UNIT 4 UNIT CONVERSION AND CALCULATIONS
Many of the problems that you will come across in science and mathematics exercises involve change from one unit to another. In this unit, procedures involved in the conversion of one unit of measurement to another shall be dealt with. Dimensional analysis, which is a general method of solving problem, which belongs to this category, will also be given attention.
Problem Solving and Dimensional Analysis
Problem solving simply means the act of finding solution to a particular problem. While Dimensional analysis is a general problem solving technique in which one uses the units of quantities to help one decides how to set up the problem.
Steps for Converting Units by Dimensional Analysis
The following steps give a summary of the approach to problem solving by dimensional analysis: (i) Write out the desired conversion from the problem statement,
Note the given unit and the desired unit. You want to convert as follows: Quantity expressed in given unit convert to quantity expressed in desired unit.
(ii) Obtain the conversion factor, (Conversion factor is a factor equal to I that converts a quantity in one unit to the same quantity in another unit) write the equivalent statement, then the conversion factor for the conversion from given unit to desired unit. You write the conversion factor as a ratio with the given unit in the denominator and the desired unit in the numerator. The conversion factor will be a number, or factor multiplied by a ratio of unit.
Conversion factor = factor x (desired unit)/(given unit)
(iii) Perform the conversion calculation. Multiply the quantity in the given unit by the conversion factor.
Quantity in given unit x factor x (desired unit)/( given unit) = quantity in desired unit
Example, how many glasses of beer will you need for a party of 12 people, if each of the invited guests would drink three glasses of beer. Then knowing the amount of glasses of beer needed, the next question is how many bottles of beer you should buy, if each bottle contains four glasses.
The calculations for the problems, in brief or at a glance are as follows:
12 persons x (3 glasses of beer)/(1 person) x (1 bottle of beer)/(4 glasses of beer) = 9 bottles of beer
Conversion of Units
One of the most important types of conversion is from one metric unit to another. Three basic steps as mention earlier were followed to accomplish this change viz.
(1) Write out the desired conversion
(2) Obtain the conversion factor and
(3) Perform the conversion calculation.
Conversion from One Metric Unit to Another
Example 1: How many milligrams are there in 7.71g?
Here we are trying to change from a metric unit having no prefix to a corresponding unit with a prefix. Step 1: The desired conversion; is 7.71g converts to mg.
Step 2: The conversion factor; what is the relationship between grams and milligrams.
Recall that; milli-is the prefix meaning 10-3 . This can be used to write an equivalent statement relating grams to milligrams.
1mg = 10-3g
The factor that converts grams to milligrams is
1mg = 10-3g (converts to g)
Step 3: The conversion calculation; This is done by multiplying what is given (7.71g) by the conversion factor (1mg/(〖10〗^3 mg)).
7.71g x 1mg/(〖10〗^3 mg) = 7.71 x 103
Example2
a) The straight line distance from a town A in the south West Nigeria and a town B in Benue State also in Nigeria is 2.98 x 105m. How far is the town in kilometers?
b) The volume of water in a household water reservoir is 1.35 x 1025ml. Calculate its volume in litres.
Solution
Step 1: The desired conversion is, 2.98 x 105m converts to km
Note that, 1km = 103m.
Step 2: the conversion factor from meters to kilometer is
1km/(〖10〗^3 m)
Also note that, the given unit is at the bottom and the desired unit is at the top.
Step 3: The conversion calculation is
2.98 x 105m x 1km/(〖10〗^3 m) = 2.98 x 102km
The correct power of 10 in the final answer was obtained by subtracting 3 form 5 (i.e 5- 3 =2); giving the power of 10 in the answer as 2
Step 1: The desired conversion is 1.35 x 1024cm3 converts to litres
One way to solve this problem is simply to replace the prefix milli with multiplication 10-3
1.35 x 1024 ml = 1.35 x 1024 x 10-3l = 1.35 x 1021l
This problem can also be solved by means of dimensional analysis. You should note that conversion factor from milliliters to litres is 〖10〗^(-3)/1ml . Thus, by dimensional analysis;
1.35 x 1024l x 10-3 l
= 1.35 x 1024 + (-3)
= 1.35 x 1021ml
Do the following self assessment exercise
Convert the following: (a) 2.58g to kg (b) 55.4cm to m (c) 8.11L to mL by means of dimensional analyses.
Many of the problems that you will come across in science and mathematics exercises involve change from one unit to another. In this unit, procedures involved in the conversion of one unit of measurement to another shall be dealt with. Dimensional analysis, which is a general method of solving problem, which belongs to this category, will also be given attention.
Problem Solving and Dimensional Analysis
Problem solving simply means the act of finding solution to a particular problem. While Dimensional analysis is a general problem solving technique in which one uses the units of quantities to help one decides how to set up the problem.
Steps for Converting Units by Dimensional Analysis
The following steps give a summary of the approach to problem solving by dimensional analysis: (i) Write out the desired conversion from the problem statement,
Note the given unit and the desired unit. You want to convert as follows: Quantity expressed in given unit convert to quantity expressed in desired unit.
(ii) Obtain the conversion factor, (Conversion factor is a factor equal to I that converts a quantity in one unit to the same quantity in another unit) write the equivalent statement, then the conversion factor for the conversion from given unit to desired unit. You write the conversion factor as a ratio with the given unit in the denominator and the desired unit in the numerator. The conversion factor will be a number, or factor multiplied by a ratio of unit.
Conversion factor = factor x (desired unit)/(given unit)
(iii) Perform the conversion calculation. Multiply the quantity in the given unit by the conversion factor.
Quantity in given unit x factor x (desired unit)/( given unit) = quantity in desired unit
Example, how many glasses of beer will you need for a party of 12 people, if each of the invited guests would drink three glasses of beer. Then knowing the amount of glasses of beer needed, the next question is how many bottles of beer you should buy, if each bottle contains four glasses.
The calculations for the problems, in brief or at a glance are as follows:
12 persons x (3 glasses of beer)/(1 person) x (1 bottle of beer)/(4 glasses of beer) = 9 bottles of beer
Conversion of Units
One of the most important types of conversion is from one metric unit to another. Three basic steps as mention earlier were followed to accomplish this change viz.
(1) Write out the desired conversion
(2) Obtain the conversion factor and
(3) Perform the conversion calculation.
Conversion from One Metric Unit to Another
Example 1: How many milligrams are there in 7.71g?
Here we are trying to change from a metric unit having no prefix to a corresponding unit with a prefix. Step 1: The desired conversion; is 7.71g converts to mg.
Step 2: The conversion factor; what is the relationship between grams and milligrams.
Recall that; milli-is the prefix meaning 10-3 . This can be used to write an equivalent statement relating grams to milligrams.
1mg = 10-3g
The factor that converts grams to milligrams is
1mg = 10-3g (converts to g)
Step 3: The conversion calculation; This is done by multiplying what is given (7.71g) by the conversion factor (1mg/(〖10〗^3 mg)).
7.71g x 1mg/(〖10〗^3 mg) = 7.71 x 103
Example2
a) The straight line distance from a town A in the south West Nigeria and a town B in Benue State also in Nigeria is 2.98 x 105m. How far is the town in kilometers?
b) The volume of water in a household water reservoir is 1.35 x 1025ml. Calculate its volume in litres.
Solution
Step 1: The desired conversion is, 2.98 x 105m converts to km
Note that, 1km = 103m.
Step 2: the conversion factor from meters to kilometer is
1km/(〖10〗^3 m)
Also note that, the given unit is at the bottom and the desired unit is at the top.
Step 3: The conversion calculation is
2.98 x 105m x 1km/(〖10〗^3 m) = 2.98 x 102km
The correct power of 10 in the final answer was obtained by subtracting 3 form 5 (i.e 5- 3 =2); giving the power of 10 in the answer as 2
Step 1: The desired conversion is 1.35 x 1024cm3 converts to litres
One way to solve this problem is simply to replace the prefix milli with multiplication 10-3
1.35 x 1024 ml = 1.35 x 1024 x 10-3l = 1.35 x 1021l
This problem can also be solved by means of dimensional analysis. You should note that conversion factor from milliliters to litres is 〖10〗^(-3)/1ml . Thus, by dimensional analysis;
1.35 x 1024l x 10-3 l
= 1.35 x 1024 + (-3)
= 1.35 x 1021ml
Do the following self assessment exercise
Convert the following: (a) 2.58g to kg (b) 55.4cm to m (c) 8.11L to mL by means of dimensional analyses.
- Teacher: TANIMU SARATU GALMA