3348 ch 2 part 1 notes and exercises
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Mathematics
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Feb 20, 2024
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Fractions, Decimals, and Percentages
Fractions, decimals, and percentages are different ways of expressing the same values. Throughout your experience in health care, you will need to use these numbers to communicate your findings.
FRACTIONS
Fractions are numbers that are expressed as parts of a whole. (Pizza slices 5/8 + 1/8 =6/8=) The top number (his five pieces, called the numerator) is the parts of the whole that we measured, and the bottom number (8, called the denominator) is the total (whole) number of pieces.
5/8 the numerator (number of PARTS)
denominator (TOTAL number of parts)
8/8=1 Simple fractions: > a whole number (ex. 3/4, 6/7, 9/10)
Compound fractions (aka mixed number fractions): represent numbers than one (1 ½,3 ¾, 56 ¼)
-can also be expressed with a numerator larger than the denominator : called improper or top- heavy fractions (ex. 3/2 is the same amt as 1 ½ ex. 15/4= 3 ¾ )
You can convert an improper fraction to a mixed number fraction by numerator by denomin to the nearest whole number and showing amt. left over as a fraction. We convert mixed number fractions to improper fractions by reversing the process. Multiply the denominator (in this example, 4) by the whole number (3) to get 12. Then add the remaining fraction (¾) to get 15/4.
Frequently, we reduce fractions to make them easier to understand and to work with. If 10 of the 20 cribs in the nursery are full, we probably would not say the nursery is 10/20 (ten-
twentieths) full. We would reduce the fraction to ½, and we would say that it is half-full.
E.
Multiplication works exactly the same way—we can multiply the fraction by whatever factor we want, as long as we do the same thing to both the numerator and the denominator.
The medical-surgical (med-surg) unit on the 2
nd
floor is 5/12 full, and the med-surg unit on the 3rd floor is 1/12 full. Workers on the 3rd floor need to shut off the ac for repairs, and the hospital decides to move (add) the patients from the 3rd floor to the 2nd floor. How many patients will be on the 2nd floor med-surg unit after the patients are moved? In this case, the addition is easy because the denominators are the same.
But let us try adding fractions where the denominator is not the same. Say the 3rd floor was 1/3 full, and the 2nd floor is ½ full. (Note: 1/3 + ½ does not equal 2/5!)
To add (or subtract) fractions with different denominators, we must multiply the fractions by some factor first so that we are adding fractions with the same denominator. (must be same # for BOTH numerator and denominator)
One of the clinics attached to the hospital system handles walk-ins and provides some urgent care services. Of the 120 patients seen last month, 10 were Asian-American, 35 were Latino or Hispanic, and 15 were African-American. Sasha wants to report these ethnicities in simple fractions.
Determine the fraction of the whole for each ethnic group and report your findings in simple fractions.
Asian-American: 10/120 = Latino or Hispanic: 35/120 =
African-American: 15/120=
Exercise 2-1
Fractions
1. Convert the following improper fractions to mixed number fractions:
a. 12/8=
b. 5/2=
c. 144/12=
2. Convert the following mixed number fractions to improper fractions:
a. 3
3/8 =
b. 13
½ =
c. 7 5/16 =
3. Reduce the fractions below to their simplest form.
a. 2/8=
b. 5/100=
c. 75/1000=
d. 12/144=
e. 6/36=
4. Add or subtract the following fractions. Report your answers in simple fractions.
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DECIMALS
Decimals are actually fractions whose denominators are some power of 10 (10, 100, 1000, etc.) and are written as a decimal point followed by the numerator.
Ex. the fraction 1/10 (one-tenth) can be written as the decimal 0.1.
(seven and sixty-four one-hundredths) is expressed as 7.64
In decimals, the whole is always divisible by 10 (for example: 10, 100, 1000). The decimal point separates
the whole from the parts. Whole numbers are to the left of the decimal point, while the parts are to the right.
Decimals, like fractions, describe parts of a whole.
Changing Fractions to Decimals
Ex. coupon gives 1/3 off of your purchase. One of the t-shirts is priced at $36.00. If it is 1/3 off, how much will I save? Change the 1/3 into a decimal, then multiply it by the price to see how much you are going to save. 1/3 = (1 divided by 3)=0.33 x $36.00 = $12.00 You are saving $12.00
Ex. a shirt is 2/5 off. It costs $39. 2 divided by 5 = 0.4 x $39.00 = $15.60 so your cost is $39. - $15.60 = $23.40
Quotient
: the result of division in this case, 0.4
Key Take Away
Convert fractions into decimals by dividing the numerator by the denominator
.
Changing the fraction to a decimal leads us to another important concept: rounding. Rounding is a method of reducing the number of digits in a number so that it is less precise but is more convenient to use. If the number is between 0 and 4, you drop the remaining digits and leave the number in the tenths place as it is. This is called rounding down. If the digit is between 5 and 9, you add one to the digit in the tenths place. This is called rounding up.
In many health care applications, converting to a decimal makes a fraction easier to use. Let us say the city of Midville, FL has three hospitals—two are larger facilities, and one is smaller. Of all the admissions last year, very few patients had a principal diagnosis of MRSA, a kind of bacterial infection that is difficult to treat with antibiotics.
What fraction (part of the whole) of patients in all three Midville hospitals had MRSA?
We know how to set up the fraction for each: 2/14,065 1/4023 1/12,200 But we would not want to try to find the common denominator of all these fractions in order to add them together. It would be much easier (though slightly less precise) to convert each fraction to a decimal, then add the decimals. Let us look at the math:
Calculating the part of the whole of the patients in Midville who were treated for MRSA using fractions would be difficult; but when we convert the fractions to decimals and use rounding, we can see that 0.00047 of all the patients (30,288) in Midville had MRSA.
Math Review
If we say 0.00047 patients of all the patients in Midville had MRSA, how many people is that?
The 4 is in the 10,000s place, so we might say 4.7 infections for every 10,000 people
. Or, we might say 47 of every 100,000 patients
were treated
for this infection.
1.
Convert the following fractions to decimals: a. 3/8
b. 13
1/2
c. 7
5/16
d. 1/160
e. 60/10000
2. In the decimal 0.012358467, the digit 1 is in the __________ place.
3. In the decimal 0.193847, the digit 7 is in the __________ place.
4. Round each decimal to the tenths place. Then round each to the hundredths place. Then round to the thousandth place. Rounding Calculator
a. 0.09513999
b. 0.551031
c. 1.342809
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PERCENTAGES
A percentage is the number of times something occurs out every 100 times
. Presentation of the percentage standardizes the data so that groups that are NOT ALIKE can be compared.
Ex. you answered 24 of 27 questions correctly on one quiz, and 30 of 35 questions right on another, which quiz did you score better on?
To calculate a percentage, divide the observations in the category by the total observations, and
multiply by 100.
How many male and female patients were discharged in February this year compared to last year? We can look at the difference (the variance) between the number of women and men in each period, as illustrated in Table 2-1, but the result may not be helpful.
It appears that there has been a larger increase in discharges of women (36) than in men (27). But is that true?
To give a more accurate analysis of the activity, we should also provide the percentage of observations and the percent variance.
For example, if we want to know what percent of the patients discharged in February 2014 were
women, we would do the following: 48% of the discharges in February 2014 were women. But what about the variance?
How many more women were discharged in 2015? Table 2-2 expands the data to include the percentages. Now, it is clear that the total number of discharges has increased by 8.6%, the percentage of women increased by 10.3%, and the percentage of men increased by 7.0%. These are descriptive statistics, so we cannot say why there is a greater percentage increase in women patients than in men. We will have to examine this data over a longer period of time and look further into the types of illnesses and treatments
that the patients have to understand the reason for the change, if it continues.
Fractions, decimals, and percentages are closely related concepts, and in practice, you will need to be able to convert between these formats frequently.
To convert from a fraction to a decimal
Divide the numerator (top number) by the denominator (bottom number).
To convert from a decimal to a fraction
Divide the decimal number by the power of 10 that it represents (e.g., .75 is hundredths, so 75/100), then simplify the fraction (75/100 = ¾).
To convert from a decimal to a percent
Multiply the decimal by 100, and add a percentage sign.
To convert from a percent to a decimal
Divide the percentage by 100, and drop the percentage sign.
Stat Tip
Dividing the percentage by 100 results in moving the decimal point two places to the left. Once you are comfortable with this method, you may want to use it as a shortcut for conversions.
To convert from a fraction to a percentage
Divide the numerator by the denominator, then multiply the result by 100, and add a percentage sign.
To convert from a percentage to a fraction
Drop the percentage sign, then divide the decimal number by the power of 10 that it represents.
Exercise 2-3 Percentages
1. Calculate the following percentages:
a. 10/50
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b. 49/100
c. 17/1000
d. 14/16
e. 1810/2000
2. Convert the following percentages to decimals:
a. 1%
b. 10%
c. 47%
d. 0.5%
3. Convert the following to the simplest fraction:
a. .5
b. 0.98
c. 0.333333333
d. 1.75
e. 90%
f. 25%