 to 12; first we need to find their biggest common divisor which is 6 here. 6 over 6 equal 1 and 12 over 6 equal 2. So, when we simplify the ratio of 6 to 12 we get the ratio of 1 to 2. While we are solving problems, we use the method of simplifying to make the problems easier.
While we are extending the ratio we multiply the both terms with the same number. For example, if we want to extend the ratio of 6 to 12 with 3, we do those calculations: 6 multiplied by 3 equal 18 and 12 multiplied by 3 equal 36. So, when we extend the ratio of 6 to 12 with 3 we find the ratio of 18 to 36. We use this property in some questions. For example, when we want to find the length that coincides with the 4 cm length in a 1 over 200-scaled map, we extend our ratio. The scale of this map is 1 over 200, so it is the ratio of 1 to 200. This map tells us that 1 cm equals 200 cm in real life. When we are given the length of 4 cm, to find what coincide with this length in real life, we extend the ratio by 4. 1 multiplied by 4 equal 4, 200 multiplied by four equal 800. With these calculations we get the ratio of 4 to 800. So, the length of 4 cm in this map coincides with 800 cm in real life.
Now let’s talk about “Proportion”. Proportion is the equality of two ratios. For example the ratio of 2 to 6 and the ratio of 3 to 9 are proportional with each other. To clarify that these two ratios are proportional with each other, we use the properties of ratio, which is simplifying and extending. When we simplify the ratio of 2 to 6 by 2 we get the result as the ratio of 1 to 3. And, when we simplify the ratio of 3 to9 by 3. Here we get the result as the ratio of 1 to 3 too. To use the extending property we find the smallest common factor of the second term of the both ratios. The smallest factor of the 6 and 9 is 45. We extend the ratio of 2 to 6 by 9 to make 6 equal to 45. And then, we find the ratio of 18 to 45. After that, we extend the ratio of 3 to 9 by 6 to make 6 equal to 45. And again we find the ratio of 18 to 45. If we simplify or extend our ratios we can see that the ratio is always the same. So, we can say that these two ratios are equal to each other. And, as we said that the equality of the two ratios means proportion so these ratios are proportional.
Let’s think about we have to ratios, which are the ratio of A to B and the ratio of C to D. And if we say that the ratio of A to B and the ratio of C to D is equal to each other it is a proportion. We can get from this proportion that A multiplied by D and B multiplied by C is equal. How did we get that? Of course by using the cross multiplication or cross product. We can use the cross product when we write our ratios as A divided by B equal to C divided by D. so if we perform the cross product; we ll multiply A by D and B by C. And, we can also use the cross multiplication when we wrote our ratios as A colon B and C colon D. If we write this ratios’ proportion, we write as A colon B equals C colon D. In this form of the proportion A and D are the terms that remain outside. And, again in this form of proportion B and C are the terms, which stay inside of the proportion; they are the numbers that stands near the equal sign. When we multiply these terms, we call it cross multiplication. So, when we do cross multiplication, we do these calculations: A multiplied by D is equal the result of B multiplied by C. Both methods give us the same answer. For example, let’s use the ratios 3 over 4 and 6 over 8. When we multiply the terms, which remain outside, 3 and 8 we get 24. And, when we multiply the terms, which stay inside, 6 and 4 we get 24 too. As a result, multiplication of first ratio’s first term and second ratio’s second term is equal to multiplication of the first ratio’s second term and second ratio’s first term.
Properties of Proportion:
1) First property of proportion is, if terms of both proportion change places with one another, the result is same. For example, if we write 4 divided by 3 instead of 3 divided by 4, and 8 divided by 6 instead of 6 divided by 6; external terms will be 6 and 4, and internal terms will be 3 and 8. But if we do cross-multiplication, the terms will still be equal, as a result of first property.
2) Second property of proportion; is that if internal terms change places with one another, ratios will again form a proportion. The internal terms are 4 and 6 in 3 divided by 4 and 6 divided by 8. If internal terms change places, ratios will be 3 divided by 6 and 4 divided by 8. But again, internal terms will be 6 and 4, and external terms will be 3 and 8. If we do cross-multiplication our result will still be the same, so they form a proportion as a result of second property.
3) Third property of proportion is that if external terms change places with one another, ratios will still form a proportion. The external terms are 3 and 8 in 3 divided by 4 and 6 divided by 8. If external terms change places, ratios will be 8 divided by 4 and 6 divided by 3. But external terms will still be 8 and 3, and internal terms will be 4 and 6. If we do cross-multiplication our result will still be the same, so they form a proportion as a result of third property.
Now, let’s study the sorts of proportions. Proportion has 2 sorts;
1) First sort is direct proportion. In direct proportion if two of the values increase, then other one will increase, too. Or if two of the values decrease, other one will decrease also. For example, if one person walks 2 meters in 6 minutes, this person walks 4 meters in 12 minutes or 6 meters in 18 minutes. Those variables are direct proportions as we can see they increase or decrease with same ratios.
2) Second sort is inverse proportion. In inverse proportion if two of the values increase, then other will decrease. Or if two of the values decrease, other one will increase. As long as number of workers increases, time will decrease due to cooperation. If number of workers decreases 2 times, time will increase 4 times.
