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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

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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

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here we go with a different kind of problem

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from

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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

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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

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how much water should be evaporated 
from so this right here

00:11:25.133 --> 00:11:27.233
is your water

00:11:27.233 --> 00:11:29.333
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

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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

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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

00:12:51.766 --> 00:12:56.500
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

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let's turn our attention now to value mixture 
problems

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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

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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