## Units used in calculations

In calculations, several factors may be involved in a term and it is important to keep tract of the units of each of the factors in order to ascertain the final unit of the term. For example, consider converting 88 kilograms to micrograms. To make this conversion, several factors are present in the term for the calculation. Suppose we make the conversion as follows:

88 kg = 88( 1000)(1000)(1000) = 88,000,000,000 fig (1)

where 88(1000)(1000)(1000) is called the term of the calculation and 88 and the 1000s are the factors of the term. As can be seen, it is quite confusing what each of the 1000s refers to. To make the calculation more tractable, it may be made by putting the units in each of the factors. Thus,

88 kg = 88 kg^1000|-^1000^-1000ffg j = 88,000,000,000 fg (2)

This second method makes the calculations more tractable, but it makes the writing long, cumbersome, and impractical when several pages of calculations are done. For example, in designs, the length of the calculations can add up to the thickness of a book. Thus, in design calculations, the first method is preferable with its attendant possible confusion of the units. Realizing its simplicity, however, we must create some method to make it tractable.

This is how it is done. Focus on the right-hand side of the equation of the first calculation. It is known that the unit of 88 is kg, and we want it converted to fig. Remember that conversions follow a sequence of units. For example, to convert kg to fig, the sequence might be any one of the following:

In Sequence (3), the conversion follows the detailed steps: first, from kg to g, then from g to mg and, finally, from mg to fig. Looking back to Equation (1), this is the sequence followed in the conversion. The first 1000 then refers to the g; the second 1000 refers to the mg; and the last 1000 refers to the fig. Note that in this scheme, the value of a given unit is exactly the equivalent of the previous unit. For example, the value 1000 for the g unit is exactly the equivalent of the previous unit, which is the kg. Also, the value 1000 for the mg unit is exactly the equivalent of its previous unit, which is the g, and so on with the fg.

In Sequence (4), the method of conversion is a short cut. This is done if the number of micrograms in a kilogram is known. Of course, we know that there are 106 micrograms in a kilogram. Thus, using this sequence to convert 88 kg to micrograms, we proceed as follows:

Example 1 Convert 88,000,000,000 fig to kg using the detailed-step and the short-cut methods.

Solution: Detailed step:

88,000,000,000 fg = 88,000,000,000( 10-3)(10-3)(10-3) = 88 kg Ans

Note that the first (10 ) refers to the mg; the second (10 ) refers to the g; and the

last (10 ) refers to the kg. There is no need to write the units specifically. Short cut:

88,000,000,000 fg = 88,000,000,000( 10-6) = 88 kg Ans Note that there are 10—6 kg in one fig.

Example 2 Convert 10 cfs to m /d.

Solution: cfs is cubic feet per second.

1ft = 1/3.281m 1 s = 1/60 min; 1 min = 1/60 hr; 1 hr = 1/24 d

Therefore,

10 cfs = 10ft3/s = 10 f —J--13-1- = 24,462 m3/d Ans 