In the previous section we learned how the metric weights are expressed as decimals. In this section we will see how metric volumes are expressed as decimals.
Before we go into the details, we must have a preliminary idea about the difference between weight and volume. Consider a refrigerator. It has a particular weight. We will be concerned about it's weight in the following situations:
• How many people will be required to carry a newly bought refrigerator from the delivery truck into the house? If the refrigerator weighs more, more people will be required to carry it. Perhaps we will require the help of neighbours.
• Does the floor have enough capacity to carry the refrigerator? If it weighs more, the floor will have to be very strong. Particularly if the floor is made of materials like timber boards.
We will be concerned about it's volume in the following situations:
• Does the room have enough space to accommodate the refrigerator? The room should have enough length and width. Other wise, when the refrigerator is placed, there will not be any space left for other activities. The room must also have enough height. So we see that, in this situation, the size (volume) of the refrigerator is causing concern.
• Does the refrigerator have enough volume to store all the food stuffs required for the family? If it is for a large family, or a restaurant, the volume must be large.
So now we know the preliminary difference between volume and weight. In higher classes, we will see more details. At present we will learn how to measure this volume. The standard unit to measure volume is litre. Fig.6.31 shows a '1 litre measure'.
It is a vessel with markings. If we want to buy 1 litre of oil, the shop keeper will take oil up to the top most mark which is the '1 litre mark'. For quantities smaller than a litre, there are smaller 'measuring vessels'. Different types of such vessels can be seen here.
Litre is similar to the kilogram that we use to measure weights. Just as the kilogram was divided into 10, 100, and 1000 equal parts, the litre is also divided. This is to measure small volumes. The names of the smaller parts are also similar. For example,
• Corresponding to decagrams, we have decalitres.
• Corresponding to centigrams we have centilitres.
So a table similar to the Table 6.1 that we saw for weights can be formed for volume also. It is shown as xxx Table 6.2 below: It is shown adjacent to the 'weights table' for better understanding.
From the table we can see that:
• Each litre is divided into 10 equal parts. Each of these parts is called a decilitre
• Each decilitre is divided into 10 equal parts. Each of these parts is called a centilitre
• Each centilitre is divided into 10 equal parts. Each of these parts is called a millilitre
So the fig.6.27 that we used to show the division of kilograms can be used here also. The only modification to be made is the changing of the names of each parts. It is shown below:
We can also arrive at the following facts
• 1 litre = 10 decilitres = 100 centilitres = 1000 millilitres
So we can do all the types of problems that we did in the case of weights. Let us see some examples:
Solved example 6.13
So we have to learn to express the volume in terms of litres and millilitres only. For example, suppose some oil in a container has a volume of 2 litres and 6 decilitres. We do not have decilitres in the set of ‘standard vessels’. But we do have litres and millilitres. After measuring off 2 litres, we must measure off 600 millilitres. Because 6 decilitres = 600 millilitres.
We must be able to do the reverse also. That is., if we are given a certain volume, we must be able to express it in terms of litres and/or millilitres. Consider an example: The reading in an electronic dispensing machine is 4.283 litres. We can use this reading to convey the idea. People will understand it. For those who want finer details, we can split it into litres and millilitres. 4.283 litres is 4 litres plus 283 millilitres.
proof:
0.283 litre = 2 decilitres + 8 centilitres + 3 millilitres
• 2 decilitres = 20 centilitres (∵ 1 decilitre = 10 centilitres)
20 centilitres = 200 millilitres (∵ 1 centilitres = 10 millilitres)
• 8 centilitres = 80 millilitres (∵ 1 centilitres = 10 millilitres)
• 3 millilitres = 3 millilitres
Total = 200 + 80 + 3 = 283 millilitres.
From the above proof, we can note the following points:
• We get a volume. in litres
♦ The tenths give us decilitres
♦ The hundredths give us centilitres
♦ The thousandths give us millilitres
• We want the 'decilitres' and the 'centilitres' to go. For that:
♦ Multiply the digit in the tenths place (the decilitres) by 100
♦ Multiply the digit in the hundredths place (the centilitres) by 10
♦ keep the digit in the thousandths place as such
• Add the three items. This will give the 'quantity after the decimal point' in millilitres.
An even easier method is to multiply the decimal part by 1000.
Some examples:
• 3.041 litres = 3 litres + [.041 × 1000] millilitres = [3 litre + 41 millilitres] = [3000 millilitres + 41 millilitres] = 3041 millilitres
• 5.002 litres = 5 litres + [.002 × 1000] millilitres = [5 litres + 2 millilitres] = [5000 millilitres + 2 millilitres] = 5002 millilitres
• 9.305 litres = 9 litres + 305 millilitres
• 2.3 litres = 2 litres + 300 millilitres
• 2.03 litres = 2 litres + 30 millilitres
Readers are advised to write the proof for each of the above examples in all the 3 methods.
So we have seen how the metric volumes are expressed as decimals. In the next section we will see the expression of metric lengths as decimals.
Before we go into the details, we must have a preliminary idea about the difference between weight and volume. Consider a refrigerator. It has a particular weight. We will be concerned about it's weight in the following situations:
• How many people will be required to carry a newly bought refrigerator from the delivery truck into the house? If the refrigerator weighs more, more people will be required to carry it. Perhaps we will require the help of neighbours.
• Does the floor have enough capacity to carry the refrigerator? If it weighs more, the floor will have to be very strong. Particularly if the floor is made of materials like timber boards.
We will be concerned about it's volume in the following situations:
• Does the room have enough space to accommodate the refrigerator? The room should have enough length and width. Other wise, when the refrigerator is placed, there will not be any space left for other activities. The room must also have enough height. So we see that, in this situation, the size (volume) of the refrigerator is causing concern.
• Does the refrigerator have enough volume to store all the food stuffs required for the family? If it is for a large family, or a restaurant, the volume must be large.
So now we know the preliminary difference between volume and weight. In higher classes, we will see more details. At present we will learn how to measure this volume. The standard unit to measure volume is litre. Fig.6.31 shows a '1 litre measure'.
 |
| Fig.6.31 |
Litre is similar to the kilogram that we use to measure weights. Just as the kilogram was divided into 10, 100, and 1000 equal parts, the litre is also divided. This is to measure small volumes. The names of the smaller parts are also similar. For example,
• Corresponding to decagrams, we have decalitres.
• Corresponding to centigrams we have centilitres.
So a table similar to the Table 6.1 that we saw for weights can be formed for volume also. It is shown as xxx Table 6.2 below: It is shown adjacent to the 'weights table' for better understanding.
 |
| Table 6.2 |
• Each litre is divided into 10 equal parts. Each of these parts is called a decilitre
• Each decilitre is divided into 10 equal parts. Each of these parts is called a centilitre
• Each centilitre is divided into 10 equal parts. Each of these parts is called a millilitre
So the fig.6.27 that we used to show the division of kilograms can be used here also. The only modification to be made is the changing of the names of each parts. It is shown below:
 |
| Fig.6.32 |
• 1 litre = 10 decilitres = 100 centilitres = 1000 millilitres
So we can do all the types of problems that we did in the case of weights. Let us see some examples:
Solved example 6.13
The water in an ordinary container was divided into
• 3 litres +
• 5 decilitres +
• 7 centilitres +
• 4 millilitres.
What is the volume of water in litres?
Solution:
• 3 litres = 3 litres
• 5 decilitres:
1 decilitre = 1⁄10 litre = 0.1 litre
∴ 5 decilitres = 0.5 litres
• 7 centilitres:
1 centilitre = 1⁄100 litre = 0.01 litre
∴ 7 centilitres = 0.07 litres
• 4 millilitres:
1 millilitre = 1⁄1000 litre = 0.001 litre
∴ 4 millilitres= 0.004 litres
Thus total volume = 3 + 0.5 + 0.07 + 0.004 = 3.574 litres
Second method:We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place. (Details here)
We have:
Number of litres = 3; decilitres = 5; centilitres = 7 and millilitres = 4
So the volume in litres = 3.574
Solved example 6.14
The volume of some oil in a container is equal to 5 litres + 3 decilitres + 8 millilitres. What is the volume of the oil in litres?
Solution:
• 5 litres = 5 litres
• 3 decilitres:
1 decilitre = 0.1 litre
∴ 3 decilitres = 0.3 litres
• 8 millilitres:
1 millilitre= 0.001 litres
∴ 8 millilitres= 0.008 litres
Thus total volume = 5 + 0.3 + 0.008 = 5.308 litres
Second method:
We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place.
We have:
Number of litres = 5; decilitres = 3; centilitres = 0 and millilitres = 8
So the volume in litres = 5.308
Solved example 6.15
The volume of a liquid is known to be 2.839 litres. Split it into litres, decilitres, centilitres and millilitres.
Solution:
Volume = 2.839 litres.
The ‘whole number part’ is 2. So there are 2 full litres.
The decimal portion 0.839 litres gives the quantity between 2 litres and 3 litres
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 litre’ in 0.839. But 1 one tenth of a litre is 1 decilitre. So there are 8 decilitres.
From (1), we can say there are 3 ‘one hundredths of a litre’ in 0.839. But 1 one hundredth of a litre is one centilitre. So there are 3 centilitres.
From (1), we can say there are 9 ‘one thousandths of a litre’ in 0.839. But 1 one thousandth of a litre is one millilitre. So there are 9 millilitres.
Thus we can write: 2.839 = 2 litres + 8 decilitres + 3 centilitres + 9 millilitres.
Second method:
We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place.
We have:
Volume. = 2.839 litres
So, Number of litres = 2; decilitres = 8; centilitres = 3 and millilitres = 9
Solved example 6.16
When some petrol was filled into a car, the reading was 3.402 litres. Split this volume into litres, decilitres, centilitres and millilitres.
Solution:
Second method:
We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place.
We have:
Volume = 3.402 litres
So, Number of litres = 3; decilitres = 4; centilitres = 0 and millilitres = 2
Solved example 6.17
Volume of a certain object is 3.256 litres. Express this in centilitres.
Solution:
Volume = 3.256 litre. That is., 3 litres + 2 decilitres + 5 centilitres + 6 millilitres.
We have to convert each item into centilitres:
• 3 litres = 30 decilitres (∵ 1 litre = 10 decilitres)
30 decilitres = 300 centilitres (∵ 1 decilitre = 10 centilitres)
• 2 decilitres = 20 centilitres (∵ 1 decilitre = 10 centilitres)
• 5 centilitres = 5 centilitres
• 6 millilitres = 0.6 centilitres (∵ 1 centilitre = 10 millilitres ⇒ 1 millilitre = 0.1 centilitres)
So we get 3.256 litres = 300 + 20 + 5 + 0.6 = 325.6 centilitres
Solved example 6.18
Volume of a certain object is 5.029 litres. Express this in millilitres
Solution:
Volume = 5.029 litres. That is., 5 litres + 0 decilitres + 2 centilitres + 9 millilitres.
Now we have to convert each item into millilitres:
• 5 litres = 50 decilitres (∵ 1 litre = 10 decilitres)
50 decilitres = 500 centilitres (∵ 1 decilitre = 10 centilitres)
500 centilitres = 5000 millilitres (∵ 1 centilitre = 10 millilitres)
[Once we understand the basics, we need not write the detailed steps. We need write only this:
5 litre = 5000 millilitres (∵ 1 litre = 1000 millilitres)]
• 0 decilitres = 0 millilitres
• 2 centilitres = 20 millilitres (∵ 1 centilitre = 10 millilitres)
• 9 millilitres = 9 millilitres
So we get 5.029 litres = 5000 + 0 + 20 + 9 = 5029 millilitres
So we have learned how to
• 3 litres +
• 5 decilitres +
• 7 centilitres +
• 4 millilitres.
What is the volume of water in litres?
Solution:
• 3 litres = 3 litres
• 5 decilitres:
1 decilitre = 1⁄10 litre = 0.1 litre
∴ 5 decilitres = 0.5 litres
• 7 centilitres:
1 centilitre = 1⁄100 litre = 0.01 litre
∴ 7 centilitres = 0.07 litres
• 4 millilitres:
1 millilitre = 1⁄1000 litre = 0.001 litre
∴ 4 millilitres= 0.004 litres
Thus total volume = 3 + 0.5 + 0.07 + 0.004 = 3.574 litres
Second method:We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place. (Details here)
We have:
Number of litres = 3; decilitres = 5; centilitres = 7 and millilitres = 4
So the volume in litres = 3.574
Solved example 6.14
The volume of some oil in a container is equal to 5 litres + 3 decilitres + 8 millilitres. What is the volume of the oil in litres?
Solution:
• 5 litres = 5 litres
• 3 decilitres:
1 decilitre = 0.1 litre
∴ 3 decilitres = 0.3 litres
• 8 millilitres:
1 millilitre= 0.001 litres
∴ 8 millilitres= 0.008 litres
Thus total volume = 5 + 0.3 + 0.008 = 5.308 litres
Second method:
We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place.
We have:
Number of litres = 5; decilitres = 3; centilitres = 0 and millilitres = 8
So the volume in litres = 5.308
Solved example 6.15
The volume of a liquid is known to be 2.839 litres. Split it into litres, decilitres, centilitres and millilitres.
Solution:
Volume = 2.839 litres.
The ‘whole number part’ is 2. So there are 2 full litres.
The decimal portion 0.839 litres gives the quantity between 2 litres and 3 litres
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 litre’ in 0.839. But 1 one tenth of a litre is 1 decilitre. So there are 8 decilitres.
From (1), we can say there are 3 ‘one hundredths of a litre’ in 0.839. But 1 one hundredth of a litre is one centilitre. So there are 3 centilitres.
From (1), we can say there are 9 ‘one thousandths of a litre’ in 0.839. But 1 one thousandth of a litre is one millilitre. So there are 9 millilitres.
Thus we can write: 2.839 = 2 litres + 8 decilitres + 3 centilitres + 9 millilitres.
Second method:
We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place.
We have:
Volume. = 2.839 litres
So, Number of litres = 2; decilitres = 8; centilitres = 3 and millilitres = 9
Solved example 6.16
When some petrol was filled into a car, the reading was 3.402 litres. Split this volume into litres, decilitres, centilitres and millilitres.
Solution:
Second method:
We know that, the number of litres fall before the decimal point, the number of decilitres fall in the tenths place, number of centilitres fall in the hundredths place and the number of millilitres fall in the thousandths place.
We have:
Volume = 3.402 litres
So, Number of litres = 3; decilitres = 4; centilitres = 0 and millilitres = 2
Solved example 6.17
Volume of a certain object is 3.256 litres. Express this in centilitres.
Solution:
Volume = 3.256 litre. That is., 3 litres + 2 decilitres + 5 centilitres + 6 millilitres.
We have to convert each item into centilitres:
• 3 litres = 30 decilitres (∵ 1 litre = 10 decilitres)
30 decilitres = 300 centilitres (∵ 1 decilitre = 10 centilitres)
• 2 decilitres = 20 centilitres (∵ 1 decilitre = 10 centilitres)
• 5 centilitres = 5 centilitres
• 6 millilitres = 0.6 centilitres (∵ 1 centilitre = 10 millilitres ⇒ 1 millilitre = 0.1 centilitres)
So we get 3.256 litres = 300 + 20 + 5 + 0.6 = 325.6 centilitres
Solved example 6.18
Volume of a certain object is 5.029 litres. Express this in millilitres
Solution:
Volume = 5.029 litres. That is., 5 litres + 0 decilitres + 2 centilitres + 9 millilitres.
Now we have to convert each item into millilitres:
• 5 litres = 50 decilitres (∵ 1 litre = 10 decilitres)
50 decilitres = 500 centilitres (∵ 1 decilitre = 10 centilitres)
500 centilitres = 5000 millilitres (∵ 1 centilitre = 10 millilitres)
[Once we understand the basics, we need not write the detailed steps. We need write only this:
5 litre = 5000 millilitres (∵ 1 litre = 1000 millilitres)]
• 0 decilitres = 0 millilitres
• 2 centilitres = 20 millilitres (∵ 1 centilitre = 10 millilitres)
• 9 millilitres = 9 millilitres
So we get 5.029 litres = 5000 + 0 + 20 + 9 = 5029 millilitres
So we have learned how to
• split a given litre volume into smaller volumes like decilitres, centilitres and millilitres.
• combine given smaller volumes into litres.
• express the given volume in any one unit.
In day to day life, we do not use decilitres and centilitres. We use only litres and millilitres. Decilitres and centilitres are used only in some special industries. Hectolitres and decalitres are also not used. For large quantities we use litres and kilolitres only.
In day to day life, we do not use decilitres and centilitres. We use only litres and millilitres. Decilitres and centilitres are used only in some special industries. Hectolitres and decalitres are also not used. For large quantities we use litres and kilolitres only.
So we have to learn to express the volume in terms of litres and millilitres only. For example, suppose some oil in a container has a volume of 2 litres and 6 decilitres. We do not have decilitres in the set of ‘standard vessels’. But we do have litres and millilitres. After measuring off 2 litres, we must measure off 600 millilitres. Because 6 decilitres = 600 millilitres.
We must be able to do the reverse also. That is., if we are given a certain volume, we must be able to express it in terms of litres and/or millilitres. Consider an example: The reading in an electronic dispensing machine is 4.283 litres. We can use this reading to convey the idea. People will understand it. For those who want finer details, we can split it into litres and millilitres. 4.283 litres is 4 litres plus 283 millilitres.
proof:
0.283 litre = 2 decilitres + 8 centilitres + 3 millilitres
• 2 decilitres = 20 centilitres (∵ 1 decilitre = 10 centilitres)
20 centilitres = 200 millilitres (∵ 1 centilitres = 10 millilitres)
• 8 centilitres = 80 millilitres (∵ 1 centilitres = 10 millilitres)
• 3 millilitres = 3 millilitres
Total = 200 + 80 + 3 = 283 millilitres.
From the above proof, we can note the following points:
• We get a volume. in litres
♦ The tenths give us decilitres
♦ The hundredths give us centilitres
♦ The thousandths give us millilitres
• We want the 'decilitres' and the 'centilitres' to go. For that:
♦ Multiply the digit in the tenths place (the decilitres) by 100
♦ Multiply the digit in the hundredths place (the centilitres) by 10
♦ keep the digit in the thousandths place as such
• Add the three items. This will give the 'quantity after the decimal point' in millilitres.
An even easier method is to multiply the decimal part by 1000.
Some examples:
• 3.041 litres = 3 litres + [.041 × 1000] millilitres = [3 litre + 41 millilitres] = [3000 millilitres + 41 millilitres] = 3041 millilitres
• 5.002 litres = 5 litres + [.002 × 1000] millilitres = [5 litres + 2 millilitres] = [5000 millilitres + 2 millilitres] = 5002 millilitres
• 9.305 litres = 9 litres + 305 millilitres
• 2.3 litres = 2 litres + 300 millilitres
• 2.03 litres = 2 litres + 30 millilitres
Readers are advised to write the proof for each of the above examples in all the 3 methods.
So we have seen how the metric volumes are expressed as decimals. In the next section we will see the expression of metric lengths as decimals.
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