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# Converting Units

For any physical measurement to make sense, it is essential to state both the magnitude and unit. For example, ‘I walked 2 kilometres.’ Both the magnitude, 2 and the unit, kilometres are essential for the measurement to make sense. Instead, if I said, ‘I walked 2’, that does not make any sense.

Another important thing to note here is that when specifying the distance walked, you have a choice among lots of different units. The sentence, ‘I walked 2 kilometres.’ is equivalent to each of the following sentences:

• I walked 2000 meters.
• I walked 200,000 centimetres.
• I walked 78740.16 inches.
• I walked 1.243 miles.

Evidently, we need to be able to convert values in a given unit to values in another unit.

## Converting metric units

The most widely used system of units today is known as the International System of Units (SI), also colloquially known as the metric system. As of the writing of this article, all countries except three – the USA, Myanmar and Liberia – have adopted the metric system or the SI units.

For your A-level maths course, you should be familiar with the SI system of units.

### Metric unit prefixes and base units

The metric system is a base 10 system of units built on a collection of base units. This translates to a system where each unit is a combination of a base unit and a prefix, and each succeeding unit is 10 times larger than the previous one. This concept will become clearer once we look at examples.

The following table demonstrates the basic prefixes used in the metric system. This is not an exhaustive list.

 Prefix symbol Value exponent kilo k 1000 10³ hecto 100 10² deca 10 - 1 deci d 0.1 centi c 0.01 milli m 0.001

The following table lists the most important base units in the metric system.

 Quantity Unit symbol Length / distance meter m Mass kilogram kg Volume litre l Time second s

Remember how we said each unit is a combination of a base unit and a prefix? Let’s look at some examples:

5 kilograms

In this example, the base unit is grams, and the prefix is kilo. From the table, we can see that the value of a kilo is 1000. So 1 kilogram is 1000 grams => 5 kilograms is 5000 grams.

The symbol for grams is g, and the symbol for the prefix kilo is k. Thus, this can also be written as

5 kg.

Looking at the table, can you convert 5 kg to centigrams?

From the table, we can see that kilo is 10³ and centi is.

Thus,

Now that we are familiar with the metric system let us look at another example.

Express 12 cm in hm.

Solution

From the expression, you should be able to figure out that cm stands for centimetres and hm for hectometres.

Now let us convert from cm to hm

12 cm =

## Converting compound units

So far, the examples we have seen have converted fundamental units with the same base units. Now we will look at how to convert derived units made up of multiple base units.

To recap the concept of fundamental and derived quantities, visit our article on physical quantities.

Given a derived unit with a given magnitude, follow the following steps to convert it into a required unit:

1. Express the unit to be converted as a combination of its fundamental units.
2. Replace each fundamental unit in the unit to be converted with its corresponding value in the unit it will be converted to.
3. Evaluate the obtained value along with its magnitude to obtain the converted value.

Let us illustrate this with an example:

Convert 8 m³ to cm³.

Solution

As described in the above steps, let us first express m³ in terms of its fundamental units.

8 m³

= 8 m × m × m

Now we know that 1 m = 100 cm, and we have to convert m × m × m to cm × cm × cm.

So,

8

= 8 × 100 cm × 100 cm × 100 cm

Now we evaluate the obtained value along with its magnitude to obtain the converted value.

8 × 100 × 100 × 100 cm³

=

From the above example, we see that 1 m³ = 106 cm³. A common mistake students make in such situations is converting 1 m³ = 100 cm³. This pattern of mistake is very common with all kinds of square and cubic unit conversions, and you should be careful to avoid this.

Let us look at another example:

Convert 90 km/h to m/s.

Solution

Again, let us first express km/h in terms of its fundamental units.

90 km/h

= 90

Now we know that 1 km = 1000m, 1 h = 60 * 60 = 3600 s, and we have to convert km/h to m/s

So,

Now we evaluate the obtained value along with its magnitude to obtain the converted value.

In the above example, we see that 1 km/h = 5/18 m/s. This is a well-known value, and you can directly use it in your examinations without the need for conversion.

A car travels 108 km in 2 hours. Find its speed in m/s.

Solution

The speed of the car is

We know that 1 km/h = 5/18 m/s.

So,

## Converting Units - Key takeaways

• For any physical measurement to make sense, it is essential to state both the magnitude and unit.
• When specifying the measurement of a physical, there is often a choice among multiple units to use.
• The metric system is a system where each unit is a combination of a base unit and a prefix, and each succeeding unit is 10 times larger than the previous one.

Given a cubic metric unit with given magnitude, follow the following steps to convert it into a required unit:

1. Express the unit to be converted as a combination of its fundamental units.
2. Replace each fundamental unit in the unit to be converted with its corresponding value in the unit to be converted to.
3. Evaluate the obtained value along with its magnitude to obtain the converted value.

To convert fluid capacity units into solid volume units, use the conversion, 1 litre = 1000cm³ .

To convert fluid capacity units into solid volume units, use the conversion, 1 litre = 1000cm³ .

## Final Converting Units Quiz

Question

What are the two required constituents of a physical measurement?

1. magnitude

2. unit

Show question

Question

What is the most commonly used system of units around the world called?

SI units

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Question

Express 50 cm in m.

0.5 m

Show question

Question

Express 5 litres in decilitres

50 dl

Show question

Question

Express 570 mg in g

0.57 g

Show question

Question

Express 150000 cm³ in m³

0.15 m³

Show question

Question

Express 36 km/h in m/s

10 m/s

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Question

Express 4.5 km/h in m/s

1.25 m/s

Show question

Question

Express 5 m/s in km/h

18 km/h

Show question

Question

Express 20 m/s in km/h

72 km/h

Show question

Question

A car is accelerating at 9 m/s². Express the acceleration in km/h².

116640 km/h²

Show question

Question

Express 450 m² in hm²

0.45 hm²

Show question

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