In the previous sections we saw two situations where fractions are required. In this section we will see another situation:
A toy designer has made a prototype of a robot as shown in the fig.5.7 below:
He wants to test how far it would 'walk'. He is doing the test for the first time. For doing the test, he marks 0 m, 1 m, 2 m, etc on a straight line on the floor. This is shown in the fig.5.8 below:
He places the robot at the zero mark and turns the power switch on. He is hoping to get the distance walked by the robot from the markings on the floor. But what if the robot fails to reach even the 1m mark?. This situation is shown in the fig.5.9 below:
What will he write in his report? That the robot did not make it to the 1 m mark? If he writes so, his superior is sure to ask: “How far did it go then?” The designer does not have a good answer because there are no marks between the zero mark and the 1 m mark.
So let us put some marks. For that we have to divide the distance into equal parts. We will divide it into 5 equal parts. This is shown in the fig.5.10 below:
The distance between 0 and 1 m is divided into 5 equal parts. So length of each part is 1/5 m. The robot has walked 4 such units. So the distance walked is 4/5 m. Now the designer has a good answer to give. He can say: “The robot walked 4/5 m”.
Similar to the above situation, we will want to measure distances smaller than 1 m in our day to day life. In such situations, fractions come to our rescue. In the fig.5.10 above, the 1 m is divided into 5 equal parts. What if the length of an object like a steel rod falls between any two marks on the fig.5.10 ?
For example, consider the fig.5.11(a) given below. The length of the steel rod is a little greater than 3/5 m. But it is less than 4/5 m.
In such cases, we need to divide the 1 m into still smaller divisions. In the fig.5.11(b), it is divided into 8 equal parts. So we find that the length of the rod is equal to 5/8 m.
A Gold merchant who wants to measure the length of a 'gold bar' may divide the meter into still smaller parts. He may divide it into 20 or even 50 parts. Because Gold is very precious, and so accurate measurements will have to be made. So we see that a 1 m distance can be divided into small fractions in a number of ways. Different people at different parts of the world will divide the meter into small divisions according to their convenience. In order to avoid such a situation, the number of divisions is standardized. And this standard is '100 divisions'. That is., a meter is always divided into 100 small divisions. Each of these 100 divisions is called a 'centimeter'. This type of division serves the purpose for most of the measurements made in our day to day life.
But the gold merchant may not be satisfied even with this 100 division. For example, the gold bar may have a length between 23 and 24 cm. That is., the length may be more than 23 cm, but less than 24 cm. So, for making more accurate measurements, each centimeter is again divided into 10 equal divisions. Each of these small divisions is called a millimeter. So a meter is divided into one thousand small divisions called millimeters. Also we can say a meter is divided into 100 small divisions called centimeters. From this we get the standard table that we use quite often:
• 1mm
• 1 cm = 10 mm
• 1 m = 100 cm = 1000 mm
Centimeters and millimeters are clearly marked in the scales and set squares that come with our standard instrument box that we use in our Geometry classes. But the scale in that box has a maximum length of about 15 cm only. With that we can measure the length of a pen that we use to write. It is about 14 centimeters
It may be noted that in many Scientific and Engineering activities, measurements need to be made with greater accuracy than even the millimeters. There are many instruments that are used for such purposes. The vernier calipers is such an instrument.
We will now see some solved examples based on the above discussions.
A toy designer has made a prototype of a robot as shown in the fig.5.7 below:
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| Fig.5.7 Prototype of a Toy robot |
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| Fig.5.8 Distances marked on the floor |
He places the robot at the zero mark and turns the power switch on. He is hoping to get the distance walked by the robot from the markings on the floor. But what if the robot fails to reach even the 1m mark?. This situation is shown in the fig.5.9 below:
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| Fig.5.9 Robot fails to reach the 1 m mark |
So let us put some marks. For that we have to divide the distance into equal parts. We will divide it into 5 equal parts. This is shown in the fig.5.10 below:
 |
| Fig.5.10 Distance from 0 to 1 m, divided into 5 equal parts |
The distance between 0 and 1 m is divided into 5 equal parts. So length of each part is 1/5 m. The robot has walked 4 such units. So the distance walked is 4/5 m. Now the designer has a good answer to give. He can say: “The robot walked 4/5 m”.
Similar to the above situation, we will want to measure distances smaller than 1 m in our day to day life. In such situations, fractions come to our rescue. In the fig.5.10 above, the 1 m is divided into 5 equal parts. What if the length of an object like a steel rod falls between any two marks on the fig.5.10 ?
For example, consider the fig.5.11(a) given below. The length of the steel rod is a little greater than 3/5 m. But it is less than 4/5 m.
 |
| Fig.5.11 |
A Gold merchant who wants to measure the length of a 'gold bar' may divide the meter into still smaller parts. He may divide it into 20 or even 50 parts. Because Gold is very precious, and so accurate measurements will have to be made. So we see that a 1 m distance can be divided into small fractions in a number of ways. Different people at different parts of the world will divide the meter into small divisions according to their convenience. In order to avoid such a situation, the number of divisions is standardized. And this standard is '100 divisions'. That is., a meter is always divided into 100 small divisions. Each of these 100 divisions is called a 'centimeter'. This type of division serves the purpose for most of the measurements made in our day to day life.
But the gold merchant may not be satisfied even with this 100 division. For example, the gold bar may have a length between 23 and 24 cm. That is., the length may be more than 23 cm, but less than 24 cm. So, for making more accurate measurements, each centimeter is again divided into 10 equal divisions. Each of these small divisions is called a millimeter. So a meter is divided into one thousand small divisions called millimeters. Also we can say a meter is divided into 100 small divisions called centimeters. From this we get the standard table that we use quite often:
• 1mm
• 1 cm = 10 mm
• 1 m = 100 cm = 1000 mm
Centimeters and millimeters are clearly marked in the scales and set squares that come with our standard instrument box that we use in our Geometry classes. But the scale in that box has a maximum length of about 15 cm only. With that we can measure the length of a pen that we use to write. It is about 14 centimeters
It may be noted that in many Scientific and Engineering activities, measurements need to be made with greater accuracy than even the millimeters. There are many instruments that are used for such purposes. The vernier calipers is such an instrument.
We will now see some solved examples based on the above discussions.
Solved example 5.3
The length of a table is found to be 74 cm. Express this as fraction of a meter.
Solution:
• We know that a meter is divided into 100 centimeters.
• So 73 cm means, the length is less than 1 m.
• 73 cm is 73 equal divisions out of 100 equal divisions
• So the measured 73 centimeters is 73/100 of the whole 1 meter.
Solved example 5.4
A mason is measuring the width of a door. The measuring tape he is using is worn out, and so readings cannot be seen clearly. But he did manage to read 80 cm, and that, there are 12 more cm divisions (after the 80 cm mark) upto the edge of the door. What is the actual width of the door ?
Solution:
• 12 divisions, each of 1 cm length is equal to 12 cm.
• This 12 cm is after the 80 cm reading. So the actual width is 80 + 12 = 92 cm.
It may be noted that, it is not allowed to use worn out measuring tapes or scales to take measurements. Use only tapes and scales which are in good condition.
So we have seen how fractions help us to measure small distances. In the next section we will discuss about Equivalent fractions.
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