In the previous section we completed the discussion on cylinders. In this section we will see Proportions. We have already had a basic discussion about proportions in Chapter 3. Sections 3, 3.1 and 3.2 gives a basic idea about proportions. We will continue that discussion here.
Consider the three rectangles shown in fig.24.1 below. There is a special relation between all the three rectangles.
To find that relation, we will form a table as shown below:
1. The lengths and width are arranged in columns. One column is assigned for each rectangle in fig.24.1.
2. The l⁄h ratio is taken for each rectangle.
3. We can see that the ratio l⁄h is the same 1.5 for all the rectangles.
4. So all the three rectangles in fig.24.1 belong to one family.
5. Now consider the rectangle in fig.24.2 below. It has length 4.5 cm and height 3.5 cm. Can this rectangle be allowed to join the family?
6. To find that, we take the ratio l⁄h. We get 4.5⁄3.5 = 9⁄7
7. 9⁄7 ≠ 1.5. So the rectangle in fig.24.2 cannot be allowed to join the family in fig.24.1
12⁄h = 1.5 ⇒ h = 12⁄1.5 = 8 cm
• So, a rectangle whose length is 12 cm can be allowed to join the family if it’s height is 8 cm
Another problem: A rectangle has a height of 9 cm. What should be it’s length so that, it can be allowed to join the family.
Solution: Let the length be l. The ratios l⁄9 must be equal to 1.5. So we can write:
l⁄9 = 1.5 ⇒ l = 9 × 1.5 = 13.5 cm
• So, a rectangle whose height is 9 cm can be allowed to join the family if it’s length is 13.5 cm
• So, the length is proportional to height.
• 1.5 is the constant of proportionality.
• First rectangle 3:2
• Second rectangle 1.5:1 ⇒ 3:2
• Third rectangle 6:4 ⇒ 3:2
• We find that in this method also, we get the same ratio for all the members of the family.
■ We can give a name to the family: The '3:2 rectangle family'
■ There are infinite number of rectangles that can be accepted as members of this family
• There are small tvs and big tvs.
• Small tvs have small rectangular screens. Big tvs have big rectangular screens.
• So we have smaller rectangles and larger rectangles.
• But all those rectangles belong to one family. It is the '16:9 rectangle family'. [The ratio 16:9 is accepted all over the world for modern TVs]
■ From this name we can obtain one information:
Whatever be the size of the screen, if we take the ratio length : height, it will be 16:9
• From this we get: l⁄h = 16⁄9 ⇒ l = (16⁄9) h
■ So we can say:
Whatever be the size of the screen, the length will always be 16⁄9 times the height
Let us consider some TV sizes commonly available in the market:
The sales executive at a TV shop points to a TV and says: “It is a 32 inch TV”.
What does he mean? Let us analyse:
1. If we measure the distance from the bottom left corner to the top right corner of the screen, it will be 32 inches. [Note that the measurement should not include the outer frame around the screen]
2. So 32 inches is the diagonal measure of the screen. This is shown in fig.24.3 below:
3. But we want the length and height of the rectangular screen. For that, we need to do some calculations:
4. So, if we see a TV of size 32 inches, it's length is 27.89 inches, and height is 15.69 inches
■ What if it is a 40 inch TV?
The calculations are shown below:
So, if we see a TV of size 40 inches, it's length is 34.86 inches, and height is 19.61 inches
• All the rectangles in fig.24.4 above have length to height ratio 16:9. In other words, in all those rectangles, the length is 16⁄9 times the height. That is., length is always proportional to height.
• They all belong to the '16:9 rectangle family'
• If this proportionality is not maintained, picture will be distorted. We saw an example of such distortions earlier in fig.7.2.
• Also there must be enough space at the sides and rear of the TV for proper ventilation.
• In short, we must examine the instruction manuals carefully before deciding which TV to buy
Consider the three rectangles shown in fig.24.1 below. There is a special relation between all the three rectangles.
 |
| Fig.24.1 |
1. The lengths and width are arranged in columns. One column is assigned for each rectangle in fig.24.1.
2. The l⁄h ratio is taken for each rectangle.
3. We can see that the ratio l⁄h is the same 1.5 for all the rectangles.
4. So all the three rectangles in fig.24.1 belong to one family.
5. Now consider the rectangle in fig.24.2 below. It has length 4.5 cm and height 3.5 cm. Can this rectangle be allowed to join the family?
 |
| Fig.24.2 |
7. 9⁄7 ≠ 1.5. So the rectangle in fig.24.2 cannot be allowed to join the family in fig.24.1
A simple problem: A rectangle has a length of 12 cm. What should be it’s height so that, it can be allowed to join the family.
Solution: Let the height be h. The ratios 12⁄h must be equal to 1.5. So we can write:12⁄h = 1.5 ⇒ h = 12⁄1.5 = 8 cm
• So, a rectangle whose length is 12 cm can be allowed to join the family if it’s height is 8 cm
Another problem: A rectangle has a height of 9 cm. What should be it’s length so that, it can be allowed to join the family.
Solution: Let the length be l. The ratios l⁄9 must be equal to 1.5. So we can write:
l⁄9 = 1.5 ⇒ l = 9 × 1.5 = 13.5 cm
• So, a rectangle whose height is 9 cm can be allowed to join the family if it’s length is 13.5 cm
• For the family of rectangles in fig.24.1, we find that l⁄h is always equal to 1.5. That is., l⁄h = 1.5.
• This can be written as l = 1.5h. So we can say:
• For the family of rectangles in fig.24.1, the length is always 1.5 times the height• So, the length is proportional to height.
• 1.5 is the constant of proportionality.
There is another method of saying the above. Let us see that method:
1. Take the ratio length : height for each rectangle in the family.• First rectangle 3:2
• Second rectangle 1.5:1 ⇒ 3:2
• Third rectangle 6:4 ⇒ 3:2
• We find that in this method also, we get the same ratio for all the members of the family.
■ We can give a name to the family: The '3:2 rectangle family'
■ There are infinite number of rectangles that can be accepted as members of this family
Such families of rectangles are often encountered in real life. The best example is the screen sizes of modern televisions. Let us see them in more detail:
• The screens of modern televisions are rectangles. • There are small tvs and big tvs.
• Small tvs have small rectangular screens. Big tvs have big rectangular screens.
• So we have smaller rectangles and larger rectangles.
• But all those rectangles belong to one family. It is the '16:9 rectangle family'. [The ratio 16:9 is accepted all over the world for modern TVs]
■ From this name we can obtain one information:
Whatever be the size of the screen, if we take the ratio length : height, it will be 16:9
• From this we get: l⁄h = 16⁄9 ⇒ l = (16⁄9) h
■ So we can say:
Whatever be the size of the screen, the length will always be 16⁄9 times the height
Let us consider some TV sizes commonly available in the market:
The sales executive at a TV shop points to a TV and says: “It is a 32 inch TV”.
What does he mean? Let us analyse:
1. If we measure the distance from the bottom left corner to the top right corner of the screen, it will be 32 inches. [Note that the measurement should not include the outer frame around the screen]
2. So 32 inches is the diagonal measure of the screen. This is shown in fig.24.3 below:
 |
| Fig.24.3 |
■ What if it is a 40 inch TV?
The calculations are shown below:
There will be infinite number of rectangles in the '16:9 rectangle family'. But manufacturers have chosen only a few for the TV screens. Those 'chosen few' have become the 'standard'. And that standard is accepted all over the world. The fig.24.4 below shows some of the standard sizes:
 |
| Fig.24.4 |
• All the rectangles in fig.24.4 above have length to height ratio 16:9. In other words, in all those rectangles, the length is 16⁄9 times the height. That is., length is always proportional to height.
• They all belong to the '16:9 rectangle family'
• If this proportionality is not maintained, picture will be distorted. We saw an example of such distortions earlier in fig.7.2.
• While calculating the length and height of a TV in this way, the aspect ratio plays an important role. Though 16:9 is the usual ratio, some manufacturers use 16:10. In such a case, the above values will change
• The above method gives the size of screen only. To check whether a TV will fit into a tight space, the size of frame around the screen should be considered• Also there must be enough space at the sides and rear of the TV for proper ventilation.
• In short, we must examine the instruction manuals carefully before deciding which TV to buy
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