FRACTIONS, PERCENTAGES, & RATIOS MODULE A-3 FRACTIONS, PERCENTAGES, & RATIOS

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OBJECTIVES At the end of this module, the student will be able to… Define terms associated with fractions. List the rules for working with fractions. Given a mathematical problem involving fractions, add, subtract, multiply or divide fractions with and without a calculator and derive the correct answer. List the rules for working with percentages. Given a mathematical problem involving percentages, add, subtract, multiply or divide percentages with and without a calculator and derive the correct answer. Define terms associated with ratios and proportions. Describe how ratios and proportions are used in solving clinical problems. Given a mathematical problem involving ratios or proportions, derive the correct answer.

FRACTIONS Definitions Size of fractions Reporting in lowest or simplest terms Simple fraction rules Mixed fraction rules Converting fractions to decimals

Definitions Top Numerator Dividend Bottom Denominator Divisor # of pieces you have total number of pieces

General Rules When Using Fractions Rule #1: The denominator of a fraction cannot be equal to zero. A zero is allowed to as a numerator of a fraction and the resulting fraction is equal to zero. Rule #3: If either the numerator or the denominator is negative, the fraction is negative. If both the numerator and denominator are negative, the fraction is positive. Rule #4: Always report fractions in the simplest of terms.

Definitions Simple fractions – A fraction that cannot be reduced any further (lowest terms) EXAMPLE: 3/4 Compound fractions – A fraction that can be reduced further by dividing both the numerator and the denominator by the same whole number. EXAMPLE: 8/4 Mixed fractions – A number written as a whole number and a fraction. EXAMPLE: 2 1/2

Size of Fractions: Same Denominator When the denominator is the same between two fractions, the fraction with the larger numerator is the larger number. 2/3 > 1/3 3/8 > 1/8 Top number is # of pieces you have. Bigger number means more pieces of pie

Size of Fractions: Different Denominators The smaller the denominator the larger the fraction 1/2 > 1/3 1/4 > 1/8 Bottom number is total number of pieces of pie. Pie cut in less pieces means bigger pieces of pie

Reporting Answers as Fractions WRONG RIGHT Both divided by: 2/4 1/2 2 3/9 1/3 3 4/12 1/3 4 Need to reduce fractions to lowest common terms.

Steps to Reduce Fractions Factor the Numerator Factor the Denominator Find the fraction mix that equals 1. Note: Any number divided by itself is equal to one.

Adding and Subtracting Fractions To add or subtract two (or more) fractions, all fractions must first have a “common denominator” Build each fraction so the denominators are equal. Building is the opposite of factoring. Add or subtract only numerators to obtain answer. Denominators do not change. Reduce the fraction as needed.

Adding Fractions Build each fraction so the denominators are equal. Add only numerators to obtain answer. Denominators do not change. Reduce the fraction as needed.

Subtracting Fractions Build each fraction so the denominators are equal. Subtract only numerators to obtain answer. Denominators do not change. Reduce the fraction as needed.

Addition Practice 3/8 + 2/8 = ________ 1/4 + 3/5 = ________ 1/6 + 1/4 = ________ 2/3 + 5/6 = ________ 1/5 + 1/2 = ________ Build each fraction so the denominators are equal. Add only numerators to obtain answer. Denominators do not change. Reduce the fraction as needed.

Subtraction Practice 3/8 - 2/8 = ________ 3/8 - 2/8 = ________ 3/4 - 3/5 = ________ 1/4 – 1/6 = ________ 2/3 - 2/6 = ________ 3/5 – 1/2 = ________ Build each fraction so the denominators are equal. Add only numerators to obtain answer. Denominators do not change. Reduce the fraction as needed.

Multiplying Fractions To multiply two (or more) fractions: Multiply the numerators together Multiply the denominators together. Reduce the fraction as needed

Practice Multiply the numerators together 5/9 x 3/4 = __________ Multiply the denominators together. Reduce the fraction as needed

Dividing Fractions Do not divide fractions. Invert the second fraction then multiply.

Practice Do not divide fractions. Invert the second fraction then multiply. 5/12 divided by 3/4 = __________ 2/3 divided by 1/4 = __________ 5/6 divided by 2/3 = __________ 4/5 divided by 5/8 = __________

Mixed Numbers Definition: A whole number and a fraction OPTION #1: Work with whole numbers and fractions as two separate problems then add back together Apply the fraction rules to the fraction portion. Work with the whole numbers separately. Put the two pieces together as the answer.

Addition and Subtraction of Mixed Numbers Apply the fraction rules to the fraction portion. Work with the whole numbers separately. Put the two pieces together as the answer.

Mixed Numbers OPTION #2 : When working with negative numbers, it may be easier to convert the mixed fractions into compound fractions before beginning the work EXAMPLE: 1 2/3 – 2 1/3 =?

Compound Fractions Definition: A fraction that can be reduced further by dividing both the numerator and the denominator by the same whole number. Our goal is to convert a mixed number into a compound fraction. Break the mixed number into its two portions (whole number and fraction). Convert the whole number to a fraction by placing it over 1. Add the two fractions following prior rules.

Converting Mixed Numbers to Compound Fractions Break the mixed number into its two portions (whole number and fraction). Convert the whole number to a fraction by placing it over 1. Add the two fractions following prior rules.

Solution An alternative way is multiply the denominator of the fraction in the mixed number by the whole number and add the numerator. The answer is expressed over the denominator of the fraction.

Multiplication and Division of Mixed Numbers Convert to a compound fraction before multiplying or dividing. Reconvert to a mixed number by dividing numerator by the denominator.

Converting Fractions to Decimals Divide the numerator by the denominator

Percentages express a value in parts of 100 One half = 50% One quarter = 25% One third = 33%

Percentage - Clinical Example Bleach solution for decontamination: 10% = 10 parts are bleach and 90 parts are water for a total of 100 parts. Ethanol for disinfection: 75% = 75 parts ethanol and 25 parts water for a total of 100 parts.

Conversion of Percentage to a Fraction 1% is equal to 1/100.

Conversion of Percentage to a Decimal 1% is equal to 0.01. Alternatively, you can move decimal place to left two places and remove the % sign.

Conversion of Decimal to a Percentage Move decimal place to right two places and add the % sign You are actually multiplying by 100 0.21 x 100 = 21%

Percentages in calculations When there is a percentage presented in an problem you have two choices: Use the % key on the calculator Convert the % to a decimal EXAMPLE: 80 x 50% = __________ 80 x 50(% key) = ______ (calculator) 80 x .5 = ___________ (no calculator)

Percentage - Clinical Example The quantity of oxygen in room air can be written: Percentage: oxygen percentage 21% Decimal: FIO2 0.21 (fractional concentration of inspired oxygen) 21 parts of the gas in the air we breath are O2 and the other 79 parts are some other gas (nitrogen, carbon dioxide…)

Ratios & Proportions

Ratios Ratios are used to make comparisons between two things. One number to another. Number of boys to girls. Number of RBCs to WBCs. There are no units of measure related to ratios, and comparisons can be made between two things that have different units. Ratios can be expressed as: Fractions (3/4) Using the word “to” (3 to 4) Using a colon (3:4) THEY ALL MEAN THE SAME THING!

Clinical Example A ratio is the relationship of one value to another I:E Ratio – Expresses the relationship between Inhalation time as it relates to exhalation time When humans ventilate, exhalation is typically longer than inhalation. The ratio of inhalation:exhalation is usually between 1:2 and 1:3: 1:2 = one part inhalation to two parts exhalation 1:3 = one part inhalation to three parts exhalation

Clinical Example There are no units of measure related to ratios If the I:E ratio is 1:2 it could mean that: Inhalation is 0.5 seconds and exhalation is 1 second Inhalation is 1 second and exhalation is 2 seconds What if the time of inhalation was 2 seconds; what would the time of exhalation be?

Inspiratory : Expiratory Time All the breath cycles below have a 1:2 I:E ratio 1 2 4 3 6 Inhalation seconds Exhalation seconds

Proportions A proportion is a name we give to two ratios that are equal. We can express a proportion as a set of fractions: a/b=c/d Or we can express a proportion using a colon a:b = c:d When two ratios are equal (proportionate) then the cross products of the ratios are equal. “a is to b as c:d”

Ratios and Proportions Ratios and proportions are used when a new quantity of a substance is desired based on an existing ratio 1 : 3 ratio - no unit of measure Example: Odds paid when gambling. If I bet $3, I could win $9. The unit of measure is attached to substances represented by the ratio not the ratio itself.

Day to Day Example Look at the directions on a box of pancake mix To increase from 2 servings to four servings, everything is increased in proportion to the original amount 1 cup increases to 2 cups 1 tsp. increases to 2 tsp ¼ cup increases to 2/4 cup or ½ cup

Clinical Example If I have an I:E ratio of 1:3 and my inspiratory time is 0.5 seconds, what is my expiratory time? If want to increase my inspiratory time to 0.7 seconds and keep the same I:E ratio, what will my new expiratory time be?