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MAT082 :: Lecture Note :: Week 07
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Quote of the Week
{Furman.edu::
Mathematical Quotations Server}
If you have an apple and I have an apple and we exchange apples
then you and I will still each have one apple. But if you have an
idea and I have an idea and we exchange these ideas, then each
of us will have two ideas.
--
George Bernard Shaw
(01856-01950)
{Irish playwright;
more...}
[ideas]
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BARS of the Week
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Phoenix AZ
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Bar Harbor ME
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Every number has an infinite number of multiples.
The multiples for a number 'n' can be found by multiplying the 'n' by 1, 2, 3 and so on.
multiples of 2 are 2, 4, 6, 8, 10, 12, ..., 200, ..., 1000, ... 2 * 1 equals 2 2 * 2 equals 4 2 * 3 equals 6 2 * 4 equals 8 2 * 5 equals 10 2 * 6 equals 12 2 * 7 equals 14 ... 2 * 100 equals 200 ... 2 * 500 equals 1000 ...Definition: Common multiples are multiples that are common to a set (collection) of two or more numbers.
multiples of 3: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, ... multiples of 4: 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ... common multiples of 3 and 4 are 12, 24, ... [exercise] No or Yes: 48 is a common multiple of 3 and 4? [answer] Yes because 48 is evenly divisable by both 3 and 4.When adding, subtracting or comparing fractions, they must a common denominator. A common denominator for two fractions can be quickly obtained by multiplying the two denominators together. Sometimes the least (or lowest) common denominator is used by finding the LCM (Least Common Multiple) for the denominators. Note: The LCM is also be called the Lowest Common Multiple.
Definition: The LCM of a set of numbers is the smallest number that evenly divides each number in the set.
common multiples of 3 and 4 are 12, 24, 36... the least common multiple is 12 because 12 < 24 < 36 < ...The LCM for a set of numbers can be found using the following technique (algorithm).
1) Do a prime factorization of each number. 2) Multiply together factors that are gathered as follows: + one of each factors that each number have in common + all factors that are unique to each numberLCM Examples
Find the LCM for 4 and 6. lcm(4, 6) = _____ prime factorization of 4: 2 2 prime factorization of 6: 2 3 lcm factors: 2 2 3 lcm(4,6) = 2 * 2 * 3 = 12 ----------------------------------------------------- Find the LCM for 27 and 45. lcm(27, 45) = _____ prime factorization of 27: 3 3 3 prime factorization of 45: 3 3 5 lcm factors: 3 3 3 5 lcm(27, 45) = 3 * 3 * 3 * 5 = 135 ----------------------------------------------------- Find the LCM for 98 and 312. lcm(98,312) = _____ prime factorization of 98: 2 7 7 prime factorization of 312: 2 2 2 3 13 lcm factors: 2 2 2 3 7 7 13 lcm(98, 312) = 2 * 2 * 2 * 3 * 7 * 7 * 13 = 15,288 ----------------------------------------------------- Find the LCM for 8, 10, 14 and 20. lcm(8, 10, 14, 20) = _____ prime factorization of 8: 2 2 2 prime factorization of 10: 2 5 prime factorization of 14: 2 7 prime factorization of 20: 2 2 5 lcm factors: 2 2 2 5 7 lcm(8, 10, 14, 20) = 2 * 2 * 2 * 5 * 7 = 280If prime factorizations have been calculated on a set of numbers, then their Greatest Common Factor (GCF) can be found. Note: When dealing with fractions, the GCF can be called the GCD (Greatest Common Divisor).
Definition: The GCF of a set of numbers is the greatest (largest) number that evenly divides into each number in the set.
The GCF for a set of numbers can be found using the following technique (algorithm).
1) Do a prime factorization on each number. 2) Multiply together all the factors that each number have in common.GCF Examples
Find the GCF for 4 and 6. gcf(4, 6) = ______ prime factorization of 4: 2 2 prime factorization of 6: 2 3 gcf factors: 2 gcf(4, 6) = 2 ----------------------------------------------------- Find the GCF for 27 and 45. gcf(27, 45) = _____ prime factorization of 45: 3 3 5 prime factorization of 27: 3 3 3 gcf factors: 3 3 gcf(27, 45) = 3 * 3 = 9 ----------------------------------------------------- Find the GCF for 98 and 312. gcf(98, 312) = _____ prime factorization of 98: 2 7 7 prime factorization of 312: 2 2 2 3 13 gcf(98, 312) = 2 ----------------------------------------------------- Find the GCF for 8, 10, 14 and 20. gcf(8, 10, 14, 20) = _____ prime factorization of 8: 2 2 2 prime factorization of 10: 2 5 prime factorization of 14: 2 7 prime factorization of 20: 2 2 5 gcf factors: 2 gcf(8, 10, 14, 20) = 2 ----------------------------------------------------- Find the GCF for 330 and 770. gcf(330, 770) = _____ prime factorization of 330: 2 3 5 11 prime factorization of 770: 2 5 7 11 gcf factors: 2 5 11 gcf(330, 770) = 2 * 5 * 11 = 110Many calculators have built-in functions to calculate LCMs and GGFs. LCM and GCF tools can also be found on the web, but you have to make sure they work correctly. {Venturaes.com:: LCM and GCF Calculator [opens new window]}
PurpleMath.com:: LCM and GCF [opens new window]
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Find the rounding-to digit. If the digit immediately to the right of the rounding-to digit is five or larger, increase the rounding-to digit by one; otherwise, leave the rounding-to digit unchanged. In both cases, all digits to the right of the rounding-to 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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A ratio is the "relationship in quantity, amount, or size between two or more things." [source::m-w.com::Merriam-Webster Online]
The general form of a ratio is as follows is
n:m, wherenis the quantity of one thing andmis 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 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::m-w.com::Merriam-Webster Online]
Given two ratios,
n:manda:b,nisaasmis tob.Two ratios are equal if the cross-products 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]
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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 one-hundreth).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 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 2005-06 in-state tuition rates by
$5per credit-hour. The 2004-05 tuition rate was$55.old rate: $55 [2004-05 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 2006-07 in-state tuition rates from
$60 per credit-hourto$65 per credit-hour.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$1.25per 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)
7is 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)
4is12%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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