ANÁLISIS FINANCIERO

The metric system exploits the convenience of the number “10.” As Table 4-1 H _{shows, we attach a}

prefix to the basic unit of measure to create a decimal multiple or submultiple. The purpose of pre- fixes is to reduce the number of zeros in the value; it is much easier to write “μg” than it is to write “0.000001 g” or “1 * 10-6 _g.”

Let us consider some examples, starting with the basic unit of “gram” (g), which is about the weight of a typical paper clip. If we have an object that weighs 1000 g, we can report the weight as such, or we can call it “1 kilogram” (1 kg). As Table 4-1 shows, 1 kg is the same as 1000 of its basic unit, the gram:

1 kg = 1000 g

If we have an object that weighs 0.001 g, we can report the weight as such, or we can call it “1 milligram” (1 mg). As Table 4-1 shows, 1 mg is the same as ₁₀₀₀1 , or one-thousandth, of its basic unit, the gram:

1 mg = 0.001 g

This chapter presents systems of measurement, their units and symbols, and strategies and procedures for converting among them.

UniTed STaTeS cUSToMary SySTeM of UniTS

The United States Customary System of Units is also called the “English Imperial” or “American” system. The most commonly used units in this system appear below.

Likewise, a microliter (μL) is a millionth of a liter:

1 μL = _1,000,0001 L = 0.000001 L = 10-6 _L A kilometer (km) is 1000 meters:

1 km = 1000 m = 103 _m

Converting Between Units

It is often necessary to convert one unit into another, whether within one system or between systems. Always remember that in moving to a larger unit, we divide; we do not multiply. Conversely, in moving to a smaller unit, we multiply; we do not divide. This is true in all unit conversions, regardless of the system. For example, in moving from “minutes” to “hours,” the number goes down, not up:

60 minutesS 1 hour In moving to the larger unit (“hours”), we have divided by “60.”

However, in moving from “hours” to “minutes,” the number goes up, not down; we multiply by “60” in moving to the smaller unit:

1 hourS 60 minutes

In the metric system, there are several equally effective ways to think through a conversion involving prefixes. We present two approaches now and two more later.

approach 1

This approach comprises two steps.

Step 1 In Table 4-1, locate the starting prefix and the target prefix and note their cor-

responding powers of 10. Subtract the target power of 10 from the starting power of 10: starting power of 10 - target power of 10 = ∆x

Step 2 Multiply the starting value by 10∆x_{, where ∆x is the difference between the} powers of 10.

Consider, for example, the conversion of “100 g” to “kg.”

prefix Symbol (How many, or how much, of the basic unit)factor power of 10 (10x_{) american Term}

giga G 1,000,000,000 9 billion

mega M 1,000,000 6 million

kilo k 1000 3 thousand

hecto h 100 2 hundred

deka (or deca) da 10 1 ten

no prefix. This is the level of the basic

unit (meter, liter, gram) 1 0 one

deci d 1/10 (0.1) -1 one-tenth centi c 1/100 (0.01) -2 one-hundredth milli m 1/1000 (0.001) -3 one-thousandth micro μ* 1/1,000,000 (0.000001) -6 one-millionth nano n 1/1,000,000,000 (0.000000001) -9 one-billionth pico p 1/1,000,000,000,000 (0.000000000001) -12 one-trillionth femto f 1/1,000,000,000,000,000 (0.000000000000001) -15 one-quadrillionth atto a 1/1,000,000,000,000,000,000 (0.000000000000000001) -18 one-quintillionth *This is the proper symbol for “micro,” although some clinics and hospitals prefer the abbreviation “mc” because the handwritten letter “μ” can be mistaken for “M” or “m”.

Step 1

prefix Unit power of 10

Starting Milli milligram -3 Target Micro microgram -6

prefix Unit power of 10

Starting none gram 0

Target Kilo kilogram 3

1000 g 100 g 0.1 mg 1 mg 1000 g 100 g 0.1 kg 1 kg

∆x = Starting power of 10 - Target power of 10 = 0 - 3 = -3

Step 2

(Starting value)* 10∆x _{= 100 * 10}-3 _{= 100 , 1000 = 0.1} Therefore, 100 g = 0.1 kg

notice that, in moving to the larger unit (g S kg), the value went down, not up; the procedure was the same as division by 1000. Here is a way to visualize the conversion:

now let us take an example that converts in the opposite direction, say, from “0.1 mg” to “μg.”

Step 1

∆x = Starting power of 10 - Target power of 10 = (-3) - (-6) = 3

Step 2

(Starting value) * 10∆x _{= 0.1 * 10}3 _{= 0.1 * 1000 = 100} Therefore, 0.1 mg = 100 μg

notice that, in moving to the smaller unit (mg S μg), the value went up, not down; the procedure was the same as multiplication by 1000. Here is a way to visualize the conversion:

Let us consider a final example that has a numeral other than “0” or “1,” say, the conversion of 0.382 mg into “μg.”

Using approach 1: Step 1.

Starting power of 10 - target power of 10 = ∆x (-3) - (-6) = 3 Step 2. 0.382 mg * 103 _{= 382 μg} Using approach 2: Step 1. larger factor smaller factor = 0.001 0.000001 = 1000 Step 2. Conversion is toward the smaller unit. Therefore, we multiply:

0.382 mg * 1000 = 382 μg approacheS 3 and 4

Dimensional analysis and the ratio method are two more approaches to solving problems of this type, but because their usefulness extends far beyond the conversion of metric prefixes, we treat them in their own later sections of this chapter.

approach 2

This approach also comprises two steps.

Step 1 In Table 4-1, locate the starting prefix and the target prefix and note their

corresponding factors. Calculate the ratio of the larger to the smaller. larger factor

smaller factor

Step 2 If the conversion is going toward the larger unit, then divide the value by the

above ratio. If it is going toward the smaller unit, then multiply.

Let us turn to the same two examples we saw in the first approach. In the conversion of “100 g” into “kg,” the larger factor is 1000 (for “kilo”), and the smaller is 1 (for “gram,” a basic unit). Therefore, the ratio of the larger to the smaller is

1000

1 = 1000

Because the conversion is going toward the larger unit, we divide the original value by the ratio, which is 1000:

100

1000 = 0.1 kg

In the second example, we carry out a conversion toward the smaller unit, that is, from “0.1 mg” to “μg”. The larger factor is 0.001 (for “milli”), and the smaller is 0.000001 (for “micro”). Therefore, the ratio of the larger to the smaller is

0.001

0.000001 = 1000

Because the conversion is going toward the smaller unit, we multiply the original value by the ratio, which is 1000: