In the previous section we saw some solved examples illustrating the application of direct proportions. In this section we will see some more examples.
Solved example 3.3
In the fig. given below, the tall red pole of height 12 m, cast a shadow of length 8 m. At that same time a nearby shorter blue pole, cast a shadow of length 6m. Find the height of the blue pole.
Solution:
This is a question of direct proportion. The direct proportion between two quantities: (1) Height of pole and (2) Length of shadow. Any of these two quantities can be taken as x, and the other as y. In all the problems that we did so far, there were more than one set of values for x and y. So we calculated y/x ratio in each case. If y/x is a constant, then we could confirm that the two quantities concerned, are in direct proportion, and that they satisfy the equation y = kx.
But in this problem, only one set of values is given. We can take the ratio y/x for that set. But how do we know that it is a constant?
By looking at the fig., we can see that, if the height increase, the length of shadow will increase. The reverse is also true: If the height decrease, the length of shadow will decrease. [But the position of the sun should remain the same. In the question, it is given that both the shadows are measured at the same time]. So this is a case of direct proportion. We can proceed as usual. The table is shown below:
The y/x ratio is also shown in the same table. The length of 6 m for the shadow, and the corresponding height x1 of the pole should alos have appropriate place in the table. And they must also give the same constant 0.67. So we have
6/x1 = 0.67 ⇒ x1 = 6/0.67 = 8.96 m
Solved example 3.4
At a construction site, different numbers of workers were present on different days. The supervisor recorded the number of workers present on each day and the total wages given to them on that day. The following table is taken from his record book:
Check whether the two quantities: (1) No. of workers present on a day and (2) The total wages on that day, are in direct proportion.
Solution:
The data is already in a tabular form. Next step is to calculate the y/x ratios in each of the columns from (iii) to (viii). For convenience, the ratios can be written in the same table in the corresponding columns. This is shown below:
We find that the resulting ratios are not constant. We have two properties to satisfy: ‘direct’ and ‘proportional’. In our present case:
• the changes are direct because, when the no. of workers increase, the total wages increase. And, when no. of workers decrease, the total wage decrease.
• But it is not proportional because, the ratio is not a constant. We cannot obtain ‘y’ (the total wages) by multiplying ‘x’ (the number of workers) with a constant.
So the two quantities are not in direct proportion. The reason for this can be stated as follows: The total number of workers present on any day will consist of two categories: ‘Masons’ who are skilled workers, and ‘Assistants’ who are unskilled workers’. Masons have a higher wage than assistants. So the total wages on a day will depend on the number of masons and assistants present on that day.
The supervisor should have written the number of masons and assistants separately for each day. Then, a person who looks at the records at a later date would get a clear understanding. However such incomplete tables are recorded on rare occassions for ‘quick reference’. And for us, such an incomplete table is a good example to compare proportionality and non-proportionality.
Another example for non-proportionality:
Consider the Height and Weight of a person. As his height increases, weight also increases. So the change is ‘direct’. But there is no constant by which we can multiply the height to get the weight. So the two quantities are not proportional.
Some times it may so happen that at a larger value of height, the weight has become lower. This may be due to special physical training, special diet, or even some illness. In such a case, the change is neither ‘direct’ nor ‘proportional’.
Solved example 3.5
A stack of bricks weigh 32.4 kg. There are 12 bricks in the stack. If another stack weigh 43.2 kg, how many bricks does it have?
Solution:
This is a case of direct proportion. Because, the bricks are all identical in the first stack. The same type of bricks are present in the second stack also. If the number of bricks increase, weight increase. The reverse is also true: If the number of bricks decrease, weight decrease. So we can proceed as usual. The table is given below:
The y/x ratio is also shown in the same table. The weight 43.2 kg and the corresponding no. of bricks should also have appropriate positions in the table. And they must give the same constant 2.70. So we have:
43.2/x1 = 2.70 ⇒ x1 = 43.2/2.7 = 16 Nos.
In the next section we will see the direct proportional changes in angles.
Solved example 3.3
In the fig. given below, the tall red pole of height 12 m, cast a shadow of length 8 m. At that same time a nearby shorter blue pole, cast a shadow of length 6m. Find the height of the blue pole.
 |
| Fig.3.2 |
This is a question of direct proportion. The direct proportion between two quantities: (1) Height of pole and (2) Length of shadow. Any of these two quantities can be taken as x, and the other as y. In all the problems that we did so far, there were more than one set of values for x and y. So we calculated y/x ratio in each case. If y/x is a constant, then we could confirm that the two quantities concerned, are in direct proportion, and that they satisfy the equation y = kx.
But in this problem, only one set of values is given. We can take the ratio y/x for that set. But how do we know that it is a constant?
By looking at the fig., we can see that, if the height increase, the length of shadow will increase. The reverse is also true: If the height decrease, the length of shadow will decrease. [But the position of the sun should remain the same. In the question, it is given that both the shadows are measured at the same time]. So this is a case of direct proportion. We can proceed as usual. The table is shown below:
 |
| Table 3.7 |
6/x1 = 0.67 ⇒ x1 = 6/0.67 = 8.96 m
Solved example 3.4
At a construction site, different numbers of workers were present on different days. The supervisor recorded the number of workers present on each day and the total wages given to them on that day. The following table is taken from his record book:
 |
| Table 3.8 |
Solution:
The data is already in a tabular form. Next step is to calculate the y/x ratios in each of the columns from (iii) to (viii). For convenience, the ratios can be written in the same table in the corresponding columns. This is shown below:
 |
| Table 3.9 |
• the changes are direct because, when the no. of workers increase, the total wages increase. And, when no. of workers decrease, the total wage decrease.
• But it is not proportional because, the ratio is not a constant. We cannot obtain ‘y’ (the total wages) by multiplying ‘x’ (the number of workers) with a constant.
So the two quantities are not in direct proportion. The reason for this can be stated as follows: The total number of workers present on any day will consist of two categories: ‘Masons’ who are skilled workers, and ‘Assistants’ who are unskilled workers’. Masons have a higher wage than assistants. So the total wages on a day will depend on the number of masons and assistants present on that day.
The supervisor should have written the number of masons and assistants separately for each day. Then, a person who looks at the records at a later date would get a clear understanding. However such incomplete tables are recorded on rare occassions for ‘quick reference’. And for us, such an incomplete table is a good example to compare proportionality and non-proportionality.
Another example for non-proportionality:
Consider the Height and Weight of a person. As his height increases, weight also increases. So the change is ‘direct’. But there is no constant by which we can multiply the height to get the weight. So the two quantities are not proportional.
Some times it may so happen that at a larger value of height, the weight has become lower. This may be due to special physical training, special diet, or even some illness. In such a case, the change is neither ‘direct’ nor ‘proportional’.
Solved example 3.5
A stack of bricks weigh 32.4 kg. There are 12 bricks in the stack. If another stack weigh 43.2 kg, how many bricks does it have?
Solution:
This is a case of direct proportion. Because, the bricks are all identical in the first stack. The same type of bricks are present in the second stack also. If the number of bricks increase, weight increase. The reverse is also true: If the number of bricks decrease, weight decrease. So we can proceed as usual. The table is given below:
 |
| Table 3.10 |
43.2/x1 = 2.70 ⇒ x1 = 43.2/2.7 = 16 Nos.
In the next section we will see the direct proportional changes in angles.
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