1. Introduction to Pharmaceutical Calculation

3. Decimals
Definition
A decimal is a fraction whose denominator is ten or any power of ten.
Many fields use the metric system of measure, which is a decimal system. At times, it is necessary to change a fraction to a decimal to simplify a procedure. Calculations using decimals are accomplished in the same manner as with whole numbers; the difference is decimal placement.
Placing each decimal point in its correct location is critical for accurate calculations.

4. Decimals
A decimal is a fraction whose denominator is ten or any power of ten. The denominator is never written since the decimal point serves to indicate place value of the numerals.
Examples:
Decimal Fraction Translation
/10 one tenth
/100 forty-five hundredths
/1000 sixty-five thousandths

5. Converting Fractions to Decimals
To convert a common fraction to decimals, divide the numerator by the denominator.
For example, to convert 1/2, 1/2 becomes 1 ÷ 2, which equals /2 = 1 ÷ 2 = 0.5
Drag each value to convert the following fraction to a decimal; not all numbers will be used.
1/3 = 1 ÷ 3 = 0.333

6. Converting decimals to fractions
The key to converting decimals to fractions is knowing which power of 10 to use. To convert a decimal to a common fraction, complete the following steps:
Example: 0.125
Step 1. Place decimal as numerator over appropriate power of 10 as the denominator.
0.125/1000: Three numbers behind the decimal point (.125) = three zeros.

7. Converting decimals to fractions
Step 2. Remove the decimal point. The decimal then turns into a fraction of 125/1000.
Step 3. Reduce to lowest terms. 125 ÷ 125 = 1 and 1000 ÷ 125 = 8
125/1000 can be reduced to 1/8
0.125 = 125/1000 = 1/8

8. Decimals and Equivalent Decimal Fractions
= 1/100,000
= 1/10,000
0.001 = 1/1,000
0.01 = 1/100
0.1 = 1/10
1 = 1/1

9. Adding/Subtracting Decimals
Calculations using decimals are accomplished in the same manner as with whole numbers. Make sure the decimal points are aligned, even in the results. Examples of each step for adding and subtracting decimals:
Step 1. Make sure all the decimals are aligned.
15.432
5.8
+3.25
answer
8.65
answer

10. Adding/Subtracting Decimals
Step 2. Add or subtract the numbers. Use zeros as place holder on the right side of the decimal if needed.
Ex:
5.800
24.482
Step 3. Place the decimal in the answer - directly beneath the other decimals.
15.432
5.8
+ 3.25

11. Multiplying Decimals
When multiplying decimals, the following steps should be followed:
Step 1. Multiply as if they were whole numbers
6.356
x 1.6
38136
6356__
101696
Step 2. Count how many numbers are in decimal places in all factors (numbers right of the decimal).
6.356 (Three numbers are right of the decimal point.)
(One number is right of the decimal point.)
Four numbers are right of decimal point.

12. Multiplying Decimals
Step 3. Place the decimal point in the answer by starting at the right of the product and moving a number of places equal to the sum of the decimal places the numbers multiplied (the number from Step 2).
Answer: (Four numbers are right of the decimal point)

13. Dividing Decimals
When dividing decimals, the following steps should be followed. Example: 12.5 ÷ 0.625
Step 1. Set up to divide, making sure that the number after the division sign (÷) is the divisor divides into 12.5
Step 2. Move the decimal point in the divisor as many places to the right as is necessary to make it a whole number.
0.625 becomes 625.

14. Dividing Decimals
Step 3 . Move the decimal point in the dividend the same number of places to the right. The dividend is the number being divided. The decimal moved three places in Step 2; therefore, the decimal in the dividend must move three places to the right.
12.5 becomes 12,500
Step 4. Place the decimal in the answer directly above the decimal in the dividend and complete the division divided into 12,500 equal 20.
12,500 ÷ 625 = 20

15. Rounding Decimals Rounding guidelines:
Perform all calculations using the whole decimal. Round the answer to desired place using the terminal number. The terminal number determines if the number is rounded up or remains the same. It follows the number you are rounding. Rules for rounding decimals:
Example: to round to the hundredths place (second decimal place), first find the terminal number. In the example, the terminal number is in the thousandth place or the third decimal place. The terminal number is 4; therefore, 6 remains the same. The rounded number is 0.06.

16. Decimals Summary
In this lesson, you performed addition, subtraction, multiplication, and division of decimals.
To succeed in using decimals:
Know the multiples of 10 on both sides of the decimal point (example tens versus tenths).
Line up the decimal points when adding and subtracting numbers with decimals.
Know where to place the decimal point when multiplying or dividing numbers with decimals.
When rounding decimals, use the terminal number to determine whether to round up or down.

17. Decimal point consideration in pharmacy
Do not write a whole number in decimal form.
- when writing a whole number, avoid writing a period followed by a zero ( 5.0 ), because periods are sometime hard to see and errors may result from misreading the number.
Ex: the period in 1.0 may be overlooked and the number could appear to be 10 instead, causing a 10-fold dosing error, which could harm the patient if overdose is given.

18. Decimal point consideration in pharmacy
When writing a fraction in its decimal form, always write a zero before the period.
- the periods are sometimes difficult to see, and .5 may be misread as 5, however if the period in 0.5 were illegible, the zero would alert the reader that a period is supposed to be there and would then verify the order.