Introduction to Pharmaceutical Calculation

Fractions Definition Parts of whole numbers Portion in relationship to a whole Component parts Numerator – whole number above the fraction line; number of parts or portion Denominator – whole number below the fraction line; number of equal parts to make a whole

Fraction Types Proper fraction - numerator is smaller than the denominator. Ex: 3/4 There are three parts of four parts possible. The value of the entire fraction is less than one. More examples: 5/9; 2/3; 4/7  

Illustration

Fraction Types 2. Improper fraction - numerator is larger than the denominator. Improper fractions are necessary in some calculations. Ex: 3/2 There is one whole (two of two parts) and one of two parts possible. The value of the entire fraction is greater than one. More examples: 13/5 ; 5/4

Fraction Types Mixed fraction – combination of a whole number and a proper fraction written together. Ex: 1 ½ ; 2 ¾ There is one whole number (two of two parts) and one of two parts possible. The value of the entire fraction is greater than one.

Common denominators When fractions have the same denominator, they are said to have a common denominator. Ex: 1/8, 3/8, and 5/8 all have a common denominator of 8. Note: There is a need in mathematics to find a common denominator. Before fractions can be added or subtracted, the denominators of all the fractions in the problem must be the same.  

Determining Common Denominators Finding a common denominator for the fractions: 1/4, 3/8, 5/16 Step 1: To determine the common denominator, first find the largest denominator. In the set of fractions: 1/4, 3/8, 5/16, the largest denominator is 16. Step 2: Check if the other denominators can be divided into the largest denominator an even number of times. Both 4 and 8 can be divided into 16. Then multiply the result by the numerator then retain the common denominator. 16 ÷ 4 = 4 x 1 = 4 then retain the common denominator = 4/16 16 ÷ 8 = 2 x 3 = 6 then retain the common denominator = 6/16 16 ÷ 16 = 1 x 5 = 5 then retain the common denominator = 5/16 Step 3: Change the fractions to have a common denominator without changing the value of the fractions.

Common denominators When the denominators cannot be divided by the same number, a common denominator can be found by multiplying one denominator by the other. For the fractions 1/3 and 1/8, the common denominator is determined by multiplying 3 by 8 then 1/3 = take the denominator 3 then multiply by 8 = 24 1/8 = take the denominator 8 then multiply by 3 = 24 For the fractions 3/4, 1/7, and 1/2, the common denominator is determined by multiplying 4 by 7. The 2 in 1/2 is a multiple of 4; any number divisible by 4 will be divisible by 2. 3/4 x 7/7 = 21/28 1/4 x 4/4 = 4/28 1/2 x 14/14 = 14/28 You may also think of a number divisible by all of the denominators, in this example, it’s 28.

Lowest Terms A fraction is at its lowest terms when the numerator and the denominator cannot be divided by the same number to arrive at a lower valued numerator and denominator. Example: 3/4 is at its lowest terms because the numerator (3) and the denominator (4) cannot be divided by the same number to lower their values.   The fraction 4/8 is not at its lowest terms because the numerator (4) and the denominator (8) can both be divided by the same number to lower their values. The largest number the numerator (4) and the denominator (8) can be divided by is 4. Therefore: 4/8 is 1/2 at its lowest terms.

Improper and Mixed Fractions To calculate with a mixed fraction, it needs to be changed to an improper fraction. Once an answer is determined, the improper fraction is normally converted back to a mixed fraction. To change an improper fraction to a mixed fraction: 1. Divide the numerator by the denominator. 2. Reduce the remaining fraction to its lowest terms. Example: 3/2 becomes 3 ÷ 2, which equals 1 1/2

Improper and Mixed Fractions Examples: To change the improper fraction 5/4 to a mixed fraction, 5 is divided by 4. 5 ÷ 4 = 1, the remainder becomes the numerator = 1/4 , so it becomes 1 ¼ 9/6 becomes 9 ÷ 6, which equals 1 3/6. 1 3/6 can be reduced to 1 1/2.

Improper and Mixed Fractions To change a mixed fraction to an improper fraction: 1. Multiply the denominator times the whole number. 2. Then add the numerator to this amount. This sum will become the new numerator and the denominator will remain the same. Example: 1 1/2 becomes 2 x 1 (whole number) + 1 (numerator). Answer: 3/2 Examples: Change mixed fraction 4 7/8 to an improper fraction 8 x 4 = 32 + 7 = 39 Answer: 39/8

Practice Problems Change the following mixed fractions to Improper fraction: 3 5/8 2 7/9 10 2/5 8 3/7 20 1/8 4 5/6

Answer 3 5/8 = 8 x 3 = 24 + 5 = 29/8 2 7/9 = 9 x 2 = 18 + 7 = 25/9

Practice Problems Change the following improper fractions to mixed fractions: 11/5 9/4 25/7 5/3 20/8 16/6

Answer 11/5 = 11 ÷ 5 = 2 1/5 9/4 = 9 ÷ 4 = 2 1/4 25/7 = 25 ÷ 7 = 3 4/7 11/5 = 11 ÷ 5 = 2 1/5 9/4 = 9 ÷ 4 = 2 1/4 25/7 = 25 ÷ 7 = 3 4/7 5/3 = 5 ÷ 3 = 1 2/3 20/8 = 20 ÷ 8 = 2 4/8 or 2 1/2 16/6 = 16 ÷ 6 = 2 4/6 or 2 1/3

Adding Fractions  Step 1a. To add or subtract fractions, find equivalent values with a common denominator for all fractions. In this example of 1/2 + 1/4, 2 divides into 4; 4 is the common denominator.  Step 1b. For the fraction ½ ; 4 (common denominator) divided by 2 multiply by the numerator ( 1 ) 4 ÷ 2 = 2 x 1 = 2, the fraction becomes 2 / 4 For the fraction ¼ : 4 ÷ 4 = 1 x 1 = 1, the fraction remains ¼

Adding fractions Step 2. Add the numerators only. 2 + 1 = 3. The denominators remain the same. The answer is 3/4. Step 3. Reduce fraction to lowest terms if needed. Convert any improper fractions to mixed fractions. In this equation, 3/4 is at lowest terms.

Sample problems Solve the problem 3/4 + 2/6 by following the step-by-step process. Step 1. Find equivalent values with a common denominator for all fractions. 3/4 and 2/6 have a common denominator of 12. 3/4 becomes 9/12 and 2/6 becomes 4/12. Step 2. Add numerators only. The denominator stays the same. 9/12 + 4/12 = 13/12 The final step is to convert the improper fraction to a proper fraction. 13/12 = 1 1/12.  

Subtracting Fractions Solve the problem 1/3 – 1/4 by following the step-by-step process. Step 1. Find equivalent values with a common denominator for all fractions. 1/3 and 1/4 have a common denominator of 12. 1/3 becomes 4/12 and 1/4 becomes 3/12. Step 2. Subtract numerators only. The denominator stays the same. 4/12 – 3/12 = 1/12. The final step is to convert the improper fraction to a proper fraction. 1/12 is at lowest terms.  

Practice problems Add the following fractions: 3/4 +7/8 +1/4 1/8 + 6/8 + 3/8 4/10 + 11/15 + 1/5 1/3 + 3/4 + 5/6 1/2 + 3/12 + 1/6 + 3/4 5/7 + 2/3

Answers 3/4 +7/8 +1/4 = 15/8 or 1 7/8 1/8 + 6/8 + 3/8 = 10/8 or 1 2/8 or 1 1/4 4/10 + 11/15 + 1/5 = 40/30 or 1 10/30 or 1 1/3 1/3 + 3/4 + 5/6 = 1 11/12 1/2 + 3/12 + 1/6 + 3/4 =20/12 or 1 8/12 or 1 2/3 5/7 + 2/3 = 29/21 or 1 8/21

Practice problems Subtract the following fractions 7/8 – 1/4 2/4 – 6/16 3/5 – 1/10 1/2 – 1/4 2 2/3 – 1 1/6 ¾ - 5/8

Answers 7/8 – 1/4 = 5/8 2/4 - 6/16 = 2/16 or 1/8 1/2 – ¼ = 1/4 2 2/3 – 1 1/6 = 9/6 or 1 3/6 or 1 ½ ¾ - 5/8 = 1/8

Multiplying Fractions Increasing the numerator increases the portion while the denominator or the whole remains the same. Increasing the denominator enlarges the whole while the portion remains the same. The following rules must be understood when working with fractions:   Multiplying or increasing only the numerator increases the value of the fraction. In the example 2/7 x 3 = 6/7, 2 parts of 7 is multiplied by 3 (whole number) and the result is 6 parts of 7. Multiplying or increasing only the denominator decreases the value of the fraction. In this example 2/7 x 1/3 = 2/21, 2 parts of 7 is multiplied by 1/3 (less than 1) and the result is 2 parts of 21.

Multiplying fractions can be simple if you follow the steps Multiplying fractions can be simple if you follow the steps. Using 1/2 x 1/3, the steps are: Step 1. Multiply all numerators together. 1 x 1 = 1  Step 2. Multiply all denominators 2 x 3 = 6 Step 3. Express the answer as a fraction = 1/6 Step 4. Reduce fraction to lowest terms (may be an improper fraction). Convert any improper fractions to mixed fractions. The fraction 1/6 is at lowest terms.

Sample problems Solve this problem step-by-step: 7/12 x 3/8 Step 1. Multiply all numerators together. 7 x 3 = 21 Step 2. Multiply all denominators together. 12 x 8 = 96 Step 3. Express the answer as a fraction (make sure the product of the numerators is over the product of the denominators) = 21/96 Step 4. Convert any improper fractions to mixed fractions (if needed) and reduce fraction to lowest terms. 21/96 can be reduced to 7/32.

Multiplying Fractions and Whole Numbers Solve the problem: 3/4 x 50. Step 1. Change the whole number to a fraction by placing the number over one. Change mixed fractions to improper fractions. 50 becomes 50/1 Step 2. Multiply numerators. Step 3. Multiply denominators. 3/4 x 50/1 = 150/4 Step 4. Reduce to lowest terms. 150/4 = 37 1/2

Dividing fractions Dividing fractions can be simple if you follow the steps. Using 1/2 ÷ 1/4 the steps are: Step 1. Invert the divisor. The divisor is the number being used to divide. The inverted divisor is called the reciprocal. The reciprocal of 1/4 is 4/1. The divisor 1/4 becomes 4 over 1. Step 2. Change the division sign to a multiplication sign. The problem becomes 1/2 x 4/1. Step 3. Multiply the fractions. 1/2 x 4/1 = 4/2 Step 4. Reduce fraction to lowest terms (may be an improper fraction). Convert any improper fractions to mixed fractions. 4/2 can be reduced to 2/1 or 2.

Dividing Fractions Solve the problem 7/8 ÷ 2/3 by following the step-by-step process. Step 1. Invert the divisor, the number being used to divide. 2/3 becomes 3/2. Step 2. Change the division sign to a multiplication sign. The problem becomes 7/8 x 3/2. Step 3. Multiply the fractions. 7/8 x 3/2 = 21/16 Step 4. Convert any improper fractions to mixed fractions (if needed) and reduce fraction to lowest terms. 21/16 can be reduced to 1 5/16.

Dividing Fractions Solve the problem 3 1/2 ÷ 1/5. Step 1. Change mixed fractions to improper fractions: 3 1/2 = 7/2 Step 2. Invert the divisor, the number being used to divide. 1/5 becomes 5/1. Step 3. Change the division sign to a multiplication sign. The problem becomes 7/2 x 5/1. Step 4. Follow the rules listed under multiplication of fractions. 7/2 x 5/1 = 35/2, which can be reduced to 17 1/2 Note: Dividing Fractions and Whole Numbers : Change the whole number to a fraction by placing the number over one>

Practice problems Multiply the following fractions 1/6 x 2/3 3/8 x 1/5

Answers 1/6 x 2/3 = 2/18 or 1/9 3/8 x 1/5 = 3/40

Practice problems Divide the following fractions: ½ ÷ ¼ 5/6 ÷ 2/3 4/9 ÷ 1/8 6/10 ÷ 1/8 1/5 ÷ 6/7 3/4 ÷ 1/3

Answers ½ ÷ ¼ = 4/2 or 2 5/6 ÷ 2/3 = 15/12 or 1 3/12 or 1 1/4 4/9 ÷ 1/8 = 32/9 6/10 ÷ 1/8 = 48/10 or 4 8/10 or 4 4/5 1/5 ÷ 6/7 = 7/30 3/4 ÷ 1/3 = 9/4 or 2 1/4

Summary Remember to use a common denominator when adding or subtracting fractions. Use improper fractions when multiplying or dividing with mixed fractions. When dividing, invert the divisor to use the reciprocal, and then multiply. Always reduce the final fraction to the lowest terms.