MATH 008 LECTURE NOTES Dr JASON SAMUELS Ch1 Whole Numbers $55 Solution: 81+9 55=81+495=576 576-540 = 36$ This alternate way to multiply is called the lattice method, because the boxes make a lattice. The product of every pair of digits goes in the corresponding box, then you add along the diagonals. Those diagonal slashes make sure every digit is added with the proper place value.
ex) if a car travels 438 miles in 6 hours, what is its speed (in miles per hour)?
ex) if 14 credits costs $3682, how much does each credit cost? ex) in one Yankee game, Jeter had 2 hits, Teixiera had 3 hits, ARod had 3 hit, Gardner had 0 hits, Cano had 1 hit, Cervelli had 3 hits. how many total hits did those players have? how many average hits did those players have? Rounding When a shirt costs $23.99, we often just say it costs $24, because that tells us the information we need. this is called rounding. the Nile river is 4180 feet long. how long is it to the nearest thousand miles? how long is it to the nearest hundred miles? how long is it to the nearest ten miles? ex) the most populular languages on the internet are English, which has 478,582,273 users, and Chinese, which has 383,212,617 users. what could be a useful way to round those numbers? using rounding, how many more English users are there? stop at the place value you need and replace the rest with 0's: truncating look to see which number is closer (and maybe round up): rounding ex) Nile to the nearest hundred miles is 4200 (not 4100) when do we round up? when the first number we are "erasing" is 5 or bigger ex) English speakers to the nearest million: 479,000,000
its possible that when you round the number, its the same number - this happens when the end digits (that you are rounding away) are 0's ex) Nile to the nearest ten miles is 4180 Exponents 4 5 can also be written as: 8x8x8x8x8x8x8 can also be written as: 10 5 can also be written as: can think about this in: - number of zeroes - words 100000000 can also be written as: calculate 7 x 10 6 4 2 3 3 can also be written as: 7 7 7 7 7 5 5 5 2 can also be written as: notation and names: 4 5 8 7 is "eight to the seventh power" OR "eight to the seventh" ex) is 2 to the 6 the same as 6 to the 2? remember that exponents describe multiplication (not addition) (they can also describe other operations, which we will see later) Order of Operations
or, doing it all together if you have $20 dollars in your pocket, is it the same to gain a dollar then double your money, or double your money then gain a dollar? ex) calculate 13-4 + 7 = 9 + 7 = 16 NOT = 13 11 = 2 ex) calculate 12 3 x 2 = 4 x 2 = 8 NOT = 12 6 = 2 ex) calculate 6 + 3(9-4) = 6 + 3(5) = 6 + 15 = 21 ex) calculate 8 4 + 7-2 x 5-2 2 = 8 4 + 7-2 x 5-4 = 2 + 7 2 x 5 4 = 2 + 7 10 4 = 9 10 4 = -1 4 = -5 ex) calculate 3 + 2 x (8 + 3 2 ) = 3 + 2 x (8+9) = 3 + 2 x (17) = 3 + 34 = 37 Do parentheses left to right Do exponents left to right Do multivision left to right Do addtraction left to right
note: simple calculators, like those on a cell phone, are NOT GOOD at order of operations. in that case, YOU have to know which part of the calculation to do first. Factoring & Multiples Consider the statement 4 goes into 12. It means that you can write a multiplication or a division fact using whole numbers: 3 4=12 OR 12 4 = 3. Each statement tells us that 4 is a factor of 12, and that 12 is a multiple of 4. Ex) list all the factors of 12 1,2,3,4,6,12 since these are the numbers that divide 12 with no remainder Ex) list all the factors of 30 Here is a nice way to save yourself time make a factor U Write factor pairs, starting with 1 1 30 2 15 3 10 5 6 Since the next factor after 5 is 6, you have all the factors. Why? If there was a missing factor in the right column, it would need a partner in the left column. To write the list of factors in order, go down the left side and up the right side like a U. Ex) list all the factors of 40 Primes A prime number is a number that has exactly two factors, 1 & itself. Ex) 12,30,40 are not prime they are called composite Ex) 2 is prime, since its factors are 1 & 2 Ex) 3,5,7 are prime. Ex) 9 is not prime 3 is a factor. Note that any number can be broken down into factors, each of which can be broken down into other factors. You can keep going until the only numbers left are prime numbers. One way to show this is to use a prime factor tree. Ex) factor 24 into prime numbers 24 The prime factorization of 24 is: 2 2 2 3 OR 2 3 3 1
We wont make a factor tree often, but it is useful to know that if we want to find a factor we should start by looking at prime numbers. Ch2 Fractions...which are hard, but not as hard as you think Calculate: In a class of 30, four-fifths did their homework. How many students did their homework? What is a fraction? What are some situations where fractions are useful?
ex) what fraction of the class today is female? ex) if you flip a coin, what fraction of the time will it come up heads? 1/2 ex) suppose you want to share a pizza evenly between 3 people. how many slices would you cut it into? what fraction does each person get? Simplifying fractions ex) 15 / 24 ex) 18 / 30 any number that goes into both 18 & 30 is called a common factor 6 is the biggest number that goes into both 18 & 30 it is called the greatest common factor (gcf)
notice how (3)(2)=6, if you divide by those numbers, you will completely reduce your fraction eventually. Just keep checking to see if any prime numbers are common factors. (In our problems, it should be enough to check the first four primes: 2,3,5,7.) ex) 12 / 25 there are no common factors between 12 & 25, so the fraction cannot be simplified "Unsimplifing" fractions ex) for 3 / 5, make an equal fraction with 20 in the denominator Comparing fractions with > < = (by unsimplifying ) which is more, 5/10 or 7/10?...for 5/10, you cut the pizza in 10 slices and eat 5. for 7/10 you cut the pizza in 10 slices and eat 7. so 7/10 is bigger with a common denominator you can compare fraction amounts by just comparing the numerator (since the slices are the same size). ex) one pizza was cut into 12 slices, and someone ate 5 slices. another pizza was cut into 8 slices and someone else ate 3 slices. who ate more? any number that you can turn both numbers into using multiplication is called a common multiple when this number is in a denominator, it is called a common denominator the smallest number that you can turn both numbers into using multiplication is called the least common multiple (lcm) when this number is in a denominator, it is called a least common denominator (lcd) you need a common denominator for other kinds of comparison questions, "how much combined" (addition) and "how much more" (subtraction). how can you find a common denominator? - multiply the demoninators together
- good: this always works - bad: you might get large numbers - find the lcm of the denominators - good: this is the smallest number, so the calculations are easier - bad: finding the lcm takes work how do you find the lcm? if the numbers do not have common factors, just multiply them together ex) if the numbers do have common factors, then you have three methods... - list: make a list of multiples, the lcm is the first number that appears on both lists ex) find the lcm of 6 & 8 6,12,18,24,30,36,42,48... 8,16,24 - thats it - gcf: find the gcf, calculate first number divided by gcf times the second number ex) find the lcm of 6 & 8 2 goes into both numbers calculate: 6/2 8=3 8 = 24 Ex) find the lcm for 12 & 18 The gcf is 6 the lcm is 12/6 18=2 18=36 [the third method is more advanced. It is described here, but we won t use it] - prime factors: factor each number into primes, then make a new number which includes all the prime factors of all numbers ex) find the lcm of 6 & 8 factor: 6=2 3, 8=2 2 2 new number must have how many 2 factors? 3. How many 3 factors? 1 lcm is 2 2 2 3 = 24 compare: 3 / 14 2 / 11 14 & 11 have no common factors, so get the lcd by multiplying: 14 11=154 compare: 5 / 9 7 / 12 9 & 12 do have common factors, so also could use an lcd method... list: 9,18,27,36... 12,24,36- thats it gcf: the gcf is 3, so 9/3 12 = 3 12=36 prime factors: 9=3 3, 12=2 2 3 lcm = 2 2 3 3 = 36
ex) compare, which is bigger? ex) compare, list from largest to smallest: 2/7, 4/9, 1/3 need a common denominator. how should we do this with more than two fractions? method 1: go one number at a time [I suggest this method] find a common denominator for 7 & 9, no common factors so multiply 7 9 = 63, now find a common denominator for 63 & 3 by gcf method, lcd=63 3/3 = 63 (or, 3 goes into 63 so its 63). Method 2: use all numbers at once if you just multiply denominators together, you get 7 9 3 = 189... this works but the numbers are large to get a smaller number, use lcm gcf: cant use gcf to find the lcm of more than two numbers list: 7,14,21,28,35,42,49,56,63,70... 9,18,27,36,45,54,63,72,81... 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63- thats it prime factor: 7=7, 9=3 3, 3=3 lcd = 3 3 7 = 63 For the harder problems, the prime factor method is very quick and useful to do these fraction compare/add/subtract problems, you need a common denominator. to find one, make two choices: - one number at a time OR all at once? - multiply OR list OR gcf (two numbers at a time) OR prime factor?
Compare (which is larger): a) b) c)
Adding & Subtracting fractions If you cut a pizza pie into 3 slices, then you eat one slice, then you eat another slice, how much of the whole pie did you eat in total? If you cut a pizza pie into 8 slices, for lunch you have 2 slices, and for dinner you have 3 slices, how much of the pie did you eat that day? If you are adding fractions, and you have the same denominator, just add the numerators (to find out how many slices!) So it is easy to add fractions as long as you have a common denominator. That s also true for subtracting. Suppose there is a partially eaten pie that was cut into 6 slices and there are 4 slices left. If you eat 1 slice, how much of the whole pie is left? If you are subtracting fractions, and you have the same denominator, just subtract the numerators (to find out how many slices!) What if the fractions do not have a common denominator? give them one! Calculate (simplify the answer) a) b) c) d) e) f)
Mixed Numbers For fractions bigger than 1, sometimes we prefer to indicate how many wholes are included as well as the fraction part left over. This form is called a mixed number. Converting from improper fraction to mixed number - Just divide Ex) 25 / 6 25 6 = 4r1 The fraction includes 4 wholes, with 1 slice left over 4 1 / 6 Converting from mixed number to improper fraction - convert the whole to a fraction, then add to combine into one fraction Ex) 4 1 / 6 = <- converted 4 into a fraction = + <- wanted to add, so make a common denominator = + = <- added There is a shortcut to this process. Since the denominator of the whole number is always 1, you will always multiply top and bottom by the other denominator. Then you add the other numerator. Here is an example. Ex) 4 = = compare/add/subtract mixed fractions compare: Method 1: convert to an improper fraction, do the calculation, then convert to mixed number - Good: always works - Bad: numbers can get large Ex) 4 5 / 8 + 6 3 / 4 Method 2: work with whole number parts then fraction parts - Good: usually easier - bad: sometimes harder
ex) calculate 5 + 3 2 / 9 ex) calculate 7 2 / 3-4 ex) calculate 3 1 / 4-1 1 / 9 3-1 = 2 1 / 4-1 / 9 = (lcd=4 9=36) ex) calculate 5 4 / 21 + 7 3 / 7 5+7=12 4 / 21 + 3 / 7 = lcd=21 so 5 4 / 21 + 7 3 / 7 = 12 13 / 21 There can be one complication with this method: ex) 4 5 / 8 + 6 3 / 4 4+6=10 5 / 8 + 3 / 4 = lcd=8 what do we do with 11 / 8? its an improper fraction, we can convert it to a mixed number: 11/8 = 1r3, so: 11 / 8 = 1 3 / 8 so the answer is 10 + 1 3 / 8 = 11 3 / 8 notice that this closely resembles carrying from whole number addition. in this case, we are regrouping from left to right (carrying) from the fraction part to the whole number part. ex) 7 2 / 7-4 3 / 5 2 / 7-3 / 5 = =?? lcd=7 5=35 we do not want to subtract a larger amount from a smaller amount. how can we deal with this?...by regrouping from right to left (borrowing) 7 10 / 35 = 6 + 1 + 10 / 35 = 6 + 35 / 35 + 10 / 35 = 6 + 45 / 35 7 10 / 35-4 21 / 35 = 6 45 / 35-4 21 / 35 6-4 = 2 45 / 35-21 / 35 = 24 / 35 so the result is 2 24 / 35
Multiplying & Dividing Fractions Recall what it means to multiply whole numbers Ex) 3 5 = 15 means: We are now going to make a picture so that the sides are fractions. Ex) Draw a box to represent one whole. Mark on one side, and on the other Figure out how much has been shaded remember that the box is one whole How many slices has the whole been cut into? 5 7 = 35 How many slices are shaded? 2 3 = 6 So: = = When multiplying two fractions, the number of slices in the first fraction is the number of slices for the length, the number of slices in the second fraction is the number of slices for the width, so the number of slices in the whole is the product. It works the same in the numerator and denominator. Note: if your calculation has some whole numbers and some fractions, you will need to write every number as a fraction for a whole number, you can write it as a fraction with 1 in the denominator Ex) 5 Note: you will have to simplify the result. It s easier to simplify before you multiply Ex) You do: a) b) 6
Now let s figure out how to do division with fractions Rewrite 6 3 as a fraction: Rewrite 6 3 by rewriting all whole numbers as fractions: Calculate (combine into one fraction, don t simplify): So these are all the same: 6 3 is the same as is the same as is the same as In particular, is the same as That shows us how we can divide fractions take the dividing fraction, flip it and multiply. The flip is called the reciprocal. This technique is called flip and multiply. Ex) There are other ways to understand why flip and multiply works, but this is probably the quickest. You do, calculate and simplify: a) b) c) d) 8 For multiplication and division with mixed numbers, always convert to improper fractions, calculate, then convert back to a mixed number. Ex) 4 1 You do: 8 1
Decimals What are decimals? What are some situations where decimals are useful? Compare: 5.269 6.4 Calculate: a) 14.625 + 4.79 b) 74.103 5.67 c) 3.2 x 15.03 d) 9.81 0.3
Place Value What are the place values for decimal numbers? Ex) for 54.8726 Which digit is in the ones place? Tens place? Tenths place? What is the place value for: 5 4 8 7 2 6 Recall: ten 1 s = 10, ten 10 s = 100, ten 100 s = 1000, etc now, go the other way: 1000 is ten 100 s, 100 is ten 10 s, 10 is ten 1 s, 1 is ten??? This is where we get into decimals 1 is ten tenths, a tenth is ten hundredths, a hundredth is ten thousandths, etc 10 x 0.1 =? 100 x 0.01 =? 1000 x 0.001 =? 100 x 0.1 =? 100 x 0.0001 =? The way we keep track of place value is with the decimal point and with zeroes. Ex) if you multiply by 100, 1 is now worth 100, 10 is worth 1000, etc. Also, 0.01 is now worth 1, etc. Shortcut: when you multiply by 100, move the decimal point two places to the right (fill in 0 s if necessary). Ex) if you multiply by 0.1 (one-tenth), 1000 becomes 100, 100 becomes 10, 10 becomes 1. Also, 1 becomes 0.1, 0.1 becomes 0.01, etc. Shortcut: when you multiply by 0.1, move the decimal point one place to the left (fill in 0 s if necessary). Recall, two ways to understand a whole number: 52 = 52 ones OR 5 tens and 2 ones (50+2) (no grouping) (grouping) Two ways to understand a decimal number: 0.257 = + + 257 thousandths OR Two tenths and five hundredths and seven thousandths OR (no grouping) (grouping)
Comparing Decimals Recall, for whole numbers: to compare 637 & 85, first check which starts in the bigger place value. Don t just compare the lead digits 6 & 8 to compare 3472 & 3452, go place value by place value until one is bigger. Now, for decimals: Ex) compare 3.72 & 3.58 Ex) compare 3.47 & 3.2846 Ex) compare 0.037 & 0.0084, first check which starts in the bigger place value. Don t just compare the lead digits 3 & 8. to compare 0.749 & 0.76, go place value by place value until one is bigger. Note: if it helps, remember that for any place without a digit, you could put a 0 there So comparing 0.749 & 0.76 is the same as comparing 0.749 & 0.760 But the most important thing step is to compare place value by place value. Adding and Subtracting decimals Recall for whole numbers: 376+86 = 529 37 = Now for decimal numbers, it s the exact same. The only change is that, to keep track of place value, you also have the decimal point. Ex) 64.56 + 27.5 = ex) 92.5 7.83 = Multiplying and Dividing decimals Recall for whole numbers: 3 x 2 = 6, but 30 x 200 = 6000. 9 3 = 3, but 9000 30 = 300. Note that we use the 0 s to keep track of place value. How can we do it in the calculation?
Now for decimals, we do the same thing, with one difference place value is indicated, not only by 0 s, but also by the decimal point, so you must keep track of both. Ex).05 x.0003 = Ex) 4.2 x 0.06 = Notice that, to turn 3 into 30, you add a zero, or move the decimal one place to the right same thing. So to turn 42 into 4.2, you remove a zero or move the decimal one place to the left same thing. So in 4.2, there is one decimal digit (as if you moved the decimal point one place to the left), and in 0.06 there are two decimal digits (as if you moved the decimal point two places to the left), so the answer has three decimal digits (as if you moved the decimal point 3 places to the left). Ex) for the problem 5.42 x 87.29, if you wanted to make sure the answer was a whole number, what would you multiply by 10 or 100 or 1000 or? You do: Ex) 3.4 x 0.6 Ex) 3.02 x 0.045 Another way to do multiplication: Now, dividing: Division is the opposite of multiplication, so it has the opposite effect Ex) 8.4 0.2
Ex) 9.3.03 Ex) 8.4.007 Ex).42 1.4
Algebra What is algebra? When or how can we use it? Ex) anyone who takes Math 150 earns 4 credits. If they start with 38 credits, how many will they have after? If they start with 3 credits, how many will they have after? If they start with 54 credits, how many will they have after? If they start with c credits, how many will they have after? Ex) a pizza is cut in 8 slices. If you eat 2 slices, how many are left? If you eat 1.7 slices, how many are left? If you eat 3 slices, how many are left? If you eat s slices, how many are left? Ex) suppose business partners make $500 on a deal, which they share evenly. If there are 4 partners, how much does each partner get? If there are 10 partners, how much does each partner get? If there are p partners, how much does each partner get? Ex) make up a situation for the expression x+5 Ex) make up a situation for the expression! Expression vs. Equation Ex) if I have x dollars, and I double my money, how much will I have? Ex) if I have $17 and I double my money, the amount I have is $3 more than James has. How much does James have?
Solving Algebra Equations We have some equation involving a variable. The goal: we want to manipulate it until we get (variable) = ~ This is called solving for the variable. The golden rule for algebra equations: if two expressions are equal, and you do equal things to them, then they are still equal. Ex) x+1=9 We want to get rid of the 1. It is attached by addition. So subtract. Ex) c+4=18 Ex) h 2 = 15 Ex) k + 4½ = 13 You do: solve for the variable Ex) t - 6 = 21 ex) x + 4.7 = 9.2 ex) "+ = Multiplication & division Ex) 3x = 6 We want to get rid of the 3. It is attached by multiplication. So divide. Ex) # = 8 Ex) 6g = 11
Ex) 3 =10 Ex) $ = 5 Either way, things cancel and you are left with z, which is what you wanted. You do: Ex) % = 24 ex) 2.2k = 66 ex) & = 7 Word problems Most important step is the first step: write an equation that tells the same story as the words.
To solve a word problem: - Indicate your quantities, and each one gets a variable - Make a math story (equation), then translate it back into words to make sure it s the same story - Solve the equation - Check the answer, does it make sense A computer technician charges $80 per hour. a) how much does 2.5 hours of work cost? b) if a job cost $300, how many hours did it take? You are driving at 60 miles per hour. How many miles do you travel in 2 hours? How many miles do you travel in 7 hours? How many miles do you travel in t hours? If you travel 540 miles, how many hours were you driving?
Another technique: use easier numbers If tickets are $5 and there are 3 of them, how much money is that? The answer is m = (5)(3) = 15 So to answer this question, use 87 instead of 3 and 9.50 instead of 5: m = (9.50)(87) = 826.50
MIDTERM REVIEW
Rate & Proportion What is a rate? When or how can it be used? What is a proportion? How or when can it be used? Try these questions: If you travel 130 miles per 2 hours, how far do you travel in 5 hours? If you earn $360 in a week when you work 30 hours, a) how much do you earn if you work 40 hours? b) how much do you have to work to earn $492? if you have time, work on these problems:
A contractor makes a construction mix with 80 pounds of sand for every 100 pounds of gravel. If the contractor has 45 pounds of sand left, how much gravel should be added? The aspect ratio of an image is the ratio of the width to the height. For movies, the standard aspect ratio is 16 to 9. If a movie theater has a 33-foot ceiling, and the screen should start 6-feet above the ground how wide should the screen be? In one class, there are 24 students, including 16 females. What is the ratio of males to students? What is the ratio of males to females? What is the ratio of females to males? If a class with 72 students has the same ratios, how many males and females are there? There are two ways to think of these problems. One: as a proportion. A proportion is an equality between two ratios (fractions). Ex) if I earn $120 in 8 hours, how much do I earn in 14 hours? Set up the equation: Solve: Two: as a rate Ex) if I earn $120 in 8 hours, how much do I earn in 14 hours? Calculate the rate:
Solve: It is generally useful to calculate the rate per each or per one. This can be represented as a fraction or decimal, and it is called the unit rate. Ex) which is cheaper: 30 cough drops for $1.80 or 80 cough drops for $4.40? Ex) which is cheaper: 4 batteries for $2.80 or 10 batteries for $3.90 or 24 batteries for $7.44? Ex) which is cheaper: a 16-ounce jar of peanut butter for $4.80 or a 24-ounce jar for $6.96? Compare the two methods: Ex) for every $100 I earn, I save $3 in the bank. If I earn $750, how much do I save? Method 1: as a proportion Method 2: as a rate Ex) a college has a ratio of 14 students to 1 faculty member. (This is called the studentto-faculty ratio.) if the college has 4200 students, how many faculty are there?
Which method is better? In simple problems, some people think proportion is easier, some think rate is easier. In all other problems, rate is easier. In fact, proportion often will not work. Ex) how many minutes are in a day?
Questions
Quiz One person has 60 kg of mass and 25 L of volume. If another person has 15 L of volume, using the same rate, what is their mass? 30 gallons of oil flow through a pipe in 4 hours. At the same rate, how long will it take for 315 gallons to flow?
Percents What is a percent? When or how can it be used? Try these questions: Ex) write the number.37 as a percent and a fraction Ex) write the number.184 as a percent Ex) write the number 8.4 as a percent Ex) write 76% as a decimal and a fraction Ex) write 7 as a fraction and a percent Ex) what is 40% of 50? Ex) during a scientific experiment, the scientist combined 10 milliliters of acid with 40 milliliters of water. What percent of the mixture was acid? What is a percent?
Ex) 70% Converting between decimal, fraction, percent From decimal to percent: -multiply by 100%, or shift the decimal two places to the right Ex).64 From percent to decimal: -divide by 100%, or shift the decimal two places to the left Ex) write 29% as a decimal Ex) 78% of the arguments that couples have are about money. Write this as a decimal. Ex) write 350% as a decimal Note that you can multiply or divide by 100% because 100% = 1 If you ever get confused or forget which way to move the decimal point, remember that percent numbers are bigger (because they are out of 100) From decimal to fraction: -identify the rightmost place value, that whole number goes in the denominator. Write the decimal as a whole number, that goes in the numerator (then simplify). Ex) convert 0.027 to a fraction Ex) convert 32.6 to a fraction From fraction to decimal: -divide (use the division algorithm) Ex) convert to a decimal From fraction to percent: - divide (to convert from a fraction), then multiply by 100% (to introduce a % ) Ex) convert to a percent
Ex) a student got 12 out of 20 questions right on a quiz. (All the questions are worth the same amount.) What was the grade: as a fraction? As a percent? From percent to fraction: -divide by 100% ( to get rid of the % ) Ex) convert 35% to a fraction (and simplify) Ex) one factory grew its production by 165%. White that as a simplified mixed number. Ex) more than ¾ of the France s energy comes from nuclear energy. Write this as (a) a decimal (b) a percent Ex) one large egg contains 15% of the recommended daily value of protein. Write this as (a) a decimal (b) a fraction Ex) after an oil spill, 1/5 of the local wildlife survived. Write this as (a) a decimal (b) a percent Solving percent problems There are 3 quantities to identify: X percent of Y is Z ( ) = * Ex) 10 is 50% of 20 Write as an equation: Ex) 25% of 120 is 30 Write as an equation: -math statements Convert to a decimal or fraction, set up an equation, then solve Ex) what is 50% of 16? Ex) what is 45% of 20?
Ex) what is 250% of 12? Ex) 4 is 25% of what number? Ex) what percent of 80 is 60? -word problems The key is to translate into an equation that tells the same story. Ex) Ex)
Ex) Ex) Ex) challenge:
Percent increase or decrease Even though it says increase or decrease, the smartest way to do these percent problems is by multiplying. To see that, let s compare possible methods. Ex) a $50 shirt is marked 30% off. What is the new price? Method 1: addition/subtraction how much is deducted from the price? 30% of $50: 0.30(50) = 15 New price = 50 15 = 35 $ Method 2: multiplication How much is left? 30% off means 70% is left 70% of the original price (because 100% - 30% = 70%) New price is 70% of the original price = 0.70 50 = 35 $ Ex) a $250 tablet is marked 20% off. What is the new price? 20% off means 80% is left (because 100% - 20% = 80%) New price is 80% of the original price = 0.80 250 = 200 $ You do: Ex) a $120 jacket is marked 40% off. What is the new price?
The multiplication method is much better when there are SEVERAL discounts. (It will also be better for algebra problems.) Ex) an $80 jacket is marked down 50%, then marked down 50% again. What is the final price? If you said it is now free does that make sense? The trick is that the second 50% represents 50% of the new, smaller price. Method 1: First discount: 0.5(80) = 40 new price: 80-40 = 40 $ Second discount: 0.5(40) = 20 New price: 40 20 = 20 $ Method 2: First discount: 50% off means 50% is left Second discount: 50% off means 50% is left Final price: 0.5 0.5 80 = 20 $ Ex) a $160 pair of shoes is marked down 40% then another 25%. What is the final price? 40% off means 60% left, 25% off means 75% left Final price = 0.60 0.75 160 = 72 $ You do: Ex) a bathing suit normally sells for $20. It is marked off 40%, and then marked off another 75%. What is the final price? challenge follow up: what is the final total discount (as a percent)? What if the value goes up by a certain percent? This is what happens with interest on a credit card or savings account. Ex) suppose you have 400 $ in the bank, and it earns 3% interest. How much do you now have in the bank? Method 1: 3% of 400 =.03 400 = 12 New value = 400+12 = 412 $ Method 2: 3% increase means there is now 103% of the original amount (because 100% + 3% = 103%) New value is 103% of the original value 1.03 400 = 412 $ You do: if $500 increases by 4%, how much money is there now?
Again, the multiplication method is much better when there are several percentage changes. Ex) suppose you have $1000 in the bank. If it increases by 10%, then by 5%, what is the new amount? Method 2: Increase by 10% means there is 110% of the previous amount, increase by 5% means there is 105% of the previous amount. Final amount = 1000 1.10 1.05 = 1155 $ Finding the percentage change Ex) if a $600 computer is marked down to $450, what is the percentage marked off? First, the computer is marked to what percentage of the original price? 450/600 =.75, or 75% that s how much is left So the discount is 100% - 75% = 25% You do: a $20 dvd is marked down to $12. What is the discount percentage? In percent change problems, keep in mind the two different percentage amounts: what percentage is added or subtracted, and what percentage is the result Try these questions