Review of Mathematical Principles Chapter 7 Review of Mathematical Principles

Chapter 7 Lesson 7.1

Learning Objectives Work basic multiplication and division problems Interpret Roman numerals correctly Apply basic rules in calculations using fractions, decimal fractions, percentages, ratios, and proportions

Arabic Numerals Number system we are most familiar with Includes fractions, decimals, and whole numbers Examples include numbers 1, 2, 3, etc. The Arabic system of numbers is what we are most familiar with. In addition to this system, there is the Roman numeral system, which is often used for the apothecaries’ system of weights and measures when prescriptions are written. Prescriptions are also written using household and metric measures. Until the United States adopts one system of medical measurement, it is essential for nurses to understand both numeral systems and all three systems used to write prescriptions.

Basic Rules of Roman Numerals 1. Whenever a Roman numeral is repeated or a smaller Roman numeral follows a larger number, the values are added together. For example: VIII (5 + 1 + 1 + 1 = 8) 2. Whenever a smaller Roman numeral appears before a larger Roman numeral, the smaller number is subtracted. For example: IX (1 subtracted from 10 = 9) Practice: How do you read XLVI? What number is CLXIII? What number is CDLVI?

Basic Rules of Roman Numerals (cont.) 3. The same numeral is never repeated more than three times in a sequence. For example: I, II, III, IV 4. Whenever a smaller Roman numeral comes between two larger Roman numerals, subtract the smaller number from the numeral following it. For example: XIX = 10 + (10-1) = 19 Before you can interpret Roman numerals, the basic numbers and rules must be memorized. Roman numerals can be written in uppercase or lowercase letters. In expressing dosages in the apothecaries system, lowercase letters are used. A dot is always placed over the lowercase i. Practice: What is the Arabic number for lxvii?

Fractions One or more equal parts of a unit Part over whole, separated by a line: 3 parts of 4 = ¾ 3 is the top number, 4 is the bottom number The “numerator,” or top number, identifies how many parts of the whole are discussed The “denominator,” or lower number, identifies how many equal parts in the whole The two parts of a fraction are called “terms.” Fractions can be written linearly or with the numerator over the denominator. The word “NUDE” can be used to remember: the “NU,” numerator, is the first or top number, and the “DE,” denominator, is the second or bottom number. To use fractions in calculations, the numerators must be in the same unit of measure, and the denominators must be in the same unit of measure.

Fractions (cont.) Fractions may be raised to higher terms by multiplying the numerator and denominator by the same number: ¾ x 3/3 = 9/12 Fractions can be reduced to lowest terms by dividing the numerator and denominator by the same number: 9/12 ÷ 3/3 = 3/4 A fraction is easiest to work with when it has been reduced to its lowest term. ¾ and 9/12 are the same number. Are 6/23 and 60/230 the same number? What can the number 342/360 be reduced to? Keep dividing the fraction in half to assist in reducing. Do you see how it would be easier to work with the fraction 19/20 versus 342/360?

Fractions (cont.) Proper fraction: numerator is smaller than denominator For example: ¾ is a proper fraction, 3 is less than 4 Improper fraction: numerator is larger than denominator For example: 8/6 is an improper fraction, 8 is greater than 6 Mixed number: whole number is combined with a proper fraction For example: 1 ⅔ is a mixed number It is often necessary to change an improper fraction to a mixed number or a mixed number to an improper fraction for purposes of calculation. To change a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator: 1 ⅔ = 3 × 1 + 2 or 5/3. How would 3 6/7 be written as an improper fraction? To change an improper fraction to a mixed number, divide the denominator into the numerator: 8/6, 6 divided into 8 goes one time with 2 left over, which equals 1 2/6 or 1 1/3. What is the mixed number for 25/4?

Fractions (cont.) To add two fractions or subtract them, the denominators must be the same number. If two fractions have the same denominator, add the numerators and put the sum over the common denominator: 2/3 + 5/3 = 7/3 If two fractions have different denominators, a common denominator must be found. The common denominator is a number that both denominators can be divided into evenly. For example: 3/4 + 4/3 What number could the two denominators 4 and 3 be divided into? An easy way to find the common denominator is to multiply the two denominators together. However, sometimes, a lower common denominator exists. Will the answer to a calculation be wrong if the lowest common denominator has not been determined? Solve for 3/4 + 4/3 = 9/12 + 16/12 or 18/24 + 32/24. Are these numbers the same when reduced?

Fractions (cont.) Multiplying fractions; multiply the numerators together and the denominators together For example: 2/4 × 3/9 = 2 × 3 (6)/ 4 × 9 (36) Tip: it is easier to reduce the fractions to lowest terms before multiplying. Therefore: ½ × 1/3 = 1/6 It is much easier to multiply or divide fractions. When multiplying a fraction, a common denominator is not required. Practice: Multiple 4/6 x 2/3. Did you reduce before multiplication or after? Did it make a difference in the answer obtained? Mixed numbers must be changed to improper fractions before multiplying or dividing. One is the denominator for all whole numbers.

Fractions (cont.) To divide two fractions, invert (or turn upside down) the fraction that is the divisor and then multiply. For example: ¾ ÷ ½ = ¾ × 2/1 = 3 × 2 / 4 × 1 or 6/4 *** 6/4 can be reduced to 3/2 or 1 ½. The second fraction is the divisor. When the divisor is inverted, essentially you are dividing the fraction. To divide a fraction, invert the divisor, simplify, and then multiply the numerators and denominators. Remember, mixed numbers must be changed to improper fractions before multiplying or dividing.

Decimals All fractions can be converted to a decimal fraction by dividing the numerator into the denominator. For example: ¾ is 3 ÷ 4 = 0.75 To add two decimal fractions, first line up the decimal points. For example: 0.345 + 2.456 = 2.801 To subtract two decimal fractions, first line up the decimal points. For example: 1.6 − 0.567 = 1.033 Decimals are added or subtracted like Arabic numbers. It is important to keep the decimals lined up. Convert 5/6 to a decimal. Add 5.834 + 0.218 Subtract 0.786 from 2.1

Decimals (cont.) Multiplying decimals 1.467 (3 decimal places) ________ .343278 (6 decimal places in answer) To multiply decimals, first multiply the two numbers the same as you would if they were whole numbers. Then count the number of decimal places in each of the two numbers that have been multiplied together (counting from right to left). Add up the total number of decimal places in the two numbers. Count off the total number of decimal places in the answer from right to left and insert the decimal point.

Decimals (cont.) To divide two decimals, first move the decimal point in the divisor enough places to the right to make it a whole number. 6 ÷ .23 (the decimal must be moved two places to the right to change .23 into “23”) 600 ÷ 23 (move the decimal two places to the right in the dividend) = 26.09 (rounded) The dividend is the number being divided. The divisor is the number that is being divided into the dividend. Decimal changes made in the divisor also have to be made in the dividend. The decimal is moved to the right to convert a decimal fraction into a whole number.

Chapter 7 Lesson 7.2

Learning Objectives Apply basic rules in calculations using fractions, decimal fractions, percentages, ratios, and proportions

Ratios and Percents A ratio is a way of expressing the relationship of one number to another or expressing a part of a whole number. The relationship is reflected by separating the numbers with a colon (e.g., 2:1). Percent (%) means parts per hundred; can be written as fractions or decimals Ratios are commonly used to express concentrations of drug in a solution. Ratios also may be expressed as fractions. What would a ratio of 6:1 staffing on a hospital floor mean? To change a fraction or a ratio to a percentage, divide the numerator by the denominator and multiply by 100. To change a mixed number to a percentage, convert the mixed number to a fraction, divide the numerator by the denominator, and multiply by 100. Percentages may be expressed as decimals, fractions, or ratios.

Proportions Way of expressing a relationship between two ratios. The two ratios are separated by a double colon (::) which means “as.” If three variables are known, the fourth can be determined. When solving for “x,” the numerators must be the same measurement and the denominators the same measurement. The numerators and denominators in the proportion must be written in the same units of measure. When a proportion is written in linear terms, such as 3:4 :: 6:8, the numbers on the outside are called “extremes,” and the numbers on the inside are called “means.” The means multiplied should equal the extremes multiplied. 3 × 8 = 6 × 4. It is incorrect to state there are 3 mg /1 mL :: 4 mcg/x mL. The milligram (mg) and microgram (mcg) measures are not equal. Therefore one of the units of measure must be converted to the other. To solve for “x,” cross multiply the means or extremes known and divide by the remaining variable to determine “x.” Knowing how to calculate ratios and proportions is one of the main foundations needed for drug dosage calculations.