Chapter Notes: Measurement of Length, Mass and Capacity

# Chapter Notes: Measurement of Length, Mass and Capacity Notes | Study Mathematics for Class 5 - Class 5

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 Table of contents Units of Measurement Conversion of Units Mass Capacity Estimate in Measures Finding Fractions of Quantities
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Units of Measurement

The set of standard units of measurement that we use is known as the metric system. It uses numbers that are multiples and sub-multiples of 10.

• The standard unit of length in the metric system is a metre (m).
• The standard unit of mass in the metric system is a gram (g).
• The standard unit of capacity in the metric system is a litre (L).

These standard units are too small for measuring certain quantities and also too large for some.
Gram is too small a unit to measure the weight of a human being, whereas metre becomes too large to measure the length of a paper clip. Litre is too large to measure the amount of liquid in a dropper and too small to measure the capacity of a petrol tanker.
So, we introduce some more units of measures.
Some are higher than the standard units and some are lower. We name them by adding prefixes to the standard unit and the prefix shows how big or small the unit is compared to the standard unit (metre, gram, litre).

To name metric units other than the standard units (metre, gram, litre) a prefix precedes the standard unit.
Since kilo means 1000, therefore, 1 kilometre (km) = 1000 metres.
As milli means 1 / 1000 , therefore, 1 millilitre (mL) = 1 / 1000 litre = 0.001 litre.
Now, we deal with all measures, i.e., length, mass and capacity and study how the various units relate with the standard unit and with each other.

Length

Your height, the distance between two cities, the distance around a park, etc. are all quantities which measure length.

The standard unit of length in the metric system is metre (m).

The chart given below shows the units of length in the metric system with metre as the reference unit.

This shows that the metric system is like the decimal system.
From the above given chart, you can conclude the following:

Conversion of Units

The conversion of various units to metre and metre to various units can be shown diagrammatically as shown below.

To change from higher unit to lower unit, you multiply.

Examples:

• 3 km = (3 × 1000) m = 3000 m
• 8 hm = (8 × 100) m = 800 m
• 5 m = (5 × 100) cm = 500 cm
• 1.03 m = (1.03 × 1000) mm = 1030 mm

To change from lower unit to higher unit, you divide.

Examples:

• 8400 m = (8400 ÷ 1000) km = 8.4 km
• 370 m = (370 ÷ 100) hm = 3.7 hm
• 2000 mm = (2000 ÷ 1000) m = 2 m
• 73 cm = (73 ÷ 100) m = 0.73 m

Conversion from One Unit to Other Units

The following diagram shows how to carry out these conversions.

Rule:

1. Moving from left to right, you multiply by 10 at each step. Move one step right, multiply by 10, move two steps right, multiply by 10 × 10 = 100 and so on.
2. Moving from right to left, you divide by 10 at each step. Move one step left, divide by 10, move two steps left, divide by 100 and so on.

Example 1: Convert:
(a) 64 hm to dam
(b) 12 km to m
(c) 12 dam to cm
(d) 0.4 cm to mm
(e) 7.2 dm to mm

(a) 64 hm = (64 × 10) dam = 640 dam
dam is one step to the right of hm, so multiply by 10.
(b) 12 km = (12 × 1000) m = 12000 m
m is three steps to the right of km, so multiply by 10 × 10 × 10 = 1000.
(c) 12 dam = (12 × 1000) cm = 12000 cm
cm is three steps to the right of dam, so multiply by 10 × 10 × 10 = 1000.
(d) 0.4 cm = (0.4 × 10) mm = 40 mm
mm is one step to the right of cm, so multiply by 10.
(e) 7.2 dm = (7.2 × 100) mm = 720 mm
mm is two steps to the right of dm, so multiply by 10 × 10 = 100.

Edurev Tips: All these are conversions from higher units to lower units, so multiply.

Example 2: Convert:
(a) 60 mm to cm
(b) 720 dam to km
(c) 85 hm to km
(d) 7340 dm to hm
(e) 12000 cm to dam

(a) 60 mm = (60 ÷ 10) cm = 6 cm
cm is one step to the left of mm, so divide by 10.
(b) 720 dam  = (720 ÷ 100) km = 7.2 km
km is two steps to the left of dam, so divide by 100.
(c) 85 hm = (85 ÷ 10) km = 8.5 km
km is one step to the left of hm, so divide by 10.
(d) 7340 dm  = (7340 ÷ 1000) hm = 7.340 hm
hm is three steps to the left of dm, so divide by 1000.
(e) 12000 cm = (12000 ÷ 1000) dam = 12 dam
dam is three steps to the left of cm, so divide by 1000.

Writing Metric Measures in Decimal Notation

Since 100 centimetres = 1 metre, we can write
264 cm =  = 2.64 m
2.64 m = 2 metres 64 centimetres
= 2 metres 6 decimetres 4 centimetres

2.64 m = 2 m + 0.6 m + 0.04 m
= 2 m + 0.6 × 10 dm + 0.04 × 100 cm
= 2 m + 6 dm + 4 cm

Again, 3174 metres = 3000 metres + 100 metres + 70 metres + 4 metres
= 3 km + 1 hm + 7 dam + 4 m

= 3 km + 0.1 km + 0.07 km + 0.004 km = 3.174 km.

The decimal point separates the kilometres from the smaller units.

More Examples:

1. 0.408 km = 0 km 4 hm 0 dam 8 m
2. 3.015 m = 3 m 0 dm 1 cm 5 mm
3. 6 dam 3 m 5 dm 7 cm 2 mm can be written as 63.572 m.
6 dam 3 m = 6 × 10 m + 3 m = 63 mTherefore, write 63, put the decimal point and after the decimal point write the figures for dm, cm and mm respectively.
4. 4 metres and 3 centimetres can be written as 4.030 m, which means 4 metres 0 decimetres 3 centimetres 0 millimetres. You can simply write it as 4.03 m as there are no millimetres. Since there is no decimetre, we put a zero at the decimetres place.

Now, we can use the above concept to devise a shortcut to make conversion between units easier.

Shortcut for Conversion between Units

Example 3: Convert 3.18 m to cm.

1 m = 100 cm
To multiply by 100, shift the decimal point 2 places to the right.
∴ 3.18 m = 3.18 × 100 cm = 318 cm

Example 4: Convert 240 dm to dam.

To divide by 100, shift the decimal point 2 places to the left.
= 2.40 dam.
Reference chart:

Example 5: Convert 3.2 m to mm.

1 m = 1000 mm
1. To multiply by 1000, shift the decimal point 3 places to the right.
2. As there are no digits after 2, add two zeros.
∴ 3.2 m = (3.2 × 1000) mm
= 3200 mm.

The same principle of conversions is used in case of units of measurement of mass and capacity. The only change is in the standard unit.

Mass

The amount of matter in an object is its mass. The standard unit that is used to measure mass in the metric system is gram (g).
The chart given below shows the units of mass in the metric system with gram as the reference unit.

Important Relationships:

1. Conversions of Units

The given diagrams show conversions from the various higher and lower units to gram and vice versa. They also depict the conversions between the various units.

• To convert higher units to lower units, multiply.
• To convert lower units to higher units, divide.

2. Conversion of various units to grams and vice versa.
Examples:

• 24 kg = (24 × 1000) g = 24000 g;
• 1.217 g = (1.217 × 1000) mg = 1217 mg
• 3200 mg = (3200 ÷ 1000) g = 3.2 g;
• 315 cg = (315 ÷ 100) g = 3.15 g

3. Conversion between Units

• 14 mg = (14 ÷ 10) cg = 1.4 cg
• 300 dg = (300 ÷ 100) dag = 3 dag
• 2417 cg = (2417 ÷ 1000) dag = 2.417 dag
• 38 kg = (38 × 100) dag = 3800 dag
• 23.7 dg = (23.7 × 10) cg = 237 cg
• 4 hg = (4 × 1000) dg = 4000 dg

Capacity

The standard unit of capacity in the metric system is litre (L). The chart given below shows the units of capacity in the metric system with litre as the reference unit.

Important Relationships:

1. Conversion of Units

The conversion from higher and lower units to litres and conversion of litres to various units as well as the interconversions between the various units are shown in the following diagrams.

• To convert higher units to lower units, multiply.
• To convert lower units to higher units, divide.

2. Conversion of various units to litres and litres to various units
Examples:

• 2.3 kL = (2.3 × 1000) L = 2300 L;
• 1.8 cL = (1.8 ÷ 100) L = 0.018 L;
• 64 hL = (64 × 100) L = 6400 L;
• 600 L = (600 ÷ 100) hL = 6 hL;
• 7315 mL = (7315 ÷ 1000) L = 7.315 L;
• 11.9 L = (11.9 × 1000) mL = 11900 mL

3. Conversions between Units
Examples:

• 25 hL = (25 × 10) daL = 250 daL
• 13.8 dL = (13.8 × 100) mL = 1380 mL
• 500 daL = (500 ÷ 100) kL = 5 kL
• 3.117 hL = (3.117 × 1000) dL = 3117 dL

The Four Basic Operations on Metric Measures

Example 6: Ritu drew a line segment of length 15 cm 4 mm. Then, she erased a portion of it. The remaining line segment measured 7 cm 6 mm. What is the length in mm of the erased line segment?

Length of the line segment in the beginning = 15 cm 4 mm
Length of line segment left after erasing = 7 cm 6 mm
Length of line segment erased = 15 cm 4 mm – 7 cm 6 mm
= 7 cm  8 m m

(a) 3 cm 4 mm and 9 cm 8 mm
(b) 14 dm 4 cm and 23 dm 7 cm
(c) 53 kg 305 g and 7 kg 828 g
(d) 8 L 718 mL and 7 L 732 mL

(a)

12 mm
= 10 mm + 2 mm
= 1 cm + 2 mm
= 13 cm 2 mm = 13.2 cm.
(b)

11 cm
= 10 cm + 1 cm
= 1 dm + 1 cm
= 38 dm 1 cm = 38.1 dm.
(c)

1133 g
= 1000 g + 133 g
= 1 kg + 133 g
= 61 kg 133 g = 61.133 kg.
(d)

1450 mL
= 1000 mL + 450 mL
= 1 L + 450 mL
= 16 L 450 mL = 16.450 L.

Example 8: Subtract:
(a) 10 m  36 cm – 5 m 83 cm
(b) 28 cm 4 mm – 13 cm 8 mm
(c) 40 kg 353 g – 17 kg 500 g
(d) 8 kL 150 L – 4 kL 850 L

(a)

36 cm < 83 cm
Borrow 1 m = 100 cm
100 cm + 36 cm
= 136 cm
= 4.53 m.
(b)

4 mm < 8 mm
Borrow 1 cm = 10 mm
10 mm + 4 mm
= 14 mm
= 14.6 cm.
(c)

353 g < 500 g
Borrow 1 kg = 1000 g
1000 g + 353 g
= 1353 g
= 22.853 kg.
(d)

150 L < 850 L
Borrow 1 kL = 1000 L
1000 L + 150 L
= 1150 L
= 3.300 kL.

2. Multiplication
Arrange numbers in columns unitwise and then multiply as you would multiply whole numbers.

Example 9: Find, in centimetres, the height of a pile of 25 books, if each book is  3 cm, 5 mm thick.

Thickness of 1 book = 3 cm 5 mm = 3.5 cm
Height of 25 books = 3.5 cm × 25
= 87.5 cm
Thus, height of the pile of 25 books
= 87.5 cm.

Example 10: A carton full of fruits weighs 6 kg 125 g. What is the weight of 12 such cartons in kg?

Weight of one carton = 6 kg 125 g
= 6.125 kg
∴ Weight of 12 cartons = (6.125 × 12) kg
= 73.500 kg
= 73 kg 500 g.

3. Division
In division also arrange the numbers in columns unitwise and then divide like whole numbers.

Example 11: Reena prepared 4 L 156 mL of orange juice. Distribute it equally among 8 children. How many mL of orange juice each child gets?

Juice Reena prepared = 4 L 156 mL = 4.156 L
When distributed among 8 children,
juice each child gets = (4.156 ÷ 8) L
= 0.51 95 L
= 519 .5 mL.

Example 12: How many 150 mL glasses can I fill with 5 bottles of soft drinks each holding 1.2 litres?

Total soft drink in 5 bottles = 1.2 L × 5 = 6.0 = 6 L

∵ 1 L = 1000 mL
Total soft drink = 6 L = 6 × 1000 = 6000 mL
Number of 150 mL of glasses that can be filled = 6000 mL ÷ 150 mL = 40
Thus, with 6 L of soft drink, I can fill 40 glasses of 150 mL.

Estimate in Measures

You cannot always be exact in measures. In our day-to-day life, we estimate the measures. To be able to estimate measures correctly, you should have a fair idea of how to relate commonplace things with the commonly used units of measures.

1. Length

• For long lengths or distances, we use kilometres (km) as unit.
Length of India = 3214 km Breadth of India = 2933 km
• For medium lengths, we use metres (m).
The length of your dining table.
• For short lengths, we use centimetres (cm).
Length of a small ruler = 15 cm.
• For very short lengths, we use millimetres (mm).
The tip of your pencil = 2 mm.

2. Mass

• For very large masses, we use tonne (t) and quintal (q).
Truck = 4 tonne,Sack of rice = 1 quintal.
• For medium masses, we use kilograms (kg).
The weight of your school bag = 5 kg.
• For small masses, we use grams (g).
An orange is about 100 g.
• For very small masses, we use milligrams (mg).
A teaspoon of salt = 5 mg.

3. Capacity

• For large capacities, we use kilolitre.
Capacity of a swimming pool = 600 kL.
• For medium capacities, we use litres.
A bathing bucket = 20 L.
• For small capacities, we use millilitres.
A bottle of medicine = 200 mL.

Finding Fractions of Quantities

Example 13: A basket contains 3 kg 705 g of mangoes. 23 of the mangoes are eaten by Mr Bhasin. Lata, his daughter, gets 25 of the remaining mangoes. What is her share in grams?

Total weight of mangoes = 3 kg 705 g
Mangoes eaten by Mr Bhasin

Mangoes left = 3.705 kg – 2.47 kg
= 1.235 kg
Mangoes eaten by Lata

= (0.494 × 1000) g
= 494 g

So, Lata ate 494 g of the mangoes.

Example 14: Anshul had 45 kg of wafers. He packed all the wafers equally into 5 small packets. How many grams of wafers were there in each packet?

Total wafers with Anshul = 4 / 5 kg

= 800 g
800 g wafers are filled in 5 small packets.
∴ Wafers in one packet = 800 ÷ 5
= 800 / 5 = 160 g
So, each packet contains 160 g of wafers.

Example 15: Madhuri drew a line segment of length 20 cm 5 mm. She accidentally erased 2 / 5 of it. What is the length of the remaining line segment in cm?

Length of the line segment drawn = 20 cm 5 mm = 20.5 cm
Length of the erased line segment =  2 / 5 of 20.5 cm
=  (2 × 4.1) cm
=  8.2 cm
Length of the remaining line segment = 20.5 cm – 8.2 cm
= 12.3 cm.

The document Chapter Notes: Measurement of Length, Mass and Capacity Notes | Study Mathematics for Class 5 - Class 5 is a part of the Class 5 Course Mathematics for Class 5.
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