5 Basic Mathematics

Ensuring Proper Doses “It is the health-care professional’s responsibility to ensure that the patient receives the proper dose of medication, and to educate the patients about the proper measurement of doses.”

Essentials of Math Basic math uses: Arabic numbers Roman numerals Fractions Decimals Percents Ratios Proportions

Arabic and Roman Numeral Systems The Arabic system is based on the numbers 0 through 9. Arabic numbers can be written as whole numbers, fractions, and decimals. Roman numerals consist of letters that represent numbers. Roman numerals are commonly used to represent units of the apothecary system.

Reading Roman Numerals Roman numerals are read by adding or subtracting the value of the letters. I = 1; V = 5; X = 10; L = 50; C = 100 When a lower valued letter follows a larger valued letter, add the letters. When a lower valued letter precedes a larger valued letter, subtract the lower valued letter from the larger.

Reading Roman Numerals IX = 10 – 1 = 9 IV = 5 – 1 = 4 XIII = 10 + 3 = 13 XL = 50 – 10 = 40 XXXIV = 10 + 10 + 10 + (5 – 1) = 30 + 4 = 34

Table 5-1 The Most Common Roman Numerals and Their Arabic Values

Fractions A fraction is one or more equal parts of a unit. The fraction below means 3 parts out of 4 total parts. Also means 3 ÷ 4 3 4 Top number: numerator Bottom number: denominator

Classification of Fractions A common fraction represents equal parts of a whole (e.g., 1/2, 2/5, 3/7, 4/9). A decimal fraction is commonly referred to as a decimal (e.g., 0.5, 1.7, 5.25, 10.79).

Classification of Fractions The numerator of a proper fraction is less than its denominator and its value is less than 1 (e.g., 1/2, 2/3, 3/4, 10/24).

Classification of Fractions The numerator of an improper fraction is greater than, or the same as, its denominator; its value is equal to or greater than 1 (e.g., 5/3, 8/4, 12/9, 75/25). A mixed fraction is a whole number and a fraction combined; its value is always greater than 1 (e.g., 1-1/2, 2-1/4, 3-1/2).

Complex Fractions A complex fraction consists of at least 1 fraction and no more than 1 whole number; its value may be less than, equal to, or greater than 1.

Adding Fractions When fractions have the same denominators, add the numerators and keep the value of the denominators the same; then reduce to lowest terms. 1/10 + 3/10 = 4/10 = 2/5

Subtracting Fractions If fractions have the same denominator, subtract the smaller numerator from the larger numerator and keep the denominator the same; then reduce to lowest terms. 6/8 – 2/8 = 4/8 = 1/2

Dissimilar Denominators If fractions do not have the same denominator, change them so they have the smallest common denominator; then subtract the numerators and leave the denominator the same.

Subtracting/Adding Fractions With Dissimilar Denominators Since 12 is a multiple of 24 (24 ÷ 2 = 12), divide both numerator and denominator of the fraction with the larger denominator by 2 to reach the smallest common denominator. Note that 24 and 12 can also be divided by 4 or 3, but 5 cannot be divided by either 4 or 3.

Multiplying Fractions To multiply fractions, first multiply the numerators, then multiply the denominators; and then reduce to lowest terms. 3/5 × 2/4 = 6/20 = 3/10

Dividing Fractions The dividend is the number being divided, and the divisor is the number that is dividing. To divide, you must invert the divisor (3/4 becomes 4/3); then multiply the fractions and reduce to lowest terms. 4/8 ÷ 2/8 = 4/8 × 8/2 = 32/16 = 2

Decimals Decimals are used within the metric system, and their denominators are understood to be 10 or a multiple of 10. The denominator is not written; instead a decimal point is added to the numerator to signify the multiple of 10.

Decimals For decimals that are less than 1, always place a zero to the left of the decimal point to avoid confusion (2/10 = 0.2; 19/100 = 0.19).

Values of Decimals Decimals decrease in value from left to right. Decimals increase in value from right to left. Each column in a decimal has its own value depending on where it is situated compared to the decimal point.

Decimal Values Hundreds Tens Decimal point Tenths Hundredths 100 10 . 0.1 0.01  Increasing value  Decreasing

Figure 5-1 Decimal values as they relate to the location of the decimal point.

Adding Decimal Fractions To add decimals, write the decimals in a column, aligning the decimal points directly under each other. 1.4 0.7 + 0.5 0.2

Subtracting Decimals To subtract decimals, write the decimals in columns, aligning the decimal points; zeros may be added after the decimal point without changing the values. 0.225 0.300 – 0.525

Multiplying Decimal Fractions Multiply the numbers; count the number of places to the right of the decimal points in both numbers; then, place the decimal point in the answer at that position. 182.844 5.4 × 33.86

Dividing Decimals Convert decimals to whole numbers by moving the decimal point in the divisor to the right; then move the decimal point in the dividend the same number of places to the right.

Dividing Decimals 4.75 ÷ 0.5 = X Move the divisor’s decimal 1 place to the right to make a whole number; then move the dividend’s decimal 1 place too. Now the equation is 47.5 ÷ 5 = 9.5.

Ratios A ratio is a mathematical expression that compares one number to another number, or expresses a part of a whole number. The expression 3:4 means “3 out of 4 parts” or 3 ÷ 4. 2/5 = 2:5 2/100 = 2:100

Proportions A proportion expresses the relationship of equality between two ratios. The two inside terms (means) when multiplied, must equal the two outside terms (extremes) when multiplied. 1:4 :: 3:12 To verify, 4 × 3 = 12, and 1 × 12 = 12.

Figure 5-2 The means and extremes of a proportion.

Percents The term percent, or the symbol %, means hundredths. Percentages may be expressed as fractions, decimals, or ratios. 60% = 60/100 = 0.60 = 60:100

Decimal Conversions To change a percent to a decimal, move the decimal point 2 places to the left. 60% = 0.60 To change a fraction to a percent, divide the numerator by the denominator, then multiply the results by 100 and add the percent sign. 1/5 = 1 ÷ 5 = 0.2; then, 0.2 × 100 = 20%