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MAT081 :: Lecture Note :: Week 07
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A ratio is the "relationship in quantity, amount, or size between two or more things." [source::mw.com::MerriamWebster Online]
The general form of a ratio is as follows is
n:m
, wheren
is the quantity of one thing andm
is the quantity of another thing. For example, if a recipe calls for 3 parts lemon and 5 parts lime, then the ratio is3:5
.Sometimes ratios are written using the word to instead of a colon. For example, 3 parts lemon to 5 parts lime.
Fractions can be written as ratios.
1/7 is 1:7 or 1 to 7 3/8 is 3:8 or 3 to 8 15/7 is 15:7 or 15 to 7Which of the following, if any, are valid ratios?
8:8 1.5:1 4:2.25 0:55 4:0 2:1A rate is a special kind of ratio, indicating a "relationship between two measurements with different units."
PurpleMath.com:: Ratios [opens new window]
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A rate is a ratio that "relate" different "units." MerriamWebster Online defines rate as "a fixed ratio between two things."
Example rates.
65 miles per hour (mpg) [ratio  65:1] You can travel 65 miles in one hour. $0.59 per pound [ratio  0.59:1] It will cost you $0.59 to buy one pound of foo. 60 beats per minute [ratio  60:1] A good heart rate for a man at rest is 60 beats in one minute. {"beat" is one complete pulsation of the heart}
It is possible for us to manually measure our heart rate. We were given a ratio of
60 beats per minute
, but a minute is a long time; therefore, we can approximate our heart rate by counting the heart beats for10
seconds and multiply the final count by6
. [There are60
seconds per minute.]TopEndSports.com:: Heart rate measurement
The term per implies a ratio (i.e. fraction i.e. division).
Receive 10 coupons per every 3 purchases. _ ratio... 10:3 fraction... 10/3 division... 3.3There are many "forms" of rates. Birth rates, death rates, tax rates, and numerous other financial rates (e.g. mortgage rates, commission rates, interest rates, inflation rates just to name a few). There are also discount rates, tipping rates, data transfer rates, phone rates, turnover rates, failure rates and so on.
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A proportion is a "statement of equality between two ratios in which the first of the four terms divided by the second equals the third divided by the fourth." [source::mw.com::MerriamWebster Online]
Given two ratios,
n:m
anda:b
,n
isa
asm
is tob
.Two ratios are equal if the crossproducts are equal.
/ * \   2:4 equals 10:20   \ * / 2 * 20 = 40 4 * 10 = 40From the book "Zero: The Biography of a Dangerous Idea"...
"To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature was as simple as understanding the mathematics of proportions."PurpleMath.com:: Proportions: Introduction [opens new window]
YouTube.com::Math Education Professor Dor Abrahamson
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The word percent means "per one hundred."
Percent values are suffixed by a
%
character.A percent is a fraction where the denominator is 100.
5% equals 5/100 15% equals 15/100 120% equals 120/100 37.5% equals 37.5/100Percent values can be written as a decimal number by multiplying the value by
0.01
.4% equals 4(.01) equals 0.04 11% equals 11(.01) equals 0.11 110% equals 110(.01) equals 1.10Decimal values can be written as percents by multiplying by 100%.
0.02 equals 0.02(100)% equals 2% 0.33 equals 0.33(100)% equals 33% 2.10 equals 2.10(100)% equals 210%Percents can be written as fractions by multiplying by
1/100
(or one onehundreth).5% equals 5(1/100) equals 5/100 25% equals 25(1/100) equals 25/100 110% equals 110(1/100) equals 110/100PurpleMath.com:: Converting Between Decimals, Fractions, and Percents [opens new window]
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The following algorithm converts a fraction into a percent.
a) divide the numerator by the denominator b) multiply the resulting decimal number by 100(%)Examples.
3/8 = 0.375; 0.375 * 100(%) = 37.5% 1/2 = 0.5; 0.5 * 100(%) = 50% 7/11 = 0.6364; 0.6364 * 100(%) = 63.64% 11/7 = 1.5714; 1.5714 * 100(%) = 157.14%The following algorithm converts a percent into a fraction.
a) write the percent as a fraction (n% = n/100) b) reduce the fraction, if possibleExamples.
25% = 25/100; 25/100 reduces to 1/4 62% = 62/100; 62/100 reduces to 31/50 202% = 202/100; 202/100 reduces to 2 1/50 1% = 1/100; 1/100 is reduced 0.2% = 0.2/100; 0.2/100 reduces to 0.1/50 or 1/500
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The percentage increase between two values is calculated as follows.
new value  original value  * 100 original valueThe percentage decrease between two values is calculated as follows.
original value  new value  * 100 original valueExamples
The Maricopa Community Colleges increased 200506 instate tuition rates by
$5
per credithour. The 200405 tuition rate was$55
.old rate: $55 [200405 rate] increase: $ 5 ["increase" implies addition] new rate: $60 [sum of $55 and $5] percentage increase: 60  55 5  =  = 0.0909 55 55 0.0909 * 100 = 9.09%The Maricopa Community Colleges increased 200607 instate tuition rates from
$60 per credithour
to$65 per credithour
.percentage increase: 65  60 5  =  = 0.0833 60 60 0.0833 * 100 = 8.33%
The Valley Metro Transit is considering decreasing bus fares by
$0.25
. Current bus rates are$1.25
per two hours of bus riding.current rate: $1.25 proposed decrease: $0.25 [the word "decrease" implies subtraction] new rate: $1.00 [difference of $1.25 and $1.00] percentage decrease: 1.25  1.00 0.25  =  = 0.2000 1.25 1.25 0.2000 * 100 = 20%PurpleMath.com:: "Percent of" Word Problems: General Increase and Decrease [opens new window]
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Percent problems come in many forms. Three popular forms are often worded as follows.
What is 'a' percent of 'b'? 'a' is what percent of 'b'? 'a' is 'b' percent of what?Here are some rules to map these type of percent word problems into equations.
"what" becomes a variable "is" becomes an equals operator "of" becomes a multiply operator1) What is
5%
of10
?n = 5% * 10 ...variable is named 'n' n = 5(.01) * 10 ...%5 converted to decimal n = 0.05 * 10 ...word "of" became times n = 0.5 ...decimal arithmetic 0.5 is 5% of 10 ...final answer2)
7
is what percent of49
?7 = n * 49 7 n * 49  =  49 49 1  = n 7 .143 = n .143 * 100 = 14.3 = n (final answer: 14.3%) 7 is 14.3% of 493)
4
is12%
of what?4 = 12% * n 4 = 12(.01) * n 4 .12 * n  =  .12 .12 33.3 = n 4 is 12% of 33.3External Hyperlinks
 PurpleMath.com:: Basic "Percent of" Word Problems [opens new window]
 WebMath.com:: Word Problems Involving Percents [opens new window]
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_______% of the five cells are red.
_______% of the five cells are green.
_______% of the ten cells are not white.
Shade 45% of the five cells.
Shade 0.5% of the five cells.
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