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MAT081 :: Lecture Note :: Week 06
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48÷2(9+3)
7+7÷7+7x77
A decimal number is written with the whole number followed by a dot (decimal point) followed by the fractional part. A decimal number falls between two integers that differ by one in value.
2 < 2.43 < 3 8 < 8.0001 < 9 99 < 99.9999 < 10A decimal fraction is a fraction where the denominator is 10 raised to a positive integer exponent.
10^{4} equals 10,000 10^{3} equals 1,000 10^{2} equals 100 10^{1} equals 10 10^{0} equals 1 10^{1} equals 1/10 equals 0.1 10^{2} equals 1/100 equals 0.01 10^{3} equals 1/1000 equals 0.001 10^{4} equals 1/10,000 equals 0.0001 ... 10^{9} equals 0 1/1,000,000,000 equals 0.000000001To the left of the decimal point are the ones, tens, hundreds, thousands, and so on. On the fractional side of decimal number are the tenths, hundreths, thousandths, and so on. There are no oneths.
Decimal numbers that lie between zero and one (and zero and a negative one) are often prefixed with a zero.
.1 = 0.1 .375 = 0.375 .55 = 0.55Trailing zeroes after the decimal point are not necessary; however, in science, engineering, statistics and other fields, trailing zeros are retained to show a level of confidence in the accuracy of the number.
When writing a decimal number in English, use the word and to represent the decimal point.
7.59 is seven and fiftynine hundreths 0.459 is four hundred fiftynine thousandths 5000.29 is five thousand and twentynine hundreths 233.056 is two hundred thirtythree and fiftysix thousandthsA NanoMoment
Note: nano is a prefix meaning onebillionth (or
10^{9}
or1/1,000,000,000
or0.000000001
).1 nano...  = 0.000000001 1,000,000,000In everydayworld, GDT replaces nano with "very, very, very small." For example, a nanofoo is a very, very, very small foo. It doesn't matter what foo is; whatever it is, it is very, very, very small.
Let's get smaller (i.e. closer to zero)...
A nanosecond is a very, very, very short second (i.e. a billionth of a second).
From Fall 2004: "Optical 'rulers' are lasers that emit pulses of light lasting just 10 femtoseconds (10 quadrillionths of a second, or 10 millionths of a billionth of a second)."
 GDT::Blog:: Nanotech SmallBlog
 GDT::BAB:: A Mathematical Singularity
 GDT::Computing::Bit:: NanosecondGrace Hopper to ASU
 GDT::BAB:: Popular PhraseNano Giga
External Hyperlink(s)
 Math.com:: Decimals
 ScienceMadeSimple.net:: Fractions to Decimal Conversions Chart
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If necessary, rewrite the problem vertically and lineup the decimal points. It is okay to pad decimal numbers with zeros.
example: 4.451 + 13.3 4.451 + 13.3  17.751 or 4.451 + 13.300  17.751 example: 5.00073 + 255.101 5.00073 + 255.10100  260.10173Subtraction works the same way; i.e., lineup the decimal points prior to doing the subtraction.
5.55  4.02 5.55  4.02  1.53
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Execute the multiply as if dealing with whole numbers (i.e. it is not necessary to lineup the decimal points).
Insert the decimal point in the product by starting at the right and moving a number of places equal to the sum of the decimal places in both numbers multiplied.
5.50 * 2.1 5.50 * 2.1  550 + 1100  11550 5.50 has 2 digits to the right of the decimal point. 2.1 has 1 digit to the right of the decimal point. 2+1 is 3; therefore, the decimal point goes left of the 3rd digit from the right 11.550 or 11.55
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If the divisor has a decimal point, then make it a whole number by moving the decimal point to the right.
Move the decimal point in the dividend to the right by the number of moves made in the divisor.
Execute a whole number division ignoring any decimal point in the dividend.
Insert a decimal point in the quotient directly above the decimal point in the dividend.
_________ 3.5 ) 15.75 45 _________ 35 ) 157.5 140  175 175  0 Insert the decimal point into the quotient. 4.5Recall that answers to division problems can be checked by multiplying the quotient (result) by the divisor. The product should equal the dividend.
4.5 * 3.5  225 +135  1575 Both 4.5 and 3.5 have 1 digit to the right of the decimal. Insert decimal point into the product 2 digits left of the rightmost digit. 15.75
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Find the roundingto digit. If the digit immediately to the right of the roundingto digit is five or larger, increase the roundingto digit by one; otherwise, leave the roundingto digit unchanged. In both cases, all digits to the right of the roundingto digit become zero.
 nearest  number ten hundred thousand  13825 13830 13800 14000 77093 77090 77100 77000 72555 72560 72600 73000  nearest  number tenth hundreth thousandth  1.28382 1.3 1.28 1.284 102.1291 102.1 102.13 102.129 0.05454 0.1 0.05 0.055PurpleMath.com:: Rounding Numbers
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Fractions are often expressed in their lowest form (i.e. simplified or reduced).
The following technique can be used to reduce fractions.
1) Do a prime factorization of the numerator. 2) Do a prime factorization of the denominator. 3) Cancel (eliminate) "like" factors.Example.
12 reduce:  44 2 * 2 * 3 prime factorizations:  2 * 2 * 112*2* 3 cancel like factors: 2*2* 11 3 final answer:  11Another reduction techique is to use the greatest common factor of the numerator and denominator.
x x ÷ gcf(x,y)  reduces to  y y ÷ gcf(x,y)Example.
12 reduce:  44 gcf(12, 44) equals 4 12 ÷ 4 3  =  44 ÷ 4 11Another fraction reducing strategy is as follows.
1) Determine if the numerator or the denominator is the simplist prime factorization and do a prime factorization. 2) Cancel the prime factors that are also factors of the fraction part that was not prime factorized.Example.
25 reduce:  205 do a prime factorization on the numerator 25 5 * 5  =  205 205 5 evenly divides 205 41 times 5 does not evenly divide 41 25 5  =  205 41An observation about cancelling like terms...
a(c) a c a  =  *  =  c dividedby c is 1 b(c) b c b
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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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