Math 247 exam #4
factor/divisor
"b"
multiple
"b"
$2 each
5 items cost $10 and 8 items cost $16.
1736 1(1000)+7(100)+3(10)+6
6 is the last digit so the entire number is even.
child 1/total 2/5 = x/80 child 2/total = 3/5 = x/80 160=5x 32 m&ms 240=5x 48 m&ms
80 m&m's will be divided between two children in the ratio 2 to 3. how many m&ms will each child receive?
F15 = {1, 15, 3, 5} F20 = {1,20,2,10,4,5} GCF(15,20) 5. what number they have in common.
GCF for 15 and 20.
25 liters: 25x3/2 = 75/2 liters or 37.5 liters 45 liters: 45x3/2 = 135/2 liters or 67.5 liters S liters: 5x3/2 = 5/2 liters of 1.5s liters this is similar to the first example but here 3 to 2 is a reduced ratio. we may hae had other numbers for glue and starch, but theyve been reduced as low as possible.
a company makes gloop by mixing glue and starch in a 3 to 2 ratio. how much glue will the company need to mix with 25 liters of starch? 45 liters of starch? s liters?
even
a counting number is even if it is divisible by 2. there is no remainder when you divide the number by 2.
people/flour -- 6/2.5 = 10/x 6x=2.5 x 10 6x=25 x=25/6 or 4 1/6 cups flour
a recipe that serves 6 people calls for 2 cups of flour. how much flour will you need to serve 10 people, assuming that the ratio of people to cups of flour remains the same?
ratio
a:b, where a and b are rational numbers, is a comparison of two quantities a and b expressed as the number of times one is greater than or less than the other
mixture a cups of red punch: 1, 2, 3, 4, 5 cups of lemon-line: 3,6,9,12,15 total number of cups: 4,8,12,16,20 x2, x3, x4, x5. mixture b cups of red punch: 3,6,9,12,15 cups of lemon-line: 5,10,15,20,25 total number of cups: 8,16,24,32,40 x2, x3, x4, x5. as long as you multiply every number by the same thing, you will get an equivalent ratio.
b) find mixtures of red punch and lemon-line soda that taste the same as mixture a. find other mixtures that taste the same as mixture b.
part-to-part
boys (part) to girls (part)
part-to-whole
boys (part) to the children in the class (whole)
we need to add 1/2 to the ice cream until we get to 3 cups. we can add 3/4 until we get to the matching partner. 18/4 or 4 1/2 cups milk. diagram on paper.
chandra made a milkshake by mixing 1/2 cup of ice cream with 3/4 cup of milk. reason about quantities to determine how many cups of milk chandra should use if she wants to use 3 cups of ice cream.
whole-to-part
children in the class (whole) to boys (part)
prime number 2,3,5,7,11,13,17
counting number other than 1 that has exactly two distinct factors/divisors; 1 and the number itself.
composite number 4,6,8,9,10,12,15
counting number that has more than two distinct factors/divisors.
10
divisible by 10 if the ones digit is divisible by 10
8
divisible by 8 if the last three digits are divisible by 8
9
divisible by 9 if the sum of its digits is divisible by 9.
F36 = {1,36,2,18,3,12,4,96}
factors for 36.
F21 = {1,21,3,7} NOTATION IMPORTANT.
find all factors of 21.
48 6 8 2 3 2 4 2 2 120 10 12 2 5 3 4 2 2 48=2^4 x 3. 120 = 2^3 x 3 x 5. GCF (48, 120) = 2^3 x 3 =24. GCF uses the primes in common on the trees and the smallest of the exponents from these common primes. 2 and 3 were on both trees. we use the single 3 and have to pick the smallest exponent from the 2's) LCM (48, 120) = 2^4 x 3 x 5 = 240. LCM uses the common primes but largest exponents on them and uses any number that wasnt in common. 2 and 3 were in common but we needed the largest exponent from 2. then we put in any other numbers not in common like the 5.
find the GCF and LCM for 48 and 120.
M15 = {15,30,45,60...} M20 = {20,40,60...} LCM (15,20) = 60. number in common.
find the LCM for 15 and 20.
M21 = {21,42,63,84,105} 21x1, 21x2, 21x3, 21x4, 21x5
find the first 4 multiples for 21.
refer to papers for this.
find the number of liters of rose oil that should be mixed with 20 liters of lily oil to make rose-lily perfume.
300 3 10 10 10 2 5 2 5 300 10 30 2 5 3 10 2 5
find the prime factorization for 300 using two different factor trees.
divides
if "a" and "b" are any integers, then "b" divides "a" if and only if the a unique integer "c" such as a =bxc.
proportion theorem
if a, b, c, and d are rational numbers and b=0 and d=0, then is a proportion if and only if ad=bc.
you add +4 gallons for every time you add +100 miles.
if joe's car goes 100 miles for every 4 gallons of gas, then how far can joe drive on 20 gallons of gas?
237,921 - odd number.
if the last digit is 0,2,4,6,8 the number is even. 1,3,5,7,9 the number is odd.
cars/students = 3/8 = x/1200 3x1200= 8x 3600 = 8x 450=x
if there are 3 cars for every 8 students at a high school, how many cars are there for 1200 students?
boys/total -- 3/5 -- 3/5 = x/45 x=27 now we are asked to find boys to the class (total). we can add 3 and 2 to get 5 and then solve the problem the same way as in part a). the 5 becomes a 45 by using 9 as the multiplier. 3x9=27 .. because 5x9=45.
if there are 45 students in the class altogether, how many boys are there?
2+3=5, but 2x3=6. adding an odd to an even will always give an odd number. think about skip counting. if you start with an odd adding 2 "skips" to the next odd, 4 will skip to the second odd, 6 will go to the next, etc. multiplying an even and odd always gives an even because every even number you multiply by is always divisible by 2. in other words, 2 is a factor so that product must be even.
if you add an odd number and an even number, what kind of number do you get? if you multiply an odd number and an even number, what kind of number do you get?
if we go over 109, we can't stop testing. 2^2=4 3^2=9 5^2=25 7^2=49 11^2=121 121 is higher than 109 so you stop there. none of the four primes worked. 109 is prime.
is 109 prime or composite?
1:3
it could be 1 boy to 3 girls, 2 boys and 6 girls... any number that gives the fraction equivalent to 1/3.
chapter/time = 1/30 = 14/x 1x=30 x 14 x=420
jane loves to read. she can read a chapter in about 30 minutes. assuming chapters are all about the same length, how long will it take her to read a book with 14 chapters?
M15= {15,30,45,60...} multiples of 15..15x1, 15x2, 15x3, 15x4, etc.
multiples of 15?
4
number divisible by 4 if the last two digits are a multiple of 4.
2
number is divisible by 2 if the ones digit is divisible by 2.
3
number is divisible by 3 if the sum of its digits is divisible by 3.
5
number is divisible by 5 if the ones digit is divisible by 5.
6
number is divisible by 6 if it is divisible by 2 and 3.
constant of proportionality
ratios are pairs of numbers but fractions are numbers. when working with quantities that vary together in a fixed ratio we called the value of the ratio this.
proportional relationship
relationship between two quantities where all coordinated pairs of values of the quantities are in the same ratio.
review 7.6 class activity.
review 7.6 class activity.
review 8.3 class activity.
review 8.3 class activity.
review slide method and class activity 8.6.
review slide method and class activity 8.6.
for 1 second: 5m/2sec = 5/2 = 2.5m per sec 15 secs: 15x2.5 or 15x5/2 = 37.5m or 75/2m. 35 secs: 35x2.5 or 35x5/2 = 87.5 or 175/2m. T secs: Tx2.5 or Tx5/2 = 2.5Tm or 5/2Tm
rich runs 5 meters every 2 seconds (and keeps a steady pace for several minutes). how far does rich run in 15, 25, and T seconds?
rose oil: ---------------------- |---------------------| lily oil ---------------------- |---8-----8-----8----| ----------24---------- we have 24 liters of lily oil in a ratio of 4 rose oil to 3 lily oil, so if we divide the 24 by 3 then each part is worth 8. that means that each part of the rose oil is also 8 so that 8x4=32 liters.
rose-lily perfume is made by mixing rose oil and lily of the valley oil in the ratio 4 to 3. how many liters of rose oil should be mixed with 24 liters of the valley oil to make rose-lily perfume?
mom/total 5/7 = x/210 sandra/total 2/7 = x/210 1050 = 7x 150 sandra/total 2/7 = x/210 420/7 60
sandra wants to buy an ipod that costs $210. her mother agreed to pay $5 for every $2 sandra saved. how much did each pay? (5+2)
proportion
statement that two pairs of amounts are in the same ratio
greatest common factor GCF (4,6) = 2 4 &6 can both be divided by 2.
the GCF is the greatest counting number that divides both a & b.
least common multiple LCM (2,3) = 6 2 x 3 = 6 3 x 2 = 6
the LCM is the least counting number that is a multiple of a & b.
boys/girls - 3/2 -- x=30 one way to solve this is to see how 2 became 20 (multiplied by 20).
the ratio is 3:2. if there are 20 girls, how many boys are in the class?
the word "to" is our biggest help. the word in the front of it goes on the left and the word behind it goes on the right. part to part > part to whole dogs:cats cats:total 7:12 12:19
there are seven dogs and twelve cats in the neighbor's yard. express the number of dogs to cats as a ratio and then the number of cats to animals as a ratio.
mixture a: 1:3 - mixture b: 3:5 (add 2 to 1 and 2 to 3) these will not taste the same or have the same color because the ratios cant reduce to the same number. even though we added the same 2 cups of red and 2 cups of lemon-line that doesnt make them equal. we should have doubled both amounts to get 2:6, which would be an equivalent ratio.
there are two containers, each holding a mixture of 1 cup red punch and 3 cups lemon-line soda. the first container is left as it is. that is mixture a. someone adds 2 cups of red punch and 2 cups of lemon-line soda to the second container. that is mixture b. a) do you think mixture a and mixture b will taste the same and have the same color? why or why not?