Scales based on powers of 10:
The most famous scale base on powers of ten in percentage, which really
means "per 100". It is much more common to see "53% of the people agree
with the president's plan" than ".53 of the people..." or "53 out of
every 100 people...". Technically, all those phrases are saying the
same thing, but percentage is the most popular.
To get a number based on a power of 10 scale, you take the small number, divide it by the big number and multiply by the power of ten, so it is small/big*scale. Sometimes we need greater precision because the
proportions are so small, the small number is tiny in comparison to the big.
When I ask a class what is the legal limit for blood alcohol while
driving, invariably someone will say "point oh eight" and most people
will agree. But .08 is wrong; .08 = 8%, and the correct answer is .08%
= .0008. I don't blame the students. The number is badly represented
and it is an easy mistake to make. Let's take a look at the number on
other scales of 10.
.08 out of 100 is the same as
.8 out of 1,000 or
8 out of 10,000 or
80 out of 100,000
80 parts out of 100,000 is a tiny proportion. To give an idea, ounce of
pure alcohol mixed into ten gallons of blood would give you 78 parts
out of 100,000, and most people have between a half gallon and a gallon
and a half of blood in their body, between 4 and 12 pints. The amount
of alcohol in a person's blood stream that is over the legal limit is
about the same amount of alcohol as found in a capful of mouthwash used
after brushing your teeth.
We will look at the per
100,000 scale for another type of statistic, measurements of mortality
rates.
Here are the number of homicides in some local cities in 2007.
Oakland: 124 homicides
Richmond: 28 homicides
San Francisco: 98 homicides
Clearly, comparing these numbers is misleading, because we know these
cities have very different numbers of citizens, so the standard way to
measure these statistics is the per 100,000 population scale, which we
find by the formula
small/big x scale
which in this case is
(# of homicides)/(city population) x 100,000
Oakland's population in 2007 is estimated at 415,000, Richmond at
106,000 and San Francisco at 825,000, so the murder rates on this
standard scale are as follows
Oakland: 124/415000 * 100000 = 29.9
Richmond: 28/106000 * 100000 = 26.4
San Francisco: 98/825000 * 100000 = 11.9
So even though more people were murdered in San Francisco than in
Richmond in 2007, the murder rate in Richmond was over twice as high,
because Richmond has barely 1/8 of the population of San Francisco.
(note: The trends for the three cities this decade are going in
different directions. Oakland's murder rate is on the rise, while
Richmond's is falling and San Francisco's has stayed about the same.)
Practice problems: (answers given in comments)
1) Here are the homicide numbers for Oakland, Richmond and San Francisco from 2004.
Oakland: 96 homicides, 399,000 population
Richmond: 40 homicides, 99,000 population
San Francisco: 96 homicides, 775,000 population
Find the murder rates from these years, rounded to the nearest tenth per
100,000 population and rank them from lowest (1st) to highest (3rd).
Showing posts with label percents. Show all posts
Showing posts with label percents. Show all posts
Wednesday, June 20, 2012
Fractions, decimals and percents.
Prime factorization and all factors.
Decimals, fractions and percents
There are three standard ways to represent rational numbers, fractions, decimals and percents. Many people are most comfortable with decimals because that is the way a calculator displays a number, but all three of the methods have their advantages and disadvantages.
Example: The number .735 is a decimal. It is common to call this "point seven three five", but its proper name would be "seven hundred thirty five thousandths". That is because it has three places after the decimal, which is the thousandth place.
Easy rule: how many ever places there are after the decimal, write a fraction, "1" over "1" followed by that many "0".
One place after the decimal corresponds to 1/10 or one tenth.
Two places after the decimal corresponds to 1/100 or one hundredth.
Three places after the decimal corresponds to 1/1000 or one thousandth.
Following this rule, .735 = 735/1000. You should be able to see that both of these numbers are divisible by 5, so this is not in lowest terms. 735/1000 = 147/200, and this is in lowest terms. (If you have the TI-30XIIs, you can type 735 [A b/c] 1000 [enter], and the answer will be 147/200.)
The word "per" is from Latin and means "out of". It always represents division. "Percent" is "out of 100". The decimal .735 is equal to 73.5%. Technically we multiply .735 by 100 to get 73.5, then by putting the percent sign at the back we have divided by 100, so the number is the same, only the representation has changed. The mechanical rule we are usually taught is.
Decimal to percent: Move the decimal point two places to the right and put a percent sign behind it.
Percent to decimal: Move the decimal point two places to the left and remove the percent sign.
That means 33% = .33 and 3% = .03. As fractions .33 =33/100 and .03 = 3/100.
Prime factorization and all factors of a number
A factor of a number n is a number that divides into n evenly.
Example: 4 is a factor of 12 because 12/4 = 3, which can also be stated as 4x3 = 12.
Counterexample: 5 is not a factor of 12 because 12/5 = 2.4, a decimal and not a whole number.
Every number has a prime factorization, which means we break it down to a product of prime numbers. A prime can only be divided evenly by 1 and itself. (By definition, 1 is not a prime. It is called a unit.)
Here is a list of the primes less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
There are infinitely many primes, so this list goes on forever.
Find the prime factorization of 45.
Step 1: Find two numbers other than 1 and 45 that multiply to 45. An easy pair is 5x9.
Step 2: 5 is prime, but 9 is not. That means we can find a pair of numbers (not 1 and 9) that multiply to 9, in this case 3x3. 3 is prime, so 45 = 5x3x3 is the prime factorization of 45.
Find all factors of 45.
When making a list of all factors, it makes sense to write them in pairs. I recommend starting with 1 and the number itself.
1 and 45
Now find the next biggest number after 1 that divides evenly into 45. 2 won't work because 45/2 = 22.5. 3 does work, because 45/3 = 15.
3 and 15
4 doesn't work but 5 does.
5 and 9.
You can try 6, 7 and 8 if you like, but none of them work. 9 works, but it's already on our list, so we can stop.
All the factors of 45 in order are 1, 3, 5, 9, 15 and 45.
Practice
a) Change .09 into a percent.
b) Change .09 into a fraction in lowest terms.
c) Change 24% into a decimal.
d) Change 24% into a fraction in lowest terms.
e) Change 3/8 into a decimal.
f) Change 3/8 into a percent.
Answers in the comments.
For practice on factorization, click on this link.
There are three standard ways to represent rational numbers, fractions, decimals and percents. Many people are most comfortable with decimals because that is the way a calculator displays a number, but all three of the methods have their advantages and disadvantages.
Example: The number .735 is a decimal. It is common to call this "point seven three five", but its proper name would be "seven hundred thirty five thousandths". That is because it has three places after the decimal, which is the thousandth place.
Easy rule: how many ever places there are after the decimal, write a fraction, "1" over "1" followed by that many "0".
One place after the decimal corresponds to 1/10 or one tenth.
Two places after the decimal corresponds to 1/100 or one hundredth.
Three places after the decimal corresponds to 1/1000 or one thousandth.
Following this rule, .735 = 735/1000. You should be able to see that both of these numbers are divisible by 5, so this is not in lowest terms. 735/1000 = 147/200, and this is in lowest terms. (If you have the TI-30XIIs, you can type 735 [A b/c] 1000 [enter], and the answer will be 147/200.)
The word "per" is from Latin and means "out of". It always represents division. "Percent" is "out of 100". The decimal .735 is equal to 73.5%. Technically we multiply .735 by 100 to get 73.5, then by putting the percent sign at the back we have divided by 100, so the number is the same, only the representation has changed. The mechanical rule we are usually taught is.
Decimal to percent: Move the decimal point two places to the right and put a percent sign behind it.
Percent to decimal: Move the decimal point two places to the left and remove the percent sign.
That means 33% = .33 and 3% = .03. As fractions .33 =33/100 and .03 = 3/100.
Prime factorization and all factors of a number
A factor of a number n is a number that divides into n evenly.
Example: 4 is a factor of 12 because 12/4 = 3, which can also be stated as 4x3 = 12.
Counterexample: 5 is not a factor of 12 because 12/5 = 2.4, a decimal and not a whole number.
Every number has a prime factorization, which means we break it down to a product of prime numbers. A prime can only be divided evenly by 1 and itself. (By definition, 1 is not a prime. It is called a unit.)
Here is a list of the primes less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
There are infinitely many primes, so this list goes on forever.
Find the prime factorization of 45.
Step 1: Find two numbers other than 1 and 45 that multiply to 45. An easy pair is 5x9.
Step 2: 5 is prime, but 9 is not. That means we can find a pair of numbers (not 1 and 9) that multiply to 9, in this case 3x3. 3 is prime, so 45 = 5x3x3 is the prime factorization of 45.
Find all factors of 45.
When making a list of all factors, it makes sense to write them in pairs. I recommend starting with 1 and the number itself.
1 and 45
Now find the next biggest number after 1 that divides evenly into 45. 2 won't work because 45/2 = 22.5. 3 does work, because 45/3 = 15.
3 and 15
4 doesn't work but 5 does.
5 and 9.
You can try 6, 7 and 8 if you like, but none of them work. 9 works, but it's already on our list, so we can stop.
All the factors of 45 in order are 1, 3, 5, 9, 15 and 45.
Practice
a) Change .09 into a percent.
b) Change .09 into a fraction in lowest terms.
c) Change 24% into a decimal.
d) Change 24% into a fraction in lowest terms.
e) Change 3/8 into a decimal.
f) Change 3/8 into a percent.
Answers in the comments.
For practice on factorization, click on this link.
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