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Kind: captions
Language: en
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Welcome to the Cypress College math
review on mixture word problems
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in this video we are going to talk about
mixture word problems
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to start off with let me ask you a question
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what do you get when you add five gallons
and 3 miles
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how do you add miles and gallons together
is it 8... no you can't do that
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the units for each term must match they
must be identical
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for example if you are taking dollars in the
first term you better have dollars in the
second term and your answer will be in
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dollars in the problems that we will be
working with which are mixture problems
were going to have something like this
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we're going to have
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the gallons of pure acid plus the gallons of
pure acid in the second
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solution to get the gallons of pure acid in
the final solution
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let's talk about the formula you would use
for solution mixture problems let's say that
in this bucket
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I have a gallon but it's a mix part of its
water and part of it is antifreeze in fact I
know that it is 50% antifreeze
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50% antifreeze what does that mean
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how much antifreeze is in the bucket how
much actual pure antifreeze is in the
bucket
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½ gal right
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how did you get that
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will let's see what you did you took 50%
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of
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1 gallon yes 50% of one gallon and what is
of in mathematics
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of is multiplication so we take 50% percent
and we multiply it
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times the amount so we take the
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concentration times the amount
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that gives us the formula that we are going
to use in all the solution
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mixture problems and what's really, really cool
about mixture problems
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is the formula is the same for every term
on both sides of the equation even over
on
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the right hand side you also can have
concentration times amount
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let's talk next about how the percentages
relate to each other let's say you had one
concentration
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that was 15% and another mixture that its
concentration was 50%
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and you have another mixture that was
let's say 40%
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which of these three is the final mixture
now I'm assuming the we're adding them
together were not like draining something
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off the traditional type where you are
adding two solutions together and getting
a third solution
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so let's say
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could the mixture at the end of the problem
be the 50% one
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could you take something this 15% pure
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and add in something that's 40% pure and
end up with something that's 50% pure so
take something is really diluted
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mix it in with something that's way more
concentrated and get
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something that's more concentrated than
either of the two no way that's impossible
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could this be the mixture at the end could
you take something that's 50% pure
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add in something that's 40% pure and get
an answer
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that's way more diluted than either of the
two mixtures you to put together that's
impossible
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of these three numbers the one that has to
be the concentration for your
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answer must be the one on the right there
the 40% because it's numerically in
between
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it's the number that's numerically in
between the other two mixtures
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that will always be the mixture or solution
that you have at the end your final solution
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we talked about how the concentrations
relate to each other now let's talk about
how the amounts relate to each other
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if I have 3 gallons of one mix
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and add it to 2 gallons of another mix
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I'm definitely going to get a mixture that is
5 gallons right
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here's the formula that
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we are working with concentration times
amount
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so we are going to focus on the amounts
here
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the way I've set it up the amounts are in
the second set of parentheses
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what if you were given one of the amounts
on the left hand side let's say it's 5 gallons
that goes in the first
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in first term in the second parenthesis
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if you're given that that's 5 gallons then
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this would be X you don't know that
amount right
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now what do you do for the amount on the
right hand side
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well we know that they add up just like 2
gallons and three gallons added to the 5
gallons right so 5 gallons plus
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x gallons is right five plus x so five plus x
would be the amount on the right hand
side
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what if instead of being given one of the
amounts on the left hand side you were
given one any amounts on the right hand
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the amount on the right hand side yeah so
it's 12 gallons
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you're given that the amount on the right
hand side
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then you're gonna use x for one of the
amounts on the left hand side
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and then you have two options
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you can either use y for the other amount
on the left hand side
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and then have a second equation that
says that x plus y is 12 solve that
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equation for Y which would be 12 minus X
and then you can replace y with 12 - x
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I prefer to just set it up originally with 1
unknown so if I know that I have 12 gallons
altogether in the final solution
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one of the amounts on the left is x the
other is 12 - x 12 is the total subtract the
part
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total minus part
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Sonya has a 55% antifreeze solution and a
10% antifreeze solution
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she wants 30 gallons of a 49% antifreeze
solution how many gallons of each must be
mixed to get the desired solution
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so here's our generic formula
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now let's fill in what we know
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let's look first at the concentrations
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we have 55% 10% and 49% which one's
mathematically in between the other two
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49 so that's gonna be your final solution
so that's the one that goes over on the
right and the other two
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concentrations go on the left hand side
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let's look next at the amount you're given
30 gallons of the 49% you put the 30 next
to the 49%
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ok so we're given the amount on the right
if we're given the amount on the right
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then one of them is the part that's the X
and the other is whole minus part yeah
total minus part so
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30 - x
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and here's the solution to that problem
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a chemist has 3 ounces of a 9% alcohol
solution
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how many oz of a 17% alcohol solution
must be added in order to get a 15%
alcohol solution
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here's our regular formula
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now let's fill in what we know
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let's look first at the concentrations
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we have 9%
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17% and 15% which one's mathematically
in between the other two
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15 oh so that's the concentration of the
final mix
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the other two can be put in either order
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be sure to change 9% to a decimal the
correct way 0.09
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now let's look at the amounts we're given 3
ounces of the 9% so put 3 ounces next to
the 9%
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so we're given one of the amounts on the
left
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X is the other amount on the left remember
these mathematically have to add up to
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get the amount at the end so the amount
of your final mix is three plus X
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here's the solution to that problem
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how many liters of pure acid must be mixed
with a 25% acid solution to get 10 liters of
a 40% acid solution
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so this one is a little different we have pure
acid what's the concentration of pure acid
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well it's all acid so it's 100% pure right so
that would be 100%
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so the concentrations were dealing with on
this problem are 100%
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25% and 40%
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here we go
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so which one is mathematically in between
40% that's the one that goes on the right
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and put the other two in and 100% as a
decimal what's that's one exactly
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the other one is 0.25
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now for the amount you're given 10 liters
of the 40% solution so put the 10 liters
over there with the 40
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so you're given the amount on the right
anytime you're given the amount on the
right
00:10:06.466 --> 00:10:10.166
one of the amounts on the left is X the
other
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one is: total minus part so 10 minus X right
total is 10 minus the
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part over here on the left hand side is X
cause we're going to end up with 10
gallons
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or liters altogether
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here's the solution to that problem
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how much water should be evaporated
from
00:10:33.066 --> 00:10:36.932
here we go with a different kind of problem
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from
00:10:38.533 --> 00:10:43.766
this is being taken away so it's not like two
mixtures being added together
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how much water should be the evaporated
from 240 gallons of a 3% salt solution to
produce a 5% salt solution
00:10:55.433 --> 00:10:59.899
so here's our generic formula for this
problem
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notice the subtraction on this problem
instead
00:11:05.566 --> 00:11:12.699
let's put in what we know
00:11:12.700 --> 00:11:16.200
alright now we have to be careful about
what we are starting with and what we're
ending with
00:11:16.200 --> 00:11:19.700
because the subtraction makes things
different
00:11:19.700 --> 00:11:25.133
how much water should be evaporated
from so this right here
00:11:25.133 --> 00:11:27.233
is your water
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that's what's being evaporated from
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what is the concentration
00:11:31.766 --> 00:11:34.199
of water how much
00:11:34.200 --> 00:11:45.033
salt is in water well unless it's seawater it's
0% salt yes we're assuming this is just
pure water so the concentration for that
00:11:45.033 --> 00:11:55.166
would be 0% so 0% as a decimal is still just
the number zero you can take 0%
00:11:55.166 --> 00:12:06.166
change it to a decimal it's still just 0.00 it's
just zero yes
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240 gallons of the 3% salt solution so
you're starting off with a 3% solution
00:12:13.166 --> 00:12:16.466
and your ending up with a 5% now once
again
00:12:16.466 --> 00:12:20.399
the concentrations aren't going to be the
same as we're used to it's not going to be
00:12:20.400 --> 00:12:23.460
mathematically in between the other two
because we're
00:12:23.500 --> 00:12:31.800
taking away this problem is different you're
taking something away your starting off
with something that's 3% pure
00:12:32.260 --> 00:12:37.700
you're gonna make it more concentrated
by evaporating the water away and now
00:12:37.700 --> 00:12:43.166
it's gonna be 5% so it does make sense
now so let's look at the amounts
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you're given 240 gallons of the 3%
00:12:48.233 --> 00:12:51.766
we don't know how much water is taken off
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now just like before we're gonna take a
look at the amount on the left hand side
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we start off with 240 gallons we take away
X gallons which means we're left with 240
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minus X gallons
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and here's the solution to the problem
00:13:19.740 --> 00:13:23.033
let's turn our attention now to value mixture
problems
00:13:23.033 --> 00:13:29.299
what's the formula that we would use for a
value mixture problem let's say you go to
the grocery store
00:13:29.300 --> 00:13:32.266
and you're looking at hamburger
00:13:32.266 --> 00:13:36.766
and the price on the package says $3.00
per pound
00:13:36.766 --> 00:13:40.332
and you buy 2 pounds of it
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how much did that cost you
00:13:44.566 --> 00:13:46.532
$6.00
00:13:46.533 --> 00:13:49.766
yeah that would cost you $6.00
00:13:49.766 --> 00:13:54.166
because you multiply the price per pound
00:13:54.166 --> 00:13:57.966
times the number of pounds to get
00:13:57.966 --> 00:14:02.532
the price look what happens to the pounds
yeah the units cancel right off the
00:14:02.533 --> 00:14:07.133
units can cancel off just like numbers and
variables can
00:14:07.133 --> 00:14:12.866
so this is the formula that you are going to
use for value mixture problems
00:14:12.866 --> 00:14:15.366
dollars per pound
00:14:15.366 --> 00:14:18.699
times pounds and once again
00:14:18.700 --> 00:14:24.066
the formula will be the same for every term
on both sides of the equation
00:14:25.400 --> 00:14:33.233
it's really nice sometimes it won't be dollars
it could be cents instead of pounds it could
be oz but you get the idea
00:14:33.500 --> 00:14:36.700
let's look at how the prices relate to each
other
00:14:36.700 --> 00:14:38.966
let's say you have
00:14:38.966 --> 00:14:42.832
one of your prices you are dealing with is
$5.00 a pound
00:14:42.833 --> 00:14:45.966
another one is $7.00 a pound
00:14:45.966 --> 00:14:51.399
and the third one is $4.00 per pound and
this is a traditional problem where you're
00:14:51.400 --> 00:14:56.866
mixing two ingredients together to get a
third final product
00:14:56.866 --> 00:14:59.980
which one is the price of your final product
00:15:01.540 --> 00:15:05.566
could you take these two
00:15:05.566 --> 00:15:12.280
and mix them together and end up with
something at the end that's only $4.00 a
pound so you take two things that are
00:15:12.280 --> 00:15:17.233
more expensive than the third and come
up with something that's cheaper
00:15:17.233 --> 00:15:19.966
no that's not going to work is it
00:15:19.966 --> 00:15:24.232
could you take two things that are less
expensive
00:15:24.233 --> 00:15:28.766
and end up with something that's more
expensive than either the other two
00:15:28.766 --> 00:15:31.366
that doesn't make any sense either
00:15:31.366 --> 00:15:38.432
so once again the price of the mix is going
to be mathematically in between
00:15:38.440 --> 00:15:46.060
the price of the two ingredients that you
put together so $5.00 a pound will be the
price of your final mix
00:15:50.340 --> 00:15:55.800
how many pounds of peanuts that sell for
$1.80 per pound should be mixed
00:15:55.800 --> 00:16:01.300
with 3 pounds of cashews that sell for
$4.50 per pound
00:16:01.300 --> 00:16:06.433
to get a mixture that sells for $2.61 per
pound
00:16:06.440 --> 00:16:08.060
here's our regular formula
00:16:09.340 --> 00:16:12.366
now let's put in what we know
00:16:12.366 --> 00:16:17.166
let's look first at the three different prices
00:16:17.166 --> 00:16:21.599
which one's mathematically in between the
other two
00:16:21.600 --> 00:16:25.866
the $2.61 yes the $2.61 must then be
00:16:25.866 --> 00:16:29.999
the price of the final mixture
00:16:30.000 --> 00:16:33.766
put in the other two prices
00:16:33.766 --> 00:16:35.732
and now let's look at the amounts
00:16:35.733 --> 00:16:45.833
you're given 3 pounds of the cashews that
that cost $4.50 per pound so put the three
next to the 4.5
00:16:45.833 --> 00:16:48.899
so you're given one of the amounts on the
left
00:16:48.900 --> 00:16:54.666
the other amount on the left is X and once
again the two amounts on the left must
00:16:54.666 --> 00:17:00.432
mathematically add up to get the amount
on the right
00:17:00.440 --> 00:17:02.120
and here's the solution to this problem
00:17:08.060 --> 00:17:12.460
in a local supermarket hamburger sells for
$3.50 per pound
00:17:12.466 --> 00:17:16.132
and ground sirloin sells for $4.20 per
pound
00:17:16.133 --> 00:17:24.399
how many pounds of each should be
mixed in order to obtain 30 pounds of a
mixture that sells for $3.78 per pound
00:17:24.400 --> 00:17:26.133
here's our regular formula
00:17:26.660 --> 00:17:28.520
now let's put in what we know
00:17:31.566 --> 00:17:34.620
let's look first at the prices
00:17:34.633 --> 00:17:37.933
which one's mathematically in between the
other two
00:17:37.933 --> 00:17:42.666
the $3.78 so that's the one that goes on
the right hand side
00:17:42.666 --> 00:17:44.460
put the other two in
00:17:47.540 --> 00:17:52.120
now let's look at the amounts which
amount were you given 30 pounds
00:17:52.120 --> 00:17:55.700
of the mixture at the end 30 pounds of the
$3.78 mixture
00:17:55.700 --> 00:17:58.766
so the 30 goes next to the $3.78
00:17:58.766 --> 00:18:02.999
ah so you're given the amount on the right
if you're given the amount on the right
00:18:03.000 --> 00:18:07.266
then one of the amounts on the left is X
which is the part
00:18:07.266 --> 00:18:09.732
so the 30 is the
00:18:09.733 --> 00:18:12.533
total
00:18:12.533 --> 00:18:16.266
the X is part of that 30
00:18:16.266 --> 00:18:21.366
what's left total minus part right total minus
00:18:21.366 --> 00:18:31.666
part so the amount that goes next to 420
would be 30 minus X these two are
interchangeable you can interchange the
00:18:31.666 --> 00:18:38.532
30 minus X with the X it works out just the
same exact way
00:18:38.540 --> 00:18:40.260
here's the solution to that problem
00:18:45.440 --> 00:18:48.266
here are some more problems for you to
practice
00:18:48.266 --> 00:18:51.532
on mixture problems so you'll get
comfortable with them
00:18:51.533 --> 00:18:55.666
I encourage you to pause the video do
each of these problems
00:18:55.666 --> 00:18:58.632
and then watch the rest of the video to see
the
00:18:58.640 --> 00:18:59.780
answers
00:19:02.720 --> 00:19:07.260
here are the answers to the extra
problems
00:19:07.266 --> 00:19:12.466
I hope this video has helped you learn
some more about mixture problems