dimensional analysis is a problem-solving approach in which you use the units in order to set up your calculation this is commonly used when converting one quantity from a given unit to another unit for example i will show you how to convert centimeters into inches using the conversion factor that one inch is equal to 2.54 centimeters in setting up your equation you're going to start with a given unit use your conversion factor as a fraction in order to calculate the equivalent in the desired unit for example let's convert 15 centimeters to inches using the conversion factor 1 inch equals 2.54 centimeters so i will start my equation by writing down the given unit which would be 15 centimeters now the conversion factor i am going to write down as a fraction and i will put in the denominator the unit that is the same as the given unit so that would be 2.54 will go to the denominator so 2.54 centimeters and then the one inch will go to the numerator and what that does is it converts your centimeters into inches because the the unit of centimeters get cancelled out and you will be left with the desired unit which is inches when you entered this in your calculator it would be 15 divided by 2.54 you will get 5.905 inches but following the rules of sig figs we have two significant figures in the given unit so our answer should have two significant figures as well so this would be rounded to 5.9 inches so our final answer is 15 centimeters is equal to 5.9 inches there may be cases when you need more than one conversion factor to go from your given unit to your desired unit for example in converting from meter to inches we can go from meters to centimeters and then from centimeters to inches because these are two conversions we will be needing two conversion factors so we write down the given unit that's 8.00 meters and then the conversion factor where we are cancelling the meters so i would use 10 to the minus 2 meters at the bottom and one cm at the top so at this point the unit's meter is canceled out leaving you with centimeter now the centimeter can be converted to inches using one inch equals 2.54 centimeter to remove the centimeter we're going to put the 2.54 centimeter at the bottom and then the one inch at the top so this will cancel out the centimeters and you will be left with the desired unit which is inch so to enter this in your calculator you would put 8 divide by open parenthesis 1 exponent minus 2 times 2.54 close parenthesis your calculator is going to give you 314.96 but to express in the correct number of sig figs which is 3 just like the given unit we will round this to 315 so 8 meters equals 315 inches now i would just like to make a note that we use the conversion factor here that one centimeter is equal to 10 to the minus 2 meters this is a common mistake is to enter 10 to the minus 2 as 10 e minus 2. so that is wrong so i i told you to enter it as 1 exponent minus 2 because when we are entering scientific notation in the calculator we are entering the number 1 here and the exponent of negative 2 here so in your calculator this would be one exponent minus 2. what is the area in centimeters squared of a square tile with a side measuring two inches so in this question it's important to note that they gave us the side of the square tile which is two inches however it's asking for the area in centimeters squared so this means that before we do our conversion we need to calculate the area first so for a square with a side measuring two inches the area is equal to 2 inches squared and that would be 4.0 inches squared now we can proceed with the conversion so we are to convert four inches squared to centimeters squared so again we start with we start with the given quantity for inches squared and then we will use the conversion between inches and centimeters so since we have inches here we will need to cancel that out and we will put the inch on the denominator and have the centimeter on the numerator if we look at our equation right now we have inch squared here and then we have inches in the denominator so they aren't exactly matched so that means that inches squared is not cancelled out by just one inch so what we can do is to make that square by squaring the whole conversion factor so now we have inch squared at the bottom which cancels out the original inch squared unit and we will be left with a unit centimeter squared so when you enter this in your calculator you will enter 4 times 2.54 squared so don't forget that the square applies to the whole conversion factor it applies to the units and it applies to the number as well so 4 times 2.54 squared gives you 26 centimeters squared let us practice unit conversion in which you are given a few choices of conversion factors to use the question is how many cubic inches is 1.5 gallons in this problem we know that we have to convert 1.5 gallons to cubic inches so these are units of volume so i will focus on the volume conversions right here and we are mostly interested in relationships between gallons and cubic inch i'm looking at the conversion factor now and i see gallons here i see cubic inch here so we don't have a direct relationship between the two however we can note that the cubic inch is given with a conversion to cubic centimeters which is equal to milliliters so one cubic inch is equal to 16.4 ml now for the gallons i have a conversion between gallons 2 liters so i think we can connect the two units by also noting that we are going to use the liter to ml conversion and that would be 1ml equals 10 to the minus 3 liters so we are to go from gallons to cubic inch so we are going from gallons right here to cubic inch right here so the route that we are going to take is to convert gallons to liters to ml to cubic inch so this is effectively one two three conversions so we will be using three conversion factors we start with the 1.5 gallons we want to get rid of gallons so i will use the relationship between gallons and liters so the gallon would be at the bottom so one gallon equals to 3.7 854 liters which i would put at the top so at this point we have done the gallons to liters conversion my unit is liters i can now convert that to ml using the relationship 1ml equals 10 to the minus 3 liters so the 10 to the minus 3 liters go at the bottom 1ml goes at the top so that converts liters to ml and our current unit is ml and finally we can convert milliliters to cubic inch by using the conversion factor where we will put the ml at the bottom so 16.4 ml and one cubic inch at the top and that cancels out the ml and we have converted ml to cubic inch so using these three conversion factors successively we went from gallons to cubic inches enter this in your calculator as so let's work first with all of the numerator so i would put 1.5 times 3.7854 divided by open parenthesis 1 exponent minus 3 times 16.4 close parenthesis you should be able to get the answer which is 350 cubic inches if an automobile is able to travel 254 miles on 11.2 gallons of gasoline what is the gas mileage in kilometers per liter so this question calls for gas mileage in kilometers per liter we don't have gas mileage but we do have the distance of 254 miles and the volume of gasoline which is 11.2 gallons so gas mileage is equal to 254 miles divided by 11.2 gallons and we need the gas mileage in kilometers per liter so note that we are actually going to do two unit conversions here the first one being from miles to kilometers and the second one is gallons to liters again we start with a given unit that's 254 miles over 11.2 gallons and we first take care of converting miles to kilometers if we look at our choices of conversion factors we can see a direct conversion between miles and kilometers here one mile equals 1.600 kilometers so i will put one mile at the bottom so that would cancel out the mile in the given unit and the 1.6093 kilometers go at the top so now we have converted miles to kilometers so next we will convert gallons to liters in our selection of conversion factors i can see that this conversion factor is useful one gallon equals 3.7854 liters so since i start out with gallons here at the bottom i want to cancel the gallons so i will put the gallons at the top in my conversion factor and then the 3.7854 liters go at the bottom so that cancels out the gallon and for the volume unit we now have liters so we now have gas mileage in kilometers per liter so what we're going to do is enter this in the calculator so everything at the top first 254 times 1.6093 divide open parenthesis 11.2 times 3.7854 the answer is 9.64 kilometers per liter the average speed of a nitrogen molecule in air is 515 meters per second what is the speed in miles per hour in this problem we are converting speed of 550 meters per second two miles per hour this involves conversion of two units meters to miles and seconds to hour so if we start with 515 meters per second for our dimensional analysis equation we can first take care of the meters and convert it to miles in our length or distance conversion factors i see miles here and its relationship with kilometers in addition to that conversion we're going to need the conversion between kilometers and meters which is one kilometer equals 10 to the third meters so for the meters to mile conversion the plan is to go from meters to kilometers and then to miles so to cancel the meters in the original unit i will use the conversion factor 1 kilometer equals 10 to the third meters and i will be placing the 10 to the third meters at the bottom and one kilometer at the top to cancel out the meters and be left with kilometers the kilometers has to be converted now to miles and i will use the conversion 1.6093 kilometers at the bottom and one mile at the top so at this point we have taken care of the conversion from meters to miles so now to convert seconds to hours we will be using time conversions that we all know and that's 60 seconds is one minute and 60 minutes is one hour in our starting unit seconds is at the bottom so our first conversion factor for the time conversion would have the seconds at the top so i will use 60 seconds over one minute that converts seconds to minutes and to convert minutes to hour i will put 60 minutes at the top and one hour at the bottom so looking at our equation here the unit left for us would be miles per hour which is what our desired unit is to calculate you will enter in your calculator all the numerator first 515 times 60 times 60 then all of the denominator divided by open parenthesis one exponent to the third times 1.6093 the calculation will give you 1150 miles per hour