Conversion of Units

Units in Mathematics is used to measure any physical quantity. Any quantity that can be measured is called a physical quantity. The fundamental physical quantities are mass, length, and time. Every physical quantity has a separate unit to measure that quantity. Each unit in Mathematics can be expressed using a dimensional formula in terms of Mass, Length, and Time.  When two quantities of the same unit are taken as a ratio, the result obtained is dimensionless. In many Mathematical problems, the quantities involved in the calculation of the final result are not measured in the same standard unit. In those cases, the conversion of units of measurement to one standard form is required. A unit conversion chart or unit conversion table is used to perform this action.

Introduction and SI Base Units

Many times in science and mathematics, it is necessary to convert standard measurement units into other measurement units. The process is known as unit conversion. It is sometimes more significant while fixing a problem. Mathematical conversions are used to complete the calculations. 

For example, suppose the length and width of a rectangle are provided in cm and metre, respectively. Then, to calculate its area in square metres, we must convert cm to metre units. As a result, one must understand the concept and procedure of unit conversion.

When determining the area of a room in metres but the length measurement in centimetres, the usage of one unit is dependent on the situation. 

In our day-to-day actions, there are many things that we measure like:

• Speed

• Temperature

• Area

• Volume

• Mass

To measure the above quantities, we need units of measure. Many times, the units used to measure aren't the same as the measurement that is being made. Then, for certain processes and applications, some common conversion procedures should be applied. Some of these units can be changed, and it is easy to understand and apply.

Different types of base units are used to create different types of unit systems. The International System of Units, or SI, is the most frequently used unit system. There are seven SI base units, from which all other SI units can be simply deduced.

The following are the seven SI base units:

• Length: m (metre)

• Mass: kg (kilogram)

• Time: s (second)

• Electric Current: A (Ampere)

• Thermodynamic Temperature: K (degrees Kelvin)

• Amount of Substance: mol (mole)

• Luminous Intensity: cd (candela)

As a result, the measurement usually is accurate with minimal mistakes. Thereafter, it is required to convert one unit to another. 

For example,  the length of a pencil cannot be measured in kilometres. Hence, the unit of a kilometre is to be converted into centimetres. 

In most cases, multiplicative conversion factors are used to convert one unit to another unit of the same quantity.

The unit conversion process involves:

• Dimensional Analysis

• Principle for quantities defined as equalities.

• The phrase for the relationship between units is given by the conversion factor.

Examples of an Instance Where Conversion of Units of Measurement is Used:

• To measure the perimeter of a scalene triangle with sides 10 cm, 0.2 m, and 15 mm. In this problem, the sides of the triangle are measured with three different units. So, unit conversion charts for length should be used for the conversion of units of measurement of length either to a metre or centimetre or millimetre.

• Finding the total weight of 3 boys weighing 45 kg, 3400 g, and 0.07 tons respectively. In this case, the mass unit conversion table should be used to convert the weights of all the three boys to one unit of measuring mass.

Unit Conversion Chart for Mass:

Any matter which occupies space is said to have mass. The mass of a substance is measured in kilograms in the International System of units. Unit conversion table for various other units of measuring mass is given below. 

Conversion of Units Example Problems (Mass):

1. Convert 7.23 kgs into milligrams.

Solution:

1 kg = 1000 g and 1 gm  = 1000 mg

So, 1 kg = 1000 x 1000 = 106 mg

7.23 kg = 7.23 x 106 mg = 7230000 mg

2. A boy visits a grocery shop and purchases 2 kg onion, 500 g brinjal, and 25 mg of elaichi. He puts these items into a bag measuring 20 g after paying the bill. What is the total weight carried by the boy in kilograms?

Solution:

Weight of the onion = 2 kg

Weight of the brinjal = 500 g

Weight of elaichi = 25 mg

Weight of the bag =  20 g

Recall what is conversion of units factor for mass. 

1 kg = 1000 g and 1 g = 1000 mg

So, 1 g = 0.001 kg and 1 mg = 0.000001 kg

Weight of onion = 2 kg

Weight of brinjal = 500 g = 500 x 0.001 = 0.5 kg

Weight of elaichi = 25 mg = 25 x 0.000001 = 0.000025 kg

Weight of the bag = 20 g = 20 x 0.001 = 0.02 kg 

Total weight = 2 + 0.5 + 0.000025 + 0.02 = 2.520025 kg

Unit Conversion for Length:

Length is one of the physical quantities which has several units of measurement. The SI unit of length is metre (m). The unit conversion table for length is as below:

Conversion of Units Example Problems (Length):

1. In one of the conversion of units example problems, it is given that an object travels 5 cm to the north, 7 m to the east, and 0.5 km to the south. What is the total distance travelled by the object in metres?

Solution:

In this problem, the unit conversion table for length is used. 

1 cm = 0.01 m and 1 km = 1000 m

So, the total distance travelled by the object is 

= 5 cm + 7 m + 0.5 km 

= 5 x 0.01 + 7 + 0.5 x 1000

= 0.05 + 7 + 500

= 507.05 m

Fun Facts:

• Unit conversion table for time also can be written by prefixing the SI unit of time i.e. seconds with required conversion factors. 

• There are certain prefixes that are generally used to represent the units in terms of the basic unit of measurement. For example, kilo refers to 103, milli refers to 10−3, mega refers to 106, micro refers to 10−6, nano refers to 10−9, centi refers to 10−2, pico refers to 10−12, etc.