In the previous section we learned how the metric volumes are expressed as decimals. In this section we will see how metric lengths are expressed as decimals.
We will first learn how to measure lengths. The standard unit to measure length is metre. We have had a basic discussion about the metre and it's subdivisions when we learned about fractions here. Fig.6.33 shows a 'tape measure'.
It is a flexible steel strip with markings. The markings start from 0 metre. After that, 1 m, 2 m and so on are marked at regular intervals. For measuring lengths less than 1 m, or lengths which fall any where in between 'whole metres', we need sub divisions.
Subdivisions are very similar to those of kilograms and litres that we saw in previous sections. Just as the kilogram and litre were divided into 10, 100, and 1000 equal parts, the metre is also divided. The names of the smaller parts are also similar. For example,
• Corresponding to decagrams, and decalitres, we have decametres
• Corresponding to centigrams and centilitres, we have centimetres
So a table similar to the Table 6.1 that we saw for weights, and the Table 6.2 that we saw for litres, a table can be formed for metres also. It is shown as Table 6.3 below: It is shown adjacent to the previous tables for better understanding.
From the table we can see that:
• Each metre is divided into 10 equal parts. Each of these parts is called a decimetre
• Each decimetre is divided into 10 equal parts. Each of these parts is called a centimetre
• Each centimetre is divided into 10 equal parts. Each of these parts is called a millimetre
So the fig.6.27 that we used to show the division of kilograms can be used here also. There, the 'whole kilogram' was represented by a square. The metre is used for measurements along a line. So it is more appropriate to represent a 'whole metre' by a line, rather than a square. This is shown in the fig.6.34 below:
We can also arrive at the following facts
• 1 metre = 10 decimetres = 100 centimetres = 1000 millimetres
[It may be noted that in (c) and (d) above, the divisions are not exactly 100 and 1000. This is because, it is not convenient to show such small divisions on the space of an ordinary paper or a computer screen. But for our present discussion, we can assume that the divisions are exactly 100 and 1000]
So we can do all the types of problems that we did in the case of weights and volumes. Let us see some examples:
Solved example 6.19
So we have to learn to express the lengths in terms of metres, centimetres and millimetres only. For example, suppose the length of a table is 2 metres and 1 decimetre. We do not have decimetres in the ordinary 'tape measure'. But we do have metres, centimetres and millimetres. After measuring off 2 metres, we must measure off 10 centimetres. Because 1 decimetre = 10 centimetres.
We must be able to do the reverse also. That is., if we are given a certain length, we must be able to express it in terms of metres, or centimetres or millimetres, or their combinations. Consider an example: On a plan, the length of a room is marked as 4.283 metres. We can use this reading to convey the idea. People will understand it. For those who want finer details, we can split it into metres, centimetres and millimetres. 4.283 metres is 4 metres + 28 centimetres + 3 millimetres.
proof:
0.283 metres = 2 decimetres + 8 centimetres + 3 millimetres
• 2 decimetres = 20 centimetres (∵ 1 decimetre = 10 centimetres)
• 8 centimetres = 8 centimetres
• 3 millimetres = 3 millimetres
Total = (20 + 8) centimetres + 3 millimetres = 28 centimetres + 3 millimetres.
From the above proof, we can note the following points:
• We get a length in metres
♦ The tenths give us decimetres
♦ The hundredths give us centimetres
♦ The thousandths give us millimetres
• We want the 'decimetres' to go. For that:
♦ Multiply the digit in the tenths place (the decimetres) by 10
♦ keep the digit in the hundredths place as such
♦ Add the above two to get the centimetre part of the answer
♦ keep the digit in the thousandths place as such. This will be the millimetre part of the answer
If we multiply the decimal part by 1000, it will be directly converted into millimetres.
Some examples:
• 3.041 metres = 3 metres + [.041 × 1000] millimetres = [3 metres + 41 millimetres] = [3000 millimetres + 41 millimetres] = 3041 millimetres
• 5.002 metres = 5 metres + [.002 × 1000] millimetres = [5 metres + 2 millimetres] = [5000 millimetres + 2 millimetres] = 5002 millimetres
• 9.305 metres = 9 metres + 30 centimetres + 5 millimetres
• 2.3 metres = 2 metres + 30 centimetres
• 2.03 metres = 2 metres + 3 centimetres
Readers are advised to write the proof for each of the above examples.
So we have seen how the metric lengths are expressed as decimals. In the next section we will see the expression of currency as decimals.
We will first learn how to measure lengths. The standard unit to measure length is metre. We have had a basic discussion about the metre and it's subdivisions when we learned about fractions here. Fig.6.33 shows a 'tape measure'.
 |
| Fig.6.33 |
Subdivisions are very similar to those of kilograms and litres that we saw in previous sections. Just as the kilogram and litre were divided into 10, 100, and 1000 equal parts, the metre is also divided. The names of the smaller parts are also similar. For example,
• Corresponding to decagrams, and decalitres, we have decametres
• Corresponding to centigrams and centilitres, we have centimetres
So a table similar to the Table 6.1 that we saw for weights, and the Table 6.2 that we saw for litres, a table can be formed for metres also. It is shown as Table 6.3 below: It is shown adjacent to the previous tables for better understanding.
 |
| Table 6.3 |
• Each metre is divided into 10 equal parts. Each of these parts is called a decimetre
• Each decimetre is divided into 10 equal parts. Each of these parts is called a centimetre
• Each centimetre is divided into 10 equal parts. Each of these parts is called a millimetre
So the fig.6.27 that we used to show the division of kilograms can be used here also. There, the 'whole kilogram' was represented by a square. The metre is used for measurements along a line. So it is more appropriate to represent a 'whole metre' by a line, rather than a square. This is shown in the fig.6.34 below:
 |
| Fig.6.34 |
• 1 metre = 10 decimetres = 100 centimetres = 1000 millimetres
[It may be noted that in (c) and (d) above, the divisions are not exactly 100 and 1000. This is because, it is not convenient to show such small divisions on the space of an ordinary paper or a computer screen. But for our present discussion, we can assume that the divisions are exactly 100 and 1000]
So we can do all the types of problems that we did in the case of weights and volumes. Let us see some examples:
Solved example 6.19
The length of a steel rod was obtained as:
• 3 metres +
• 5 decimetres +
• 7 centimetres +
• 4 millimetres.
What is the length of the rod in metres?
Solution:
• 3 metres = 3 metres
• 5 decimetres:
1 decimetre = 1⁄10 metre = 0.1 metre
∴ 5 decimetres = 0.5 metres
• 7 centimetres:
1 centimetre = 1⁄100 metre = 0.01 metre
∴ 7 centimetres = 0.07 metres
• 4 millimetres:
1 millimetre = 1⁄1000 metre = 0.001 metre
∴ 4 millimetres = 0.004 metres
Thus total length = 3 + 0.5 + 0.07 + 0.004 = 3.574 metres
Second method:We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place. (Details here)
We have:
Number of metres = 3; decimetres = 5; centimetres = 7 and millimetres = 4
So the length in metres = 3.574
Solved example 6.20
The length of a room is equal to 5 metres + 3 decimetres + 8 millimetres. What is the length of the room in metres?
Solution:
• 5 metres = 5 metres
• 3 decimetres:
1 decimetre = 0.1 metre
∴ 3 decimetres = 0.3 metres
• 8 millimetres:
1 millimetre = 0.001 metres
∴ 8 millimetres = 0.008 metres
Thus total length = 5 + 0.3 + 0.008 = 5.308 metres
Second method:
We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place.
We have:
Number of metres = 5; decimetres = 3; centimetres = 0 and millimetres = 8
So the length in metres = 5.308
Solved example 6.21
The length of a table is known to be 2.839 metres. Split it into metres, decimetres, centimetres and millimetres.
Solution:
Length = 2.839 metres.
The ‘whole number part’ is 2. So there are 2 full metres.
The decimal portion 0.839 metres gives the quantity between 2 metres and 3 metres
This can be split as: 8/10 + 3/100 + 9/1000 - - - (1)
[proof:
• 8/10 = 800/1000
• 3/100 = 30/1000
• 9/1000 = 9/1000
• Total = 800/1000 + 30/1000 + 9/1000 = 839/1000 = 0.839]
From (1), we can say there are 8 ‘one tenths of a metre’ in 0.839. But 1 one tenth of a metre is 1 decimetre. So there are 8 decimetres.
From (1), we can say there are 3 ‘one hundredths of a metre’ in 0.839. But 1 one hundredth of a metre is one centimetre. So there are 3 centimetres.
From (1), we can say there are 9 ‘one thousandths of a metre’ in 0.839. But 1 one thousandth of a metre is one millimetre. So there are 9 millimetres.
Thus we can write: 2.839 = 2 metres + 8 decimetres + 3 centimetres + 9 millimetres.
Second method:
We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place.
We have:
Length. = 2.839 metres
So, Number of metres = 2; decimetres = 8; centimetres = 3 and millimetres = 9
Solved example 6.22
On a building plan, the width of a room is written as 3.402 metres. Split this length into metres, decimetres, centimetres and millimetres.
Solution:
Second method:
We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place.
We have:
Length. = 3.402 metres
So, Number of metres = 3; decimetres = 4; centimetres = 0 and millimetres = 2
Solved example 6.17
Length of a certain object is 3.256 metres. Express this in centimetres.
Solution:
Length = 3.256 metres. That is., 3 metres + 2 decimetres + 5 centimetres + 6 millimetres.
We have to convert each item into centimetres:
• 3 metres = 30 decimetres (∵ 1 metre = 10 decimetres)
30 decimetres = 300 centimetres (∵ 1 decimetre = 10 centimetres)
• 2 decimetres = 20 centimetres (∵ 1 decimetre = 10 centimetres)
• 5 centimetres = 5 centimetres
• 6 millimetres = 0.6 centimetres (∵ 1 centimetre = 10 millimetres ⇒ 1 millimetre = 0.1 centimetres)
So we get 3.256 metres = 300 + 20 + 5 + 0.6 = 325.6 centimetres
Solved example 6.23
Length of a certain object is 5.029 metres. Express this in millimetres.
Solution:
Length = 5.029 metres. That is., 5 metres + 0 decimetres + 2 centimetres + 9 millimetres.
Now we have to convert each item into millimetres:
• 5 metres = 50 decimetres (∵ 1 metre = 10 decimetres)
50 decimetres = 500 centimetres (∵ 1 decimetre = 10 centimetres)
500 centimetres = 5000 millimetres (∵ 1 centimetre = 10 millimetres)
[Once we understand the basics, we need not write the detailed steps. We need write only this:
5 metre = 5000 millimetres (∵ 1 metre = 1000 millimetres)]
• 0 decimetres = 0 millimetres
• 2 centimetres = 20 millimetres (∵ 1 centimetres = 10 millimetres)
• 9 millimetres = 9 millimetres
So we get 5.029 metres = 5000 + 0 + 20 + 9 = 5029 millimetres
So we have learned how to
• 3 metres +
• 5 decimetres +
• 7 centimetres +
• 4 millimetres.
What is the length of the rod in metres?
Solution:
• 3 metres = 3 metres
• 5 decimetres:
1 decimetre = 1⁄10 metre = 0.1 metre
∴ 5 decimetres = 0.5 metres
• 7 centimetres:
1 centimetre = 1⁄100 metre = 0.01 metre
∴ 7 centimetres = 0.07 metres
• 4 millimetres:
1 millimetre = 1⁄1000 metre = 0.001 metre
∴ 4 millimetres = 0.004 metres
Thus total length = 3 + 0.5 + 0.07 + 0.004 = 3.574 metres
Second method:We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place. (Details here)
We have:
Number of metres = 3; decimetres = 5; centimetres = 7 and millimetres = 4
So the length in metres = 3.574
Solved example 6.20
The length of a room is equal to 5 metres + 3 decimetres + 8 millimetres. What is the length of the room in metres?
Solution:
• 5 metres = 5 metres
• 3 decimetres:
1 decimetre = 0.1 metre
∴ 3 decimetres = 0.3 metres
• 8 millimetres:
1 millimetre = 0.001 metres
∴ 8 millimetres = 0.008 metres
Thus total length = 5 + 0.3 + 0.008 = 5.308 metres
Second method:
We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place.
We have:
Number of metres = 5; decimetres = 3; centimetres = 0 and millimetres = 8
So the length in metres = 5.308
Solved example 6.21
The length of a table is known to be 2.839 metres. Split it into metres, decimetres, centimetres and millimetres.
Solution:
Length = 2.839 metres.
The ‘whole number part’ is 2. So there are 2 full metres.
The decimal portion 0.839 metres gives the quantity between 2 metres and 3 metres
This can be split as: 8/10 + 3/100 + 9/1000 - - - (1)
[proof:
• 8/10 = 800/1000
• 3/100 = 30/1000
• 9/1000 = 9/1000
• Total = 800/1000 + 30/1000 + 9/1000 = 839/1000 = 0.839]
From (1), we can say there are 8 ‘one tenths of a metre’ in 0.839. But 1 one tenth of a metre is 1 decimetre. So there are 8 decimetres.
From (1), we can say there are 3 ‘one hundredths of a metre’ in 0.839. But 1 one hundredth of a metre is one centimetre. So there are 3 centimetres.
From (1), we can say there are 9 ‘one thousandths of a metre’ in 0.839. But 1 one thousandth of a metre is one millimetre. So there are 9 millimetres.
Thus we can write: 2.839 = 2 metres + 8 decimetres + 3 centimetres + 9 millimetres.
Second method:
We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place.
We have:
Length. = 2.839 metres
So, Number of metres = 2; decimetres = 8; centimetres = 3 and millimetres = 9
Solved example 6.22
On a building plan, the width of a room is written as 3.402 metres. Split this length into metres, decimetres, centimetres and millimetres.
Solution:
Second method:
We know that, the number of metres fall before the decimal point, the number of decimetres fall in the tenths place, number of centimetres fall in the hundredths place and the number of millimetres fall in the thousandths place.
We have:
Length. = 3.402 metres
So, Number of metres = 3; decimetres = 4; centimetres = 0 and millimetres = 2
Solved example 6.17
Length of a certain object is 3.256 metres. Express this in centimetres.
Solution:
Length = 3.256 metres. That is., 3 metres + 2 decimetres + 5 centimetres + 6 millimetres.
We have to convert each item into centimetres:
• 3 metres = 30 decimetres (∵ 1 metre = 10 decimetres)
30 decimetres = 300 centimetres (∵ 1 decimetre = 10 centimetres)
• 2 decimetres = 20 centimetres (∵ 1 decimetre = 10 centimetres)
• 5 centimetres = 5 centimetres
• 6 millimetres = 0.6 centimetres (∵ 1 centimetre = 10 millimetres ⇒ 1 millimetre = 0.1 centimetres)
So we get 3.256 metres = 300 + 20 + 5 + 0.6 = 325.6 centimetres
Solved example 6.23
Length of a certain object is 5.029 metres. Express this in millimetres.
Solution:
Length = 5.029 metres. That is., 5 metres + 0 decimetres + 2 centimetres + 9 millimetres.
Now we have to convert each item into millimetres:
• 5 metres = 50 decimetres (∵ 1 metre = 10 decimetres)
50 decimetres = 500 centimetres (∵ 1 decimetre = 10 centimetres)
500 centimetres = 5000 millimetres (∵ 1 centimetre = 10 millimetres)
[Once we understand the basics, we need not write the detailed steps. We need write only this:
5 metre = 5000 millimetres (∵ 1 metre = 1000 millimetres)]
• 0 decimetres = 0 millimetres
• 2 centimetres = 20 millimetres (∵ 1 centimetres = 10 millimetres)
• 9 millimetres = 9 millimetres
So we get 5.029 metres = 5000 + 0 + 20 + 9 = 5029 millimetres
So we have learned how to
• split a given metre length into smaller lengths like decimetres, centimetres and millimetres.
• combine given smaller lengths into metres.
• express a given length in any one unit.
In day to day life, we do not use decimetres. We use only metres, centimetres and millimetres. Decimetres are used only in some special cases like surveying of large areas of land. Hectometres and decametres are also not used. For large distances we use kilometres only.
In day to day life, we do not use decimetres. We use only metres, centimetres and millimetres. Decimetres are used only in some special cases like surveying of large areas of land. Hectometres and decametres are also not used. For large distances we use kilometres only.
So we have to learn to express the lengths in terms of metres, centimetres and millimetres only. For example, suppose the length of a table is 2 metres and 1 decimetre. We do not have decimetres in the ordinary 'tape measure'. But we do have metres, centimetres and millimetres. After measuring off 2 metres, we must measure off 10 centimetres. Because 1 decimetre = 10 centimetres.
We must be able to do the reverse also. That is., if we are given a certain length, we must be able to express it in terms of metres, or centimetres or millimetres, or their combinations. Consider an example: On a plan, the length of a room is marked as 4.283 metres. We can use this reading to convey the idea. People will understand it. For those who want finer details, we can split it into metres, centimetres and millimetres. 4.283 metres is 4 metres + 28 centimetres + 3 millimetres.
proof:
0.283 metres = 2 decimetres + 8 centimetres + 3 millimetres
• 2 decimetres = 20 centimetres (∵ 1 decimetre = 10 centimetres)
• 8 centimetres = 8 centimetres
• 3 millimetres = 3 millimetres
Total = (20 + 8) centimetres + 3 millimetres = 28 centimetres + 3 millimetres.
From the above proof, we can note the following points:
• We get a length in metres
♦ The tenths give us decimetres
♦ The hundredths give us centimetres
♦ The thousandths give us millimetres
• We want the 'decimetres' to go. For that:
♦ Multiply the digit in the tenths place (the decimetres) by 10
♦ keep the digit in the hundredths place as such
♦ Add the above two to get the centimetre part of the answer
♦ keep the digit in the thousandths place as such. This will be the millimetre part of the answer
If we multiply the decimal part by 1000, it will be directly converted into millimetres.
Some examples:
• 3.041 metres = 3 metres + [.041 × 1000] millimetres = [3 metres + 41 millimetres] = [3000 millimetres + 41 millimetres] = 3041 millimetres
• 5.002 metres = 5 metres + [.002 × 1000] millimetres = [5 metres + 2 millimetres] = [5000 millimetres + 2 millimetres] = 5002 millimetres
• 9.305 metres = 9 metres + 30 centimetres + 5 millimetres
• 2.3 metres = 2 metres + 30 centimetres
• 2.03 metres = 2 metres + 3 centimetres
Readers are advised to write the proof for each of the above examples.
So we have seen how the metric lengths are expressed as decimals. In the next section we will see the expression of currency as decimals.
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